{ "fingerprint": "5bf83bc757e5849f74df6505fa48d93e52b8fbbc0e74c7baa56f23c48886f370", "embedding_model": "text-embedding-3-large", "session": { "session_id": "896deaa4", "label": "Multi-path collection", "entries": [ { "text": "To find the fifth term, I need to identify the common difference of the arithmetic sequence and add it to the fourth term.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "The common difference is the same for any consecutive pair of terms, so I can use any of them to find it.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "For example, using the first and second terms, I can write $x-y = x+y + d$, where $d$ is the common difference.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "Solving for $d$, I get $d = -2y$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "Using another pair of terms, such as the second and third, I can check if this value of $d$ is consistent.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "I have $xy = x-y + d$, so substituting $d = -2y$, I get $xy = x-y - 2y$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "Simplifying, I get $xy = x - 3y$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "This seems like a reasonable equation, so I will assume that $d = -2y$ is correct.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "Now, to find the fifth term, I need to add $d$ to the fourth term.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "The fourth term is $x/y$, so the fifth term is $x/y + d = x/y - 2y$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "To express this as a common fraction, I need to find a common denominator for $x/y$ and $-2y$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "The least common denominator is $y$, so I can multiply the numerator and denominator of $-2y$ by $y$ to get $-2y^2/y$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 0, "metadata": { "path_label": "the_first_four_terms_in_an_arithmetic_sequence_are_x_y_x__2023-03-19T02-54-52.529607|chosen" } }, { "text": "I need to find three consecutive positive even numbers whose squares add up to $12296$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Let me call them $2n-2$, $2n$, and $2n+2$, where $n$ is a positive integer.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Then I have the equation $(2n-2)^2+(2n)^2+(2n+2)^2=12296$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "I can expand the squares and simplify the equation: $4n^2-8n+4+4n^2+4n^2+8n+4=12296$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Combining like terms, I get $12n^2+8=12296$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Subtracting $8$ from both sides, I get $12n^2=12288$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Dividing both sides by $12$, I get $n^2=1024$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Taking the square root of both sides, I get $n=\\pm 32$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "Since $n$ has to be positive, I choose $n=32$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "This means that the three consecutive positive even numbers are $2n-2=62$, $2n=64$, and $2n+2=66$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 1, "metadata": { "path_label": "the_sum_of_the_squares_of_three_consecutive_positive_even_nu__2023-02-16T04-32-46.248012|chosen" } }, { "text": "I recognize that this problem involves a triangle and its medians, which are the segments that connect each vertex to the midpoint of the opposite side.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "I also know that the three medians of a triangle are concurrent, meaning they all intersect at one point, which is called the centroid.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "The centroid has some interesting properties: it divides each median into a ratio of 2:1, and it is also the center of mass of the triangle.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "This means that if I draw a third median, say $BF$, it will also pass through $P$ and be divided into $BP:PF=2:1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "Since $CE$ is a median, $E$ must be the midpoint of $AB$, and similarly $D$ must be the midpoint of $BC$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "This implies that $DE$ is parallel to $AC$, and also half as long.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "Therefore, $AC=2DE=5$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "Now, I want to find the area of $AEDC$, which is a quadrilateral with two opposite sides parallel.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "I can use the formula for the area of a trapezoid, which is the average of the bases times the height.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 2, "metadata": { "path_label": "in_triangle_abc_medians_ad_and_ce_intersect_at_p__2023-02-16T04-11-44.196256|chosen" } }, { "text": "I notice that both equations have the same constant term, $1,$ which means that the product of the roots in each equation is also $1.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 3, "metadata": { "path_label": "let_alpha_and_beta_be_the_roots_of_x_2_px_1_0__2023-02-15T20-47-11.895085|chosen" } }, { "text": "That is, $\\alpha \\beta = 1$ and $\\gamma \\delta = 1.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 3, "metadata": { "path_label": "let_alpha_and_beta_be_the_roots_of_x_2_px_1_0__2023-02-15T20-47-11.895085|chosen" } }, { "text": "This suggests that I can use the difference of squares identity to simplify the expression I want to find.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 3, "metadata": { "path_label": "let_alpha_and_beta_be_the_roots_of_x_2_px_1_0__2023-02-15T20-47-11.895085|chosen" } }, { "text": "Specifically, I can rewrite $(\\alpha - \\gamma)(\\beta - \\gamma)$ as $(\\alpha \\beta - \\gamma (\\alpha + \\beta) + \\gamma^2),$ and $(\\alpha + \\delta)(\\beta + \\delta)$ as $(\\alpha \\beta + \\delta (\\alpha + \\beta) + \\delta^2).$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 3, "metadata": { "path_label": "let_alpha_and_beta_be_the_roots_of_x_2_px_1_0__2023-02-15T20-47-11.895085|chosen" } }, { "text": "Then, multiplying these two expressions, I get $(\\alpha \\beta - \\gamma (\\alpha + \\beta) + \\gamma^2)(\\alpha \\beta + \\delta (\\alpha + \\beta) + \\delta^2)$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 3, "metadata": { "path_label": "let_alpha_and_beta_be_the_roots_of_x_2_px_1_0__2023-02-15T20-47-11.895085|chosen" } }, { "text": "To solve this problem, I need to recall the relationship between the side lengths and the angles of a triangle, using the law of cosines.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 4, "metadata": { "path_label": "the_sides_of_a_triangle_have_lengths_11_15_and_k_wher__2023-02-15T20-42-41.482288|chosen" } }, { "text": "The law of cosines says that for any triangle with sides $a, b,$ and $c$ and the angle $C$ opposite to $c,$ we have $c^2 = a^2 + b^2 - 2ab \\cos C.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 4, "metadata": { "path_label": "the_sides_of_a_triangle_have_lengths_11_15_and_k_wher__2023-02-15T20-42-41.482288|chosen" } }, { "text": "If the triangle is obtuse, then the angle $C$ is greater than $90^\\circ,$ which means that $\\cos C$ is negative.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 4, "metadata": { "path_label": "the_sides_of_a_triangle_have_lengths_11_15_and_k_wher__2023-02-15T20-42-41.482288|chosen" } }, { "text": "This implies that $c^2 > a^2 + b^2,$ which is equivalent to $c > \\sqrt{a^2 + b^2}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 4, "metadata": { "path_label": "the_sides_of_a_triangle_have_lengths_11_15_and_k_wher__2023-02-15T20-42-41.482288|chosen" } }, { "text": "This is a problem about the angles subtended by arcs of the circle.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 5, "metadata": { "path_label": "three_points_are_chosen_uniformly_at_random_on_a_circle_wha__2023-02-15T05-40-45.535235|chosen" } }, { "text": "For example, if we choose three points A, B, and C on the circle, then the angle subtended by arc AB at the center is twice the angle subtended by arc AB at any point on the circle.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 5, "metadata": { "path_label": "three_points_are_chosen_uniformly_at_random_on_a_circle_wha__2023-02-15T05-40-45.535235|chosen" } }, { "text": "Similarly, the angle subtended by arc BC at the center is twice the angle subtended by arc BC at any point on the circle, and so on.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 5, "metadata": { "path_label": "three_points_are_chosen_uniformly_at_random_on_a_circle_wha__2023-02-15T05-40-45.535235|chosen" } }, { "text": "To find the hundreds digit of the product, I need to multiply the last three digits of each factor and look at the result.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "The last three digits of $5$ are $005$, of $6$ are $006$, of $7$ are $007$, of $8$ are $008$, of $9$ are $009$, and of $10$ are $010$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "Multiplying $005$ by $006$ gives $030$, which has a zero in the hundreds place.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "Multiplying $030$ by $007$ gives $210$, which has a two in the hundreds place.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "Multiplying $210$ by $008$ gives $1680$, which has a six in the hundreds place.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "Multiplying $1680$ by $009$ gives $15120$, which has a one in the hundreds place.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 6, "metadata": { "path_label": "what_is_the_hundreds_digit_in_the_following_product_5_tim__2023-02-15T05-24-43.013022|chosen" } }, { "text": "I notice that the expression inside the square root signs are linear functions of x, so I wonder if there is a way to use some algebra or calculus to simplify the problem.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 7, "metadata": { "path_label": "find_the_maximum_of_sqrt_x_27_sqrt_13_x_sqrt__2023-02-14T22-24-57.687736|chosen" } }, { "text": "One idea is to try to complete the square for each term, but that might get messy and I don't see an obvious way to do that.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 7, "metadata": { "path_label": "find_the_maximum_of_sqrt_x_27_sqrt_13_x_sqrt__2023-02-14T22-24-57.687736|chosen" } }, { "text": "Another idea is to use the fact that the square root function is concave, which means that it gets flatter as x increases.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 7, "metadata": { "path_label": "find_the_maximum_of_sqrt_x_27_sqrt_13_x_sqrt__2023-02-14T22-24-57.687736|chosen" } }, { "text": "To be divisible by $3$, a number must have a sum of digits that is a multiple of $3$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 8, "metadata": { "path_label": "a_textit_palindrome_is_a_positive_integer_which_reads_th__2023-02-14T05-43-37.834532|chosen" } }, { "text": "For a $4$-digit palindrome, the first and fourth digits must be the same, and the second and third digits must be the same.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 8, "metadata": { "path_label": "a_textit_palindrome_is_a_positive_integer_which_reads_th__2023-02-14T05-43-37.834532|chosen" } }, { "text": "So, the sum of digits is $2a + 2b$, where $a$ is the first digit and $b$ is the second digit.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 8, "metadata": { "path_label": "a_textit_palindrome_is_a_positive_integer_which_reads_th__2023-02-14T05-43-37.834532|chosen" } }, { "text": "This problem involves counting the number of sequences that start with three integers between 1 and 10 and then follow a recursive rule.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 9, "metadata": { "path_label": "let_s_be_the_set_of_all_ordered_triple_of_integers_a_1_a__2023-02-14T05-21-47.631169|chosen" } }, { "text": "The first thing I notice is that the recursive rule involves the absolute value of the difference of two previous terms, which means that the sequence can never become negative.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 9, "metadata": { "path_label": "let_s_be_the_set_of_all_ordered_triple_of_integers_a_1_a__2023-02-14T05-21-47.631169|chosen" } }, { "text": "The only way to get $a_n=0$ is if either $a_{n-1}=0$ or $a_{n-2}=a_{n-3}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 9, "metadata": { "path_label": "let_s_be_the_set_of_all_ordered_triple_of_integers_a_1_a__2023-02-14T05-21-47.631169|chosen" } }, { "text": "I want to find the smallest positive integer $X$ that satisfies the given conditions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "I know that any multiple of 3 has the form $3k$ for some integer $k$, so $X$ has the form $3k + 2$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "I also know that any multiple of 5 has the form $5n$ for some integer $n$, so the number with the same units digit as $X$ has the form $5n + 4$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "Since the units digits of $X$ and $5n + 4$ are the same, I can equate them and get $3k + 2 \\equiv 5n + 4 \\pmod{10}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This means that the difference between $3k + 2$ and $5n + 4$ is a multiple of 10.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "I can simplify this difference by subtracting 4 from both sides and get $3k - 2 \\equiv 5n \\pmod{10}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "Now I need to find the smallest values of $k$ and $n$ that make this equation true.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "I can try some values of $k$ and see what they imply for $n$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "If $k = 0$, then $3k - 2 = -2$ and $5n = -2 + 10m$ for some integer $m$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This implies that $n = -0.4 + 2m$, which is not an integer.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "So $k = 0$ does not work.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "If $k = 1$, then $3k - 2 = 1$ and $5n = 1 + 10m$ for some integer $m$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This implies that $n = 0.2 + 2m$, which is also not an integer.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "So $k = 1$ does not work either.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "If $k = 2$, then $3k - 2 = 4$ and $5n = 4 + 10m$ for some integer $m$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This implies that $n = 0.8 + 2m$, which is again not an integer.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "So $k = 2$ does not work as well.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "If $k = 3$, then $3k - 2 = 7$ and $5n = 7 + 10m$ for some integer $m$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This implies that $n = 1.4 + 2m$, which is not an integer.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "So $k = 3$ does not work.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "If $k = 4$, then $3k - 2 = 10$ and $5n = 10 + 10m$ for some integer $m$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This implies that $n = 2 + 2m$, which is an integer.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "So $k = 4$ works.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "This means that the smallest value of $X$ is $3k + 2 = 3 \\cdot 4 + 2 = 14$.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 10, "metadata": { "path_label": "a_positive_integer_x_is_2_more_than_a_multiple_of_3_its_u__2023-02-14T05-11-30.080705|chosen" } }, { "text": "I notice that this expression looks like a base-3 representation of a number, except that the coefficients $a_i$ can also be -1, not just 0 or 1.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I wonder what happens when $a_i=-1$ for some $i$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I realize that this means subtracting a power of 3 from the number, which could make it negative.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I want to avoid negative numbers, since the problem asks for nonnegative integers.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I think about how to prevent the expression from becoming negative.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I see that if I have a -1 coefficient for some power of 3, I need to have a 1 coefficient for a higher power of 3 to cancel it out.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I also see that if I have a 0 coefficient for some power of 3, I can have either 0 or 1 for a higher power of 3, since it does not affect the sign of the number. [*: I conclude that the only way to make the expression nonnegative is to have 1 as the highest nonzero coefficient, and then have any combination of -1, 0, or 1 for the lower coefficients.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I count how many ways I can do this.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I notice that for each power of 3, I have three choices for the coefficient, except for the highest nonzero one, which must be 1.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I also notice that the highest nonzero coefficient could be anywhere from $a_0$ to $a_7$, depending on how large the number is.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I realize that this means I have 8 different cases to consider, one for each possible position of the highest nonzero coefficient.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I start with the case where $a_0=1$ and the rest are 0.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I see that this gives me only one number, which is 1.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I move on to the case where $a_1=1$ and the rest are either -1, 0, or 1.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I see that this gives me $3^1=3$ numbers, since I have three choices for $a_0$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I continue this pattern, and I find that for each case, I have $3^i$ numbers, where $i$ is the position of the highest nonzero coefficient.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I add up all these numbers for each case, and I get \\[1+3+3^2+\\cdots+3^7=3280.\\]", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I realize that I have not counted the case where all the coefficients are 0, which gives me another number, which is 0.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I add 1 to my previous sum, and I get \\[1+1+3+3^2+\\cdots+3^7=3281.\\]", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "I check that this is one of the answer choices, and it is.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 11, "metadata": { "path_label": "how_many_nonnegative_integers_can_be_written_in_the_form_a__2023-02-13T23-58-42.465028|chosen" } }, { "text": "To find the common divisors of $90$ and $m$, I need to factorize both numbers into prime factors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 12, "metadata": { "path_label": "two_numbers_90_and_m_share_exactly_three_positive_diviso__2023-02-19T07-44-01.230745|chosen" } }, { "text": "I know that $90 = 2 \\times 3^2 \\times 5$, so the prime factors of $90$ are $2, 3,$ and $5$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 12, "metadata": { "path_label": "two_numbers_90_and_m_share_exactly_three_positive_diviso__2023-02-19T07-44-01.230745|chosen" } }, { "text": "I also know that the number of divisors of a number is equal to the product of one more than the exponents of its prime factors.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 12, "metadata": { "path_label": "two_numbers_90_and_m_share_exactly_three_positive_diviso__2023-02-19T07-44-01.230745|chosen" } }, { "text": "For example, $90$ has $(1 + 1) \\times (2 + 1) \\times (1 + 1) = 12$ divisors, which are $1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 12, "metadata": { "path_label": "two_numbers_90_and_m_share_exactly_three_positive_diviso__2023-02-19T07-44-01.230745|chosen" } }, { "text": "I notice that the polynomial has a factor of $x-4$, so one of its zeroes is $x=4$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "That means I can use synthetic division to divide the polynomial by $x-4$ and get a quotient of degree 2.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "To do that, I write down the coefficients of the polynomial, leaving out the $x$'s: $(1, -1, -m^2 - m, 2m^2 + 4m + 2)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "Then I write down the zero, $x=4$, on the left, and draw a vertical line and a horizontal line to separate the coefficients and the quotient.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I bring down the first coefficient, $1$, and write it under the horizontal line.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "Then I multiply $1$ by $4$ and write the result, $4$, above the second coefficient, $-1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I add $-1$ and $4$ and get $3$, and write it under the horizontal line.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "Then I multiply $3$ by $4$ and write the result, $12$, above the third coefficient, $-m^2 - m$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I add $-m^2 - m$ and $12$ and get $-m^2 - m + 12$, and write it under the horizontal line.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "Then I multiply $-m^2 - m + 12$ by $4$ and write the result, $-4m^2 - 4m + 48$, above the last coefficient, $2m^2 + 4m + 2$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I add $2m^2 + 4m + 2$ and $-4m^2 - 4m + 48$ and get $-2m^2 + 50$, and write it under the horizontal line.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "This is the remainder of the division, and it should be zero, since $x-4$ is a factor of the polynomial.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "So I set $-2m^2 + 50 = 0$ and solve for $m$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I subtract $50$ from both sides and get $-2m^2 = -50$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I divide both sides by $-2$ and get $m^2 = 25$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I take the square root of both sides and get $m = \\pm 5$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "These are two possible values of $m$, but I need to check if they satisfy the other condition, that all of the zeroes of the polynomial are integers.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "To do that, I look at the quotient of the division, which is the polynomial $x^2 + 3x - m^2 - m + 12$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I want to factor this polynomial and find its zeroes.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I notice that it has a constant term of $-m^2 - m + 12$, which is equal to $-25 - 5 + 12 = -18$ when $m = 5$, and $-25 + 5 + 12 = -8$ when $m = -5$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I also notice that it has a leading coefficient of $1$, so if it has integer zeroes, they must be factors of the constant term.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "So I list the factors of $-18$ and $-8$ and see if any pair of them add up to the middle coefficient, $3$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "The factors of $-18$ are $\\pm 1, \\pm 2, \\pm 3, \\pm 6, \\pm 9, \\pm 18$.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "The only pair that adds up to $3$ is $-3$ and $6$.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "So when $m = 5$, the polynomial $x^2 + 3x - m^2 - m + 12$ can be factored as $(x-3)(x+6)$, and its zeroes are $x=3$ and $x=-6$, which are integers.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "The factors of $-8$ are $\\pm 1, \\pm 2, \\pm 4, \\pm 8$.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 13, "metadata": { "path_label": "the_polynomial_g_x_x_3_x_2_m_2_m_x_2m_2_4m__2023-02-19T05-33-41.910774|chosen" } }, { "text": "I need to find a pattern or a rule that describes which lockers are open and which are closed at any given time.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 14, "metadata": { "path_label": "a_bored_student_walks_down_a_hall_that_contains_a_row_of_clo__2023-02-19T05-18-11.998786|chosen" } }, { "text": "I notice that the first time the student walks down the hall, he opens every odd locker, and the second time he walks down the hall, he opens every locker that is a multiple of 4.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 14, "metadata": { "path_label": "a_bored_student_walks_down_a_hall_that_contains_a_row_of_clo__2023-02-19T05-18-11.998786|chosen" } }, { "text": "I wonder if there is a connection between the number of times the student walks down the hall and the factors of the locker numbers.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 14, "metadata": { "path_label": "a_bored_student_walks_down_a_hall_that_contains_a_row_of_clo__2023-02-19T05-18-11.998786|chosen" } }, { "text": "I notice that the given expressions all have the form $n!+(n+1)!$, where $n$ ranges from $1$ to $8$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "I wonder if I can simplify them or factor them somehow.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "I realize that I can use the distributive property to write $n!+(n+1)!$ as $n!(1+n+1)$, or $(n+2)n!$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "This means that each expression is a multiple of $n!$ and $n+2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "For example, $1!+2! = 3\\cdot 1! = 3$, $2!+3! = 4\\cdot 2! = 8$, $3!+4! = 5\\cdot 3! = 30$, and so on.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "I want to find the least common multiple of these expressions, so I need to find the smallest positive integer that is a multiple of all of them.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "I can use the prime factorization of each expression to help me do that.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 15, "metadata": { "path_label": "the_least_common_multiple_of_1_2_2_3_3_4_4__2023-02-19T05-01-13.351578|chosen" } }, { "text": "I notice that the problem involves the product of the proper positive integer factors of $n$, which means all the factors of $n$ except $n$ itself.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "I also notice that the problem involves the number of positive divisors of $n$, which I'll call $x$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "I wonder if there is a way to relate the product of the proper factors of $n$ to $n$ and $x$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "I recall that the product of all the factors of $n$ is $n^{x/2}$, since each factor can be paired with its complement, and their product is $n$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "Therefore, the product of the proper factors of $n$ is $n^{x/2}/n$, which is $n^{(x-2)/2}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "This means that the product of the proper factors of $n$ can be written as $n^{(ax+b)/c}$, where $a=1$, $b=-2$, and $c=2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "The greatest common factor of $a$, $b$, and $c$ is $1$, as required.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "The answer is $a+b+c=1+(-2)+2=1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 16, "metadata": { "path_label": "the_product_of_the_proper_positive_integer_factors_of_n_ca__2023-02-18T06-31-44.798292|chosen" } }, { "text": "I notice that $\\binom{n}{2}$ is the number of ways to choose two elements from a set of $n$ elements, which can also be written as $\\frac{n(n-1)}{2}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 17, "metadata": { "path_label": "for_how_many_integers_n_where_2_le_n_le_100_is_binom__2023-02-18T06-24-52.906714|chosen" } }, { "text": "To see when this fraction is odd, I need to consider the parity of the numerator and the denominator.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 17, "metadata": { "path_label": "for_how_many_integers_n_where_2_le_n_le_100_is_binom__2023-02-18T06-24-52.906714|chosen" } }, { "text": "The numerator, $n(n-1)$, is always even, since it is the product of two consecutive integers, one of which must be even.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 17, "metadata": { "path_label": "for_how_many_integers_n_where_2_le_n_le_100_is_binom__2023-02-18T06-24-52.906714|chosen" } }, { "text": "To solve this problem, I need to think about the graphs of the two functions $y = \\sin 6 \\pi x$ and $y = x$ and how they intersect.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 18, "metadata": { "path_label": "find_the_number_of_real_solutions_to_sin_6_pi_x_x__2023-02-18T03-41-54.285490|chosen" } }, { "text": "The graph of $y = \\sin 6 \\pi x$ is a periodic wave that oscillates between $-1$ and $1$ with a period of $\\frac{1}{3}$, meaning it completes one cycle every $\\frac{1}{3}$ units.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 18, "metadata": { "path_label": "find_the_number_of_real_solutions_to_sin_6_pi_x_x__2023-02-18T03-41-54.285490|chosen" } }, { "text": "The graph of $y = x$ is a straight line with a slope of $1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 18, "metadata": { "path_label": "find_the_number_of_real_solutions_to_sin_6_pi_x_x__2023-02-18T03-41-54.285490|chosen" } }, { "text": "I can sketch these graphs on the same coordinate plane to get an idea of how many times they intersect.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 18, "metadata": { "path_label": "find_the_number_of_real_solutions_to_sin_6_pi_x_x__2023-02-18T03-41-54.285490|chosen" } }, { "text": "This problem involves finding the volume of a region in space defined by inequalities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "A good way to start is to try to visualize the region or sketch it on paper.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "I notice that the inequalities involve only the sum and difference of $x, y,$ and $z$, so I can try to simplify the problem by making a change of variables.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "Let $u = x + y$ and $v = x - y$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "Then the inequalities become $|u + z| + |u - z| \\le 8$ and $u, v, z \\ge 0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "This looks simpler, but I still need to find the relationship between $u, v,$ and $z$ and $x, y,$ and $z$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "I can solve for $x$ and $y$ in terms of $u$ and $v$ by adding and subtracting them: $x = \\frac{u + v}{2}$ and $y = \\frac{u - v}{2}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "Then I can find the Jacobian of the transformation, which is the determinant of the matrix of partial derivatives of $x, y,$ and $z$ with respect to $u, v,$ and $z$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "The Jacobian is $\\begin{vmatrix} \\frac{1}{2} & \\frac{1}{2} & 0 \\\\ \\frac{1}{2} & -\\frac{1}{2} & 0 \\\\ 0 & 0 & 1 \\end{vmatrix} = -\\frac{1}{2}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "The absolute value of the Jacobian is $\\frac{1}{2}$, which means that the volume element in the $u, v, z$ space is half of the volume element in the $x, y, z$ space.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "So to find the volume of the original region, I can integrate the volume element in the $u, v, z$ space and multiply by $\\frac{1}{2}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "But first, I need to find the bounds of integration for $u, v,$ and $z$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "I can do this by looking at the inequalities and seeing how they restrict the values of $u, v,$ and $z$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "For example, since $u, v, z \\ge 0$, I know that $u + z$ and $u - z$ are both nonnegative, so $|u + z| + |u - z| = u + z + u - z = 2u$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "This means that $2u \\le 8$, or $u \\le 4$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "Similarly, I can see that $v \\le 4$ and $z \\le 4$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "These are the upper bounds of integration for $u, v,$ and $z$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "The lower bounds are all zero, since $u, v, z \\ge 0$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "So I can write the integral as $\\frac{1}{2} \\int_0^4 \\int_0^4 \\int_0^4 du dv dz$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "This is a triple integral of a constant function, so it is just the product of the lengths of the intervals of integration.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "That is, $\\frac{1}{2} \\int_0^4 \\int_0^4 \\int_0^4 du dv dz = \\frac{1}{2} \\cdot 4 \\cdot 4 \\cdot 4 = 32$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 19, "metadata": { "path_label": "find_the_volume_of_the_region_in_space_defined_by_x_y__2023-02-17T06-16-12.050676|chosen" } }, { "text": "I notice that the area of $\\triangle ABC$ is given by half the base times the height, so I can use that to find $AD.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 20, "metadata": { "path_label": "we_have_triangle_triangle_abc_where_ab_ac_and_ad_is__2023-02-17T04-29-17.679930|chosen" } }, { "text": "Since $BC = 12,$ the base of $\\triangle ABC$ is $12$ and the area is $180,$ so I can write $\\frac{1}{2} \\cdot 12 \\cdot AD = 180$ and solve for $AD.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 20, "metadata": { "path_label": "we_have_triangle_triangle_abc_where_ab_ac_and_ad_is__2023-02-17T04-29-17.679930|chosen" } }, { "text": "I get $AD = 30,$ so the height of $\\triangle ABC$ is $30.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 20, "metadata": { "path_label": "we_have_triangle_triangle_abc_where_ab_ac_and_ad_is__2023-02-17T04-29-17.679930|chosen" } }, { "text": "Now I want to find the area of $ABDE.$ I notice that $ABDE$ is a trapezoid, since $AB \\parallel DE$ by given.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 20, "metadata": { "path_label": "we_have_triangle_triangle_abc_where_ab_ac_and_ad_is__2023-02-17T04-29-17.679930|chosen" } }, { "text": "The area of a trapezoid is given by half the sum of the bases times the height, so I can use that formula to find the area of $ABDE.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 20, "metadata": { "path_label": "we_have_triangle_triangle_abc_where_ab_ac_and_ad_is__2023-02-17T04-29-17.679930|chosen" } }, { "text": "I notice that the problem is asking about a congruence modulo 23, which means that I only care about the remainder of the division of both sides by 23.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "So I can simplify the problem by finding the equivalent remainders of 3874x+481 and 1205 mod 23.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "To do that, I can use the fact that if $a\\equiv b \\pmod{m}$ and $c\\equiv d \\pmod{m}$, then $a+c\\equiv b+d \\pmod{m}$ and $ac\\equiv bd \\pmod{m}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "This means that I can add or subtract multiples of 23 from any term in the congruence and it will not change the remainder. [* 3874x+481\\equiv 1205 \\pmod{23}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "I can subtract 481 from both sides to get rid of the constant term: 3874x\\equiv 724 \\pmod{23}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "I can also divide both sides by 4, since 4 is relatively prime to 23, and use the fact that $4^{-1}\\equiv 6 \\pmod{23}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 21, "metadata": { "path_label": "how_many_three_digit_positive_integers_x_satisfy_3874x_48__2023-02-17T03-41-40.518355|chosen" } }, { "text": "This problem involves finding the maximum value of an expression involving square roots and inequalities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I wonder if there is a way to simplify the expression or rewrite it in a more convenient form.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I notice that the expression is symmetric in $a$ and $b,$ so it might help to use the fact that $a + b = 16$ to eliminate one of them.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "For example, I can rewrite the expression as $\\sqrt{c(b - c)} + \\sqrt{c(16 - b - c)}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Now I only have two variables to deal with, $b$ and $c.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "However, I still have the constraint that $a \\ge b \\ge c \\ge 0,$ which implies that $16 - b \\ge b \\ge c \\ge 0.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "This means that both terms inside the square roots are nonnegative, which is good.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I wonder if there is a way to apply some inequality to the expression to get an upper bound.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I recall that the arithmetic mean-geometric mean (AM-GM) inequality states that for any nonnegative numbers $x$ and $y,$ we have $\\frac{x + y}{2} \\ge \\sqrt{xy}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "This inequality is useful for finding upper bounds for products of nonnegative numbers.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I also notice that the expression I want to maximize is the sum of two square roots of products of nonnegative numbers.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Maybe I can apply the AM-GM inequality to each term separately and then add them up.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Let me try that.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "By AM-GM, I have $\\sqrt{c(b - c)} \\le \\frac{c + (b - c)}{2} = \\frac{b}{2}$ and $\\sqrt{c(16 - b - c)} \\le \\frac{c + (16 - b - c)}{2} = \\frac{16 - b}{2}.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Adding these two inequalities, I get $\\sqrt{c(b - c)} + \\sqrt{c(16 - b - c)} \\le \\frac{b}{2} + \\frac{16 - b}{2} = 8.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "This means that the maximum possible value of the expression is at most 8.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "But is 8 actually achievable?", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "To check that, I need to find values of $b$ and $c$ that satisfy the original constraints and make both inequalities become equalities.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "For the first inequality to be an equality, I need $c = b - c,$ or $2c = b.$", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "For the second inequality to be an equality, I need $c = 16 - b - c,$ or $2c = 16 - b.$", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Combining these two equations, I get $b = 8.$", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Then, plugging in $b = 8$ into either equation, I get $c = 4.$", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "These values of $b$ and $c$ satisfy the original constraints, since $16 - b \\ge b \\ge c \\ge 0.$", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "Therefore, the maximum value of the expression is 8, and it occurs when $b = 8$ and $c = 4.$", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 22, "metadata": { "path_label": "let_a_b_and_c_be_real_numbers_such_that_a_ge_b_ge__2023-02-17T02-49-03.729968|chosen" } }, { "text": "I want to find the value of $m$ that makes the equation $10x^2 - mx + 420 = 0$ have integer solutions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "One way to do this is to use the quadratic formula, which says that the solutions are given by $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$, where $a = 10$, $b = -m$, and $c = 420$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "In order for the solutions to be integers, the expression under the square root, $b^2 - 4ac$, must be a perfect square.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "So I need to find the smallest positive value of $m$ that makes $b^2 - 4ac$ a perfect square.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "I can try some values of $m$ and see what happens.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "If $m = 1$, then $b^2 - 4ac = (-1)^2 - 4(10)(420) = 1 - 16800 = -16799$, which is not a perfect square.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "If $m = 2$, then $b^2 - 4ac = (-2)^2 - 4(10)(420) = 4 - 16800 = -16796$, which is also not a perfect square.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 23, "metadata": { "path_label": "what_is_the_smallest_positive_value_of_m_so_that_the_equat__2023-02-16T06-12-41.898635|chosen" } }, { "text": "I notice that this equation relates $f(x)$ and $f\\left(\\dfrac{1}{x}\\right)$, so I wonder what happens if I replace $x$ with $\\dfrac{1}{x}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 24, "metadata": { "path_label": "suppose_f_x_is_a_rational_function_such_that_3f_left_df__2023-02-13T02-29-10.303237|chosen" } }, { "text": "Then I get $3f(x)+2xf\\left(\\dfrac{1}{x}\\right)=\\dfrac{1}{x^2}$. [*: This looks like a system of two equations in two unknowns, $f(x)$ and $f\\left(\\dfrac{1}{x}\\right)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 24, "metadata": { "path_label": "suppose_f_x_is_a_rational_function_such_that_3f_left_df__2023-02-13T02-29-10.303237|chosen" } }, { "text": "If I multiply the first equation by $x$ and the second equation by $3$, I get $3xf\\left(\\dfrac{1}{x}\\right)+2f(x)=x^3$ and $9f(x)+6xf\\left(\\dfrac{1}{x}\\right)=\\dfrac{3}{x^2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 24, "metadata": { "path_label": "suppose_f_x_is_a_rational_function_such_that_3f_left_df__2023-02-13T02-29-10.303237|chosen" } }, { "text": "Now I can add these two equations and get $11f(x)+9xf\\left(\\dfrac{1}{x}\\right)=x^3+\\dfrac{3}{x^2}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 24, "metadata": { "path_label": "suppose_f_x_is_a_rational_function_such_that_3f_left_df__2023-02-13T02-29-10.303237|chosen" } }, { "text": "I need to find a line that passes through $A$ and divides the area of $ABCD$ in half.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "The area of $ABCD$ is the sum of the areas of $\\triangle ABC$ and $\\triangle ACD.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "I can use the formula for the area of a triangle, which is $\\frac{1}{2}$ times the base times the height.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "For $\\triangle ABC,$ the base is $BC$ and the height is the distance from $A$ to $BC.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "I can use the distance formula to find the length of $BC.$ It is $\\sqrt{(3-1)^2+(3-2)^2}=\\sqrt{5}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "I can use the slope-intercept form of the equation of a line to find the equation of $BC.$ It is $y-2=\\frac{1}{2}(x-1),$ which simplifies to $y=\\frac{1}{2}x+\\frac{3}{2}.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 25, "metadata": { "path_label": "let_a_0_0_b_1_2_c_3_3_and_d_4_0__2023-02-13T01-30-27.710108|chosen" } }, { "text": "I want to find the maximum value of a quadratic expression in $x,$ $y,$ and $z,$ subject to a linear constraint.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "This looks like a problem where I can use the method of Lagrange multipliers, which is a technique from calculus that helps find extrema of functions subject to constraints.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "The idea is to find a scalar $\\lambda$ such that the gradient of the objective function is parallel to the gradient of the constraint function, which means that the objective function is not changing in the direction of the constraint.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "The gradient of the objective function is $\\nabla(xy + xz + yz) = (y + z, x + z, x + y).$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "The gradient of the constraint function is $\\nabla(x + 2y + z) = (1, 2, 1).$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "So I want to find $\\lambda$ and $(x, y, z)$ such that $(y + z, x + z, x + y) = \\lambda(1, 2, 1).$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "This gives me three equations: \\[y + z = \\lambda, \\quad x + z = 2\\lambda, \\quad x + y = \\lambda.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "I also have the original constraint: \\[x + 2y + z = 4.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "I can try to solve this system of four equations and four unknowns by elimination or substitution.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 26, "metadata": { "path_label": "let_x_y_z_be_real_numbers_such_that_x_2y_z_4__2023-02-13T00-24-39.406214|chosen" } }, { "text": "I need to find a two-digit integer that is divisible by $n$ and ends with $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 27, "metadata": { "path_label": "a_two_digit_integer_is_divisible_by_n_and_its_last_digit_i__2023-02-13T00-17-38.112409|chosen" } }, { "text": "This means that $n$ has to be a single-digit number from $0$ to $9$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 27, "metadata": { "path_label": "a_two_digit_integer_is_divisible_by_n_and_its_last_digit_i__2023-02-13T00-17-38.112409|chosen" } }, { "text": "I also want $n$ to be as large as possible, so I will start from $9$ and work my way down.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 27, "metadata": { "path_label": "a_two_digit_integer_is_divisible_by_n_and_its_last_digit_i__2023-02-13T00-17-38.112409|chosen" } }, { "text": "If $n = 9$, then the two-digit integer must be a multiple of $9$ and end with $9$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 27, "metadata": { "path_label": "a_two_digit_integer_is_divisible_by_n_and_its_last_digit_i__2023-02-13T00-17-38.112409|chosen" } }, { "text": "I notice that 10,000 is a power of 10, so I wonder if I can use modular arithmetic to simplify the calculation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "I recall that modular arithmetic is a way of working with remainders after division by a fixed modulus, such as 10,000.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "I also recall some basic rules of modular arithmetic, such as: if $a \\equiv b \\pmod{m}$ and $c \\equiv d \\pmod{m}$, then $a + c \\equiv b + d \\pmod{m}$ and $a \\times c \\equiv b \\times d \\pmod{m}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "These rules mean that I can add or multiply numbers and then take the remainder, or I can take the remainders first and then add or multiply them. [*^{*^}] For example, $15 \\times 17 \\equiv 5 \\times 7 \\pmod{10}$, because both 15 and 17 have a remainder of 5 and 7, respectively, when divided by 10.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "I can apply this idea to the problem by writing $301 = 300 + 1$, and then using the binomial theorem to expand $301^4 = (300 + 1)^4$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "The binomial theorem says that $(300 + 1)^4 = \\sum_{k = 0}^4 \\binom{4}{k} 300^k 1^{4 - k}$, where $\\binom{4}{k}$ is the binomial coefficient or the number of ways to choose k objects out of 4.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "I can simplify this expression by noting that $1^{4 - k} = 1$ for any $k$, and that $\\binom{4}{k} 300^k$ is a multiple of 10,000 for any $k \\geq 2$, since it contains at least four factors of 10.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "Therefore, the only terms that matter for the remainder are the ones with $k = 0$ and $k = 1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "These terms are $\\binom{4}{0} 300^0 1^4 = 1$ and $\\binom{4}{1} 300^1 1^3 = 1200$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "Adding these terms, I get $301^4 \\equiv 1 + 1200 = 1201 \\pmod{10,000}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "This means that the remainder when $301^4$ is divided by 10,000 is 1201.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 28, "metadata": { "path_label": "what_is_the_remainder_when_301_4_is_divided_by_10_000__2023-02-13T00-11-47.439568|chosen" } }, { "text": "To maximize the expression, I want to make the first term as large as possible and the second term as small as possible.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "That means I should assign the largest value to the base and the exponent of the power, and the smallest value to the coefficient and the subtrahend.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "The largest value among 0, 1, 2, and 3 is 3, so I should use 3 for either $a$ or $b$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "However, I can't use 0 for the other variable, because then the power would be zero and I would lose the whole term.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "So I have to use either 1 or 2 for the other variable.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "Since 3 to any power is larger than any power of 2, I should use 3 for $a$ and either 1 or 2 for $b$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "The smallest value among 0, 1, 2, and 3 is 0, so I should use 0 for either $c$ or $d$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "However, I can't use 0 for $c$, because then the coefficient would be zero and I would lose the whole term.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "So I have to use 0 for $d$ and either 1 or 2 for $c$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "Now I have two possible choices: either $c = 1$ and $b = 2$, or $c = 2$ and $b = 1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "I need to compare these two choices and see which one gives a larger result.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "If $c = 1$ and $b = 2$, then the expression is $1 \\cdot 3^2 - 0 = 9 - 0 = 9$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "If $c = 2$ and $b = 1$, then the expression is $2 \\cdot 3^1 - 0 = 6 - 0 = 6$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "Clearly, 9 is larger than 6, so the first choice is better.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "Therefore, the maximum possible value of the expression is 9, and it occurs when $a = 3$, $b = 2$, $c = 1$, and $d = 0$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 29, "metadata": { "path_label": "in_the_expression_c_cdot_a_b_d_the_values_of_a_b__2023-02-12T12-52-43.347821|chosen" } }, { "text": "This is an infinite series of the form $$\\sum_{n=1}^{\\infty} a_n,$$ where $$a_n = \\frac{3n-1}{2^n}.$$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 30, "metadata": { "path_label": "compute_sum__n_1_infty_frac_3n_1_2_n__2023-02-12T03-43-57.446152|chosen" } }, { "text": "One way to approach this problem is to try to find a closed-form expression for the partial sums $$S_n = \\sum_{k=1}^{n} a_k,$$ and then take the limit as $$n \\to \\infty.$$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 30, "metadata": { "path_label": "compute_sum__n_1_infty_frac_3n_1_2_n__2023-02-12T03-43-57.446152|chosen" } }, { "text": "To find a formula for $$S_n,$$ I will use a technique called telescoping, which involves manipulating the terms of the series to create a lot of cancellations.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 30, "metadata": { "path_label": "compute_sum__n_1_infty_frac_3n_1_2_n__2023-02-12T03-43-57.446152|chosen" } }, { "text": "I need to find the length of the chord, so I will start by drawing a diagram of the circles and the chord.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 31, "metadata": { "path_label": "circles_c_1_and_c_2_are_externally_tangent_and_they_are__2023-02-12T03-22-25.819076|chosen" } }, { "text": "I will label the centers of the circles $O_1,O_2,$ and $O_3,$ and the points of tangency $A,B,$ and $C.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 31, "metadata": { "path_label": "circles_c_1_and_c_2_are_externally_tangent_and_they_are__2023-02-12T03-22-25.819076|chosen" } }, { "text": "I will also let $r$ be the radius of $C_3.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 31, "metadata": { "path_label": "circles_c_1_and_c_2_are_externally_tangent_and_they_are__2023-02-12T03-22-25.819076|chosen" } }, { "text": "Since the circles are tangent, I can write some equations for the distances between the centers and the radii: $O_1O_2=4+10=14, O_1O_3=r-4, O_2O_3=r-10.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 31, "metadata": { "path_label": "circles_c_1_and_c_2_are_externally_tangent_and_they_are__2023-02-12T03-22-25.819076|chosen" } }, { "text": "I need to find a way to relate the angles and the sides of the triangle.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 32, "metadata": { "path_label": "the_lengths_of_the_sides_of_a_triangle_are_consecutive_integ__2023-02-12T03-01-36.806559|chosen" } }, { "text": "I recall that the law of cosines says that for any triangle with sides a, b, c and angles A, B, C opposite to them, respectively, the following equation holds: a^2 = b^2 + c^2 - 2bc cos A.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 32, "metadata": { "path_label": "the_lengths_of_the_sides_of_a_triangle_are_consecutive_integ__2023-02-12T03-01-36.806559|chosen" } }, { "text": "I can use this formula to write three equations for my triangle, since I know the sides are consecutive integers: (n+2)^2 = n^2 + (n+1)^2 - 2n(n+1) cos C, (n+1)^2 = n^2 + (n+2)^2 - 2n(n+2) cos B, and n^2 = (n+1)^2 + (n+2)^2 - 2(n+1)(n+2) cos A.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 32, "metadata": { "path_label": "the_lengths_of_the_sides_of_a_triangle_are_consecutive_integ__2023-02-12T03-01-36.806559|chosen" } }, { "text": "I notice that the equation involves only the absolute values of $x$ and $y$, so it must be symmetric about both the $x$-axis and the $y$-axis.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 33, "metadata": { "path_label": "let_s_be_the_set_of_points_x_y_in_the_cartesian_plane__2023-02-11T19-01-33.921056|chosen" } }, { "text": "That means I can focus on just one quadrant and then multiply the length by four to get the total length.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 33, "metadata": { "path_label": "let_s_be_the_set_of_points_x_y_in_the_cartesian_plane__2023-02-11T19-01-33.921056|chosen" } }, { "text": "I choose the first quadrant, where $x$ and $y$ are both positive.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 33, "metadata": { "path_label": "let_s_be_the_set_of_points_x_y_in_the_cartesian_plane__2023-02-11T19-01-33.921056|chosen" } }, { "text": "I need to count the number of ways to assign the meal types to the people, subject to the constraint that only one person gets what they ordered.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 34, "metadata": { "path_label": "nine_people_sit_down_for_dinner_where_there_are_three_choice__2023-02-11T06-11-48.993410|chosen" } }, { "text": "I can think of this as a permutation problem, where I arrange the nine meal types in a row, and then match them up with the people in their fixed seats.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 34, "metadata": { "path_label": "nine_people_sit_down_for_dinner_where_there_are_three_choice__2023-02-11T06-11-48.993410|chosen" } }, { "text": "To avoid overcounting, I will fix the position of the one meal that matches the order, and then permute the remaining eight meals in the remaining eight positions.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 34, "metadata": { "path_label": "nine_people_sit_down_for_dinner_where_there_are_three_choice__2023-02-11T06-11-48.993410|chosen" } }, { "text": "There are nine possible choices for the matching meal, so I will multiply by nine to account for all of them.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 34, "metadata": { "path_label": "nine_people_sit_down_for_dinner_where_there_are_three_choice__2023-02-11T06-11-48.993410|chosen" } }, { "text": "Now, how many ways can I permute the remaining eight meals, so that none of them match the orders?", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 34, "metadata": { "path_label": "nine_people_sit_down_for_dinner_where_there_are_three_choice__2023-02-11T06-11-48.993410|chosen" } }, { "text": "I know that a regular hexagon can be divided into six equilateral triangles by connecting the center with the vertices.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 35, "metadata": { "path_label": "let_abcdef_be_a_regular_hexagon_let_g_h_i_j__2023-02-11T06-08-02.136775|chosen" } }, { "text": "I also know that the area of an equilateral triangle with side length $s$ is $\\frac{s^2 \\sqrt{3}}{4}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 35, "metadata": { "path_label": "let_abcdef_be_a_regular_hexagon_let_g_h_i_j__2023-02-11T06-08-02.136775|chosen" } }, { "text": "So, the area of the larger hexagon $ABCDEF$ is six times the area of one of these triangles, which is $\\frac{3s^2 \\sqrt{3}}{2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 35, "metadata": { "path_label": "let_abcdef_be_a_regular_hexagon_let_g_h_i_j__2023-02-11T06-08-02.136775|chosen" } }, { "text": "To find the area of the smaller hexagon, I need to find the length of one of its sides.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 35, "metadata": { "path_label": "let_abcdef_be_a_regular_hexagon_let_g_h_i_j__2023-02-11T06-08-02.136775|chosen" } }, { "text": "I notice that the smaller hexagon is formed by the intersections of the segments $\\overline{AH}$, $\\overline{BI}$, $\\overline{CJ}$, $\\overline{DK}$, $\\overline{EL}$, and $\\overline{FG}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 35, "metadata": { "path_label": "let_abcdef_be_a_regular_hexagon_let_g_h_i_j__2023-02-11T06-08-02.136775|chosen" } }, { "text": "I need to find the greatest common divisor of $m$ and $100$, which means I need to factor both numbers into prime factors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 36, "metadata": { "path_label": "how_many_integers_m_are_there_such_that_0_m_100_and_gc__2023-02-11T06-01-42.358934|chosen" } }, { "text": "I know that $100 = 2^2 \\cdot 5^2$, so I can write $m = 2^a \\cdot 5^b \\cdot n$, where $n$ is any other factor that is relatively prime to $100$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 36, "metadata": { "path_label": "how_many_integers_m_are_there_such_that_0_m_100_and_gc__2023-02-11T06-01-42.358934|chosen" } }, { "text": "The possible values of $a$ and $b$ are $0, 1, 2$, and the possible values of $n$ are $1, 3, 7, 9, 11, 13, \\dots, 99$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 36, "metadata": { "path_label": "how_many_integers_m_are_there_such_that_0_m_100_and_gc__2023-02-11T06-01-42.358934|chosen" } }, { "text": "To find the greatest common divisor of $m$ and $100$, I need to take the smallest exponent of each prime factor, so $\\gcd(m,100) = 2^{\\min(a,2)} \\cdot 5^{\\min(b,2)}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 36, "metadata": { "path_label": "how_many_integers_m_are_there_such_that_0_m_100_and_gc__2023-02-11T06-01-42.358934|chosen" } }, { "text": "To solve this problem, I need to find a polynomial equation with rational coefficients that has $\\sqrt[3]{7} + \\sqrt[3]{49}$ as a root.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 37, "metadata": { "path_label": "let_p_x_0_be_the_polynomial_equation_of_least_possible__2023-02-11T05-00-46.747840|chosen" } }, { "text": "I also need to find the degree of this polynomial, which is the number of roots it has, and then multiply them all together.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 37, "metadata": { "path_label": "let_p_x_0_be_the_polynomial_equation_of_least_possible__2023-02-11T05-00-46.747840|chosen" } }, { "text": "One possible strategy is to try to eliminate the cube roots by cubing both sides of the equation $x = \\sqrt[3]{7} + \\sqrt[3]{49}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 37, "metadata": { "path_label": "let_p_x_0_be_the_polynomial_equation_of_least_possible__2023-02-11T05-00-46.747840|chosen" } }, { "text": "If I do that, I get $x^3 = 7 + 3\\sqrt[3]{7^2}\\sqrt[3]{49} + 3\\sqrt[3]{7}\\sqrt[3]{49^2} + 49.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 37, "metadata": { "path_label": "let_p_x_0_be_the_polynomial_equation_of_least_possible__2023-02-11T05-00-46.747840|chosen" } }, { "text": "I need to find a way to relate the number of digits after the decimal point of $\\frac{1}{k}$ to some property of $k$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "I recall that $\\frac{1}{k}$ has a terminating decimal expansion if and only if $k$ is of the form $2^a 5^b$, where $a$ and $b$ are nonnegative integers.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "In that case, the number of digits after the decimal point of $\\frac{1}{k}$ is equal to the maximum of $a$ and $b$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "So, if I want $\\frac{1}{k}$ to have exactly $n$ digits after the decimal point, I need to find the smallest $k$ such that $\\max(a,b) = n$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "That means I can either choose $a = n$ and $b = 0$, or $a = 0$ and $b = n$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "In either case, $k = 2^n$ or $k = 5^n$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "I want to minimize $k$, so I should choose the smaller base, which is $2$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "Therefore, $f(n) = 2^n$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "Now I need to count how many positive integer divisors $f(2010) = 2^{2010}$ has.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "I know that the number of divisors of a positive integer is equal to the product of one more than the exponents of its prime factorization.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "Since $f(2010)$ is a power of $2$, its only prime factor is $2$ with an exponent of $2010$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "So, the number of divisors of $f(2010)$ is $2010 + 1 = 2011$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 38, "metadata": { "path_label": "for_positive_integers_n_let_f_n_return_the_smallest_po__2023-02-07T15-51-41.652234|chosen" } }, { "text": "I notice that the sum is the sum of all possible products of two distinct numbers from the set $\\{a_1, a_2, \\dots, a_{100}\\}.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "I also notice that each product is either $1$ or $-1,$ depending on whether the two factors have the same or opposite signs.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "Therefore, the sum is equal to the number of pairs with the same sign minus the number of pairs with opposite signs.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "I want to minimize the sum, so I want to minimize the number of pairs with the same sign and maximize the number of pairs with opposite signs.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "To do that, I need to make the signs of the numbers as evenly distributed as possible.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "That means I want to have exactly $50$ numbers equal to $1$ and $50$ numbers equal to $-1.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "In that case, the number of pairs with the same sign is $\\binom{50}{2} + \\binom{50}{2} = 2\\binom{50}{2}.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "The number of pairs with opposite signs is $50 \\cdot 50 = 2500.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "Therefore, the minimum positive value of the sum is $2\\binom{50}{2} - 2500.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "I can simplify this expression by using the formula $\\binom{n}{2} = \\frac{n(n-1)}{2}.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "Then, $2\\binom{50}{2} - 2500 = 2 \\cdot \\frac{50 \\cdot 49}{2} - 2500 = 50 \\cdot 49 - 2500 = 2450 - 2500 = -50.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 39, "metadata": { "path_label": "each_of_a_1_a_2_dots_a__100_is_equal_to_1_or__2023-02-07T15-25-28.413005|chosen" } }, { "text": "I know that the angle bisector of $\\angle A$ divides the opposite side $BC$ into two segments that are proportional to the adjacent sides $AB$ and $AC.$ That is, $\\frac{BD}{DC} = \\frac{AB}{AC}.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "I also know that by the cosine rule, $BC^2 = AB^2 + AC^2 - 2AB \\cdot AC \\cdot \\cos \\angle A.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Plugging in the given values, I get $BC^2 = 9 + 36 - 2 \\cdot 3 \\cdot 6 \\cdot \\frac{1}{8} = 45 - \\frac{9}{2} = \\frac{81}{2}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Taking the positive square root, I get $BC = \\frac{9}{\\sqrt{2}}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Now, I can use the angle bisector theorem to find $BD$ and $DC.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "If $\\frac{BD}{DC} = \\frac{AB}{AC},$ then $BD = \\frac{AB}{AC} \\cdot DC = \\frac{3}{6} \\cdot DC = \\frac{1}{2} \\cdot DC.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Adding $BD$ and $DC$ gives $BC,$ so I have $BD + DC = BC.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Substituting $BD = \\frac{1}{2} \\cdot DC$ and $BC = \\frac{9}{\\sqrt{2}},$ I get $\\frac{1}{2} \\cdot DC + DC = \\frac{9}{\\sqrt{2}}.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "Simplifying, I get $\\frac{3}{2} \\cdot DC = \\frac{9}{\\sqrt{2}}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 40, "metadata": { "path_label": "in_triangle_abc_ab_3_ac_6_and_cos_angle_a__2023-02-07T13-33-24.598666|chosen" } }, { "text": "To find the 3-digit palindromes, I need to think about what the first and last digits can be.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "Since the first and last digits are the same, they must both be even or both be odd.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "If they are both even, they can only be $0, 2, 4, 6, 8$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "But $0$ is not allowed as the first digit, so that leaves $2, 4, 6, 8$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "If they are both odd, they can be $1, 3, 5, 7, 9$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "So there are $4 + 5 = 9$ possible choices for the first and last digits.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "For each choice, the middle digit can be anything from $0$ to $9$, so there are $10$ possible choices for the middle digit.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "Therefore, the total number of 3-digit palindromes is $9 \\times 10 = 90$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "But I only want the ones that are multiples of $3$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "To check if a number is a multiple of $3$, I can add up its digits and see if the sum is divisible by $3$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "For example, $323$ is a palindrome, and its digits add up to $3 + 2 + 3 = 8$, which is not divisible by $3$, so $323$ is not a multiple of $3$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "But $363$ is a palindrome, and its digits add up to $3 + 6 + 3 = 12$, which is divisible by $3$, so $363$ is a multiple of $3$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "So I need to find a way to count how many palindromes have a digit sum that is divisible by $3$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "One way to do this is to look at the possible digit sums for each choice of the first and last digits.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 41, "metadata": { "path_label": "a_textit_palindrome_is_an_integer_that_reads_the_same_fo__2023-02-07T11-45-39.318116|chosen" } }, { "text": "I notice that the equation has the form of the sum of four squared terms, each of which is nonnegative.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 42, "metadata": { "path_label": "given_that_0_le_x_3_le_x_2_le_x_1_le_1_and_1_x_1_2_x__2023-02-11T02-04-15.718402|chosen" } }, { "text": "This makes me think of the arithmetic mean - quadratic mean inequality, which states that for any nonnegative numbers $a_1, a_2, \\dots, a_n,$ we have $\\frac{a_1 + a_2 + \\cdots + a_n}{n} \\le \\sqrt{\\frac{a_1^2 + a_2^2 + \\cdots + a_n^2}{n}}.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 42, "metadata": { "path_label": "given_that_0_le_x_3_le_x_2_le_x_1_le_1_and_1_x_1_2_x__2023-02-11T02-04-15.718402|chosen" } }, { "text": "Applying this inequality to the four terms in the equation, I get $\\frac{(1-x_1)+(x_1-x_2)+(x_2-x_3)+x_3}{4} \\le \\sqrt{\\frac{(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2}{4}}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 42, "metadata": { "path_label": "given_that_0_le_x_3_le_x_2_le_x_1_le_1_and_1_x_1_2_x__2023-02-11T02-04-15.718402|chosen" } }, { "text": "Simplifying the numerator, I get $\\frac{1}{4} \\le \\sqrt{\\frac{(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2}{4}}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 42, "metadata": { "path_label": "given_that_0_le_x_3_le_x_2_le_x_1_le_1_and_1_x_1_2_x__2023-02-11T02-04-15.718402|chosen" } }, { "text": "To solve for $x,$ I need to first find $f(f(x))$ and then equate it to $x.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "To find $f(f(x)),$ I need to plug in $f(x)$ as the input of $f.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "That means $f(f(x)) = \\frac{2f(x)+9}{f(x)-7}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "Now I need to simplify this expression by substituting $f(x) = \\frac{2x+9}{x-7}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "I get $f(f(x)) = \\frac{2\\frac{2x+9}{x-7}+9}{\\frac{2x+9}{x-7}-7}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "To simplify further, I need to multiply the numerator and denominator by $x-7.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "I get $f(f(x)) = \\frac{2(2x+9)+9(x-7)}{2x+9-7(x-7)}.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "Expanding and combining like terms, I get $f(f(x)) = \\frac{4x+18+9x-63}{2x+9-7x+49} = \\frac{13x-45}{-5x+58}.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "Now I can equate this to $x$ and cross-multiply to get a quadratic equation.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "I get $x(-5x+58) = 13x-45.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "Expanding and simplifying, I get $-5x^2+58x = 13x-45.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "Subtracting $13x$ from both sides, I get $-5x^2+45x = -45.$ [*,] Adding $45$ to both sides, I get $-5x^2+45x+45 = 0.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "To solve for $x,$ I can either factor or use the quadratic formula.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "I notice that this equation is divisible by $-5,$ so I can simplify it by dividing both sides by $-5.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "I get $x^2-9x-9 = 0.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "This looks easier to factor, since I only need to find two numbers that multiply to $-9$ and add to $-9.$", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 43, "metadata": { "path_label": "let_f_x_frac_2x_9_x_7_find_the_product_of_all_solut__2023-02-11T01-41-44.728506|chosen" } }, { "text": "A horizontal line of symmetry is a line that divides a shape into two congruent halves that are mirror images across the line.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 44, "metadata": { "path_label": "how_many_of_the_letters_in_mathcounts_have_a_horizontal_line__2023-02-10T06-45-30.326635|chosen" } }, { "text": "To find out how many letters in MATHCOUNTS have a horizontal line of symmetry, I need to look at each letter and see if I can find such a line.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 44, "metadata": { "path_label": "how_many_of_the_letters_in_mathcounts_have_a_horizontal_line__2023-02-10T06-45-30.326635|chosen" } }, { "text": "I can ignore repeated letters, since they will have the same result.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 44, "metadata": { "path_label": "how_many_of_the_letters_in_mathcounts_have_a_horizontal_line__2023-02-10T06-45-30.326635|chosen" } }, { "text": "So I only need to consider M, A, T, H, C, O, U, N, and S.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 44, "metadata": { "path_label": "how_many_of_the_letters_in_mathcounts_have_a_horizontal_line__2023-02-10T06-45-30.326635|chosen" } }, { "text": "I see a difference of two fractions involving sine functions, so I wonder if there is a way to simplify them by using some trigonometric identities or transformations.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "One identity that comes to mind is the double angle formula for sine: $\\sin 2\\theta = 2 \\sin \\theta \\cos \\theta$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "I can use this formula to rewrite $\\sin 70^\\circ$ as $\\sin 2 \\cdot 35^\\circ = 2 \\sin 35^\\circ \\cos 35^\\circ$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "Then I can plug this into the second fraction and get $-2 \\sin 70^\\circ = -4 \\sin 35^\\circ \\cos 35^\\circ$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "Now I have $\\frac{1}{2 \\sin 10^\\circ} - 4 \\sin 35^\\circ \\cos 35^\\circ$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "I notice that both fractions have a factor of 2 in the denominator, so I can combine them under a common denominator of $4 \\sin 10^\\circ$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 45, "metadata": { "path_label": "compute_frac_1_2_sin_10_circ_2_sin_70_circ__2023-02-10T06-43-11.673254|chosen" } }, { "text": "I notice that the polynomial has alternating coefficients of 2 and 3, and that the degree is odd.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I wonder if there is a pattern or a factorization that could simplify the expression.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I try to group the terms by pairs, starting from the highest degree: \\[(2x^{2001} + 3x^{2000}) + (2x^{1999} + 3x^{1998}) + \\dots + (2x + 3) = 0.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I notice that each pair has a common factor of $x^{2k}$, where $k$ is a non-negative integer, so I can factor it out: \\[x^{2000}(2x + 3) + x^{1998}(2x + 3) + \\dots + (2x + 3) = 0.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "Now I see that the whole expression has a common factor of $2x + 3$, so I can factor it out as well: \\[(2x + 3)(x^{2000} + x^{1998} + \\dots + 1) = 0.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I recall that a product of factors is zero if and only if at least one of the factors is zero, so I can set each factor equal to zero and solve for $x$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I start with the simpler factor, $2x + 3 = 0$. I subtract 3 from both sides and divide by 2 to get $x = -\\frac{3}{2}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "This is one real root of the original polynomial.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I move on to the other factor, $x^{2000} + x^{1998} + \\dots + 1 = 0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "This looks like a sum of even powers of $x$, so I wonder if I can use a substitution to simplify it.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I let $y = x^2$, so $y^k = x^{2k}$ for any positive integer $k$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "Then the equation becomes $y^{1000} + y^{999} + \\dots + 1 = 0$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "This looks like a geometric series, so I can use the formula for the sum of a finite geometric series: \\[y^{1000} + y^{999} + \\dots + 1 = \\frac{y^{1001} - 1}{y - 1}.\\]", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "To find the zeros of this expression, I need to find the values of $y$ that make the numerator zero or the denominator undefined.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 46, "metadata": { "path_label": "find_the_number_of_real_roots_of_2x_2001_3x_2000_2__2023-02-10T06-07-41.135380|chosen" } }, { "text": "I want to find the value of $n$ that makes $\\frac{n+1}{13-n}$ a positive prime number.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 47, "metadata": { "path_label": "what_is_the_only_integer_value_of_n_for_which_frac_n_1__2023-02-10T05-55-33.669703|chosen" } }, { "text": "This means that $\\frac{n+1}{13-n}$ has to be an integer greater than 1, and it has to have no positive factors other than itself and 1.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 47, "metadata": { "path_label": "what_is_the_only_integer_value_of_n_for_which_frac_n_1__2023-02-10T05-55-33.669703|chosen" } }, { "text": "Since $\\frac{n+1}{13-n}$ is a fraction, I can simplify it by finding the greatest common factor of the numerator and the denominator and dividing both by it.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 47, "metadata": { "path_label": "what_is_the_only_integer_value_of_n_for_which_frac_n_1__2023-02-10T05-55-33.669703|chosen" } }, { "text": "The greatest common factor of $n+1$ and $13-n$ is the same as the greatest common factor of $n+1$ and $13-n + (n+1) = 14$, since adding or subtracting multiples of one term does not change the greatest common factor.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 47, "metadata": { "path_label": "what_is_the_only_integer_value_of_n_for_which_frac_n_1__2023-02-10T05-55-33.669703|chosen" } }, { "text": "I need to find the smallest positive integer that is divisible by every number from 1 to 25, except for two consecutive numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "One way to approach this problem is to use the concept of the least common multiple (LCM), which is the smallest number that is a multiple of a given set of numbers.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "To find the LCM of a set of numbers, I can use prime factorization and take the highest power of each prime factor that appears in the set.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "For example, the LCM of 4, 6, and 15 is 2^2 * 3 * 5 = 60, because 4 = 2^2, 6 = 2 * 3, and 15 = 3 * 5, and the highest power of each prime factor is 2^2, 3, and 5.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "Similarly, the LCM of all the numbers from 1 to 25 would be 2^4 * 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23, because these are all the prime numbers in that range and their highest powers.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "However, this is not the answer, because I need to exclude two consecutive numbers from the set.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "Which two numbers should I exclude?", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 48, "metadata": { "path_label": "a_group_of_25_friends_were_discussing_a_large_positive_integ__2023-02-10T05-42-39.064561|chosen" } }, { "text": "I notice that this is a cubic equation, so I expect the roots to be related by some symmetric expressions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 49, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_the_cubic_polynomial_x_3__2023-02-10T05-39-16.513188|chosen" } }, { "text": "For example, I know that $a + b + c = 0,$ $ab + bc + ca = -1,$ and $abc = 1$ by Vieta's formulas.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 49, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_the_cubic_polynomial_x_3__2023-02-10T05-39-16.513188|chosen" } }, { "text": "I also notice that the expression I want to find is symmetric in $a, b, c,$ so I wonder if I can rewrite it using these formulas.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 49, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_the_cubic_polynomial_x_3__2023-02-10T05-39-16.513188|chosen" } }, { "text": "I try to expand the expression and see what happens: \\[a(b - c)^2 + b(c - a)^2 + c(a - b)^2 = ab^2 - 2abc + ac^2 + bc^2 - 2abc + ba^2 + ca^2 - 2abc + cb^2.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 49, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_the_cubic_polynomial_x_3__2023-02-10T05-39-16.513188|chosen" } }, { "text": "I can simplify this by combining like terms and using $abc = 1$: \\[a(b - c)^2 + b(c - a)^2 + c(a - b)^2 = ab^2 + ba^2 + bc^2 + cb^2 + ca^2 + ac^2 - 6.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 49, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_the_cubic_polynomial_x_3__2023-02-10T05-39-16.513188|chosen" } }, { "text": "I want to find the values of $n$ that make the number of apples after selling one a multiple of $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "I know that the number of apples before selling one is a multiple of $6$, so I can write it as $6k$ for some positive integer $k$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "Then, after selling one apple, the number of apples is $6k - 1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "To make this a multiple of $n$, I need $6k - 1$ to be divisible by $n$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "This means that $6k - 1 \\equiv 0 \\pmod{n}$, or equivalently, $6k \\equiv 1 \\pmod{n}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "This is a modular equation that I can solve for $k$ using the inverse of $6$ modulo $n$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "The inverse of $6$ modulo $n$ is the number $x$ such that $6x \\equiv 1 \\pmod{n}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "This number exists if and only if $6$ and $n$ are coprime, meaning they have no common factors other than $1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "So, the possible values of $n$ are those that are coprime with $6$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "Since $n$ is less than $10$, I can list the values of $n$ from $1$ to $9$ and check which ones are coprime with $6$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "I can use the Euclidean algorithm to check for coprimality, or I can just look at the prime factorization of $n$ and see if it has any factors of $2$ or $3$, which are the only prime factors of $6$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 50, "metadata": { "path_label": "the_number_of_apples_that_sophia_has_is_a_multiple_of_6_w__2023-02-08T23-18-56.697199|chosen" } }, { "text": "I know that Pascal's triangle is formed by adding the two numbers above each entry, starting with 1 on the edges.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 51, "metadata": { "path_label": "what_is_the_second_smallest_four_digit_number_in_pascal_s_tr__2023-02-08T20-41-43.547834|chosen" } }, { "text": "So the first row is 1, the second row is 1 1, the third row is 1 2 1, and so on.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 51, "metadata": { "path_label": "what_is_the_second_smallest_four_digit_number_in_pascal_s_tr__2023-02-08T20-41-43.547834|chosen" } }, { "text": "To find the four-digit numbers, I need to look at the rows that have more than four entries, because the first and last entries are always 1.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 51, "metadata": { "path_label": "what_is_the_second_smallest_four_digit_number_in_pascal_s_tr__2023-02-08T20-41-43.547834|chosen" } }, { "text": "The first such row is the fifth row, which is 1 4 6 4 1.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 51, "metadata": { "path_label": "what_is_the_second_smallest_four_digit_number_in_pascal_s_tr__2023-02-08T20-41-43.547834|chosen" } }, { "text": "I need to find the radius of the cylinder, which is also half of its diameter and height.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I will draw a diagram to help me visualize the problem.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I will label the radius of the cylinder as r, the radius of the cone as 5, and the height of the cone as 12.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I will also draw a right triangle inside the cone, with one leg as the radius of the cone, the other leg as the height of the cone, and the hypotenuse as the slant height of the cone.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I can use the Pythagorean theorem to find the slant height of the cone.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "It is sqrt(5^2 + 12^2) = sqrt(169) = 13.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I will label the slant height as 13.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I notice that the cylinder is tangent to the cone at two points, one at the base and one at the top.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "I will draw a line segment from the center of the cone's base to the point of tangency at the top.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 52, "metadata": { "path_label": "a_right_circular_cylinder_with_its_diameter_equal_to_its_hei__2023-02-08T10-17-49.337399|chosen" } }, { "text": "This problem involves logarithms, trigonometry, and inequalities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 53, "metadata": { "path_label": "if_0_le_theta_le_4_pi_find_all_values_of_theta_whi__2023-02-08T06-09-53.961709|chosen" } }, { "text": "I need to recall the definition of logarithms, some basic properties of logarithms, and some trigonometric identities.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 53, "metadata": { "path_label": "if_0_le_theta_le_4_pi_find_all_values_of_theta_whi__2023-02-08T06-09-53.961709|chosen" } }, { "text": "The definition of logarithms says that $\\log_b a = c$ if and only if $b^c = a.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 53, "metadata": { "path_label": "if_0_le_theta_le_4_pi_find_all_values_of_theta_whi__2023-02-08T06-09-53.961709|chosen" } }, { "text": "So, I can rewrite the given equation as $\\left(\\frac{1}{2} \\sin 2 \\theta\\right)^{1/2} = \\sin \\theta.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 53, "metadata": { "path_label": "if_0_le_theta_le_4_pi_find_all_values_of_theta_whi__2023-02-08T06-09-53.961709|chosen" } }, { "text": "This might make it easier to work with.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 53, "metadata": { "path_label": "if_0_le_theta_le_4_pi_find_all_values_of_theta_whi__2023-02-08T06-09-53.961709|chosen" } }, { "text": "I need to find all the possible pairs of coins and their sums, and then count how many distinct sums there are.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "I can use a table to organize the pairs and sums.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "For each coin, I will list the other coins that can be paired with it, and the corresponding sums.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "I will start with the penny, which has a value of $0.01.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "A penny can be paired with another penny, a nickel, a dime, or a quarter.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "The sums of these pairs are $0.02, 0.06, 0.11, and 0.26$ respectively.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "I will write these sums in the table, and mark them with a check mark to indicate that they are distinct.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "Next, I will move on to the nickel, which has a value of $0.05$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "A nickel can be paired with another nickel, a dime, or a quarter.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "The sums of these pairs are $0.10, 0.15, and 0.30$ respectively.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "I will write these sums in the table, and mark them with a check mark if they are distinct.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 54, "metadata": { "path_label": "a_boy_has_the_following_seven_coins_in_his_pocket_2_penni__2023-02-08T04-39-54.525890|chosen" } }, { "text": "I know that the measure of each interior angle of a regular $n~\\mbox{gon}$ is given by the formula $\\frac{(n-2)\\pi}{n}$ radians.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 55, "metadata": { "path_label": "let_p_1_be_a_regular_r_mbox_gon_and_p_2_be_a_regular__2023-02-08T03-27-07.652437|chosen" } }, { "text": "So, the problem is asking me to find $r$ and $s$ such that $\\frac{(r-2)\\pi}{r} = \\frac{59}{58} \\cdot \\frac{(s-2)\\pi}{s}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 55, "metadata": { "path_label": "let_p_1_be_a_regular_r_mbox_gon_and_p_2_be_a_regular__2023-02-08T03-27-07.652437|chosen" } }, { "text": "I can simplify this equation by multiplying both sides by $\\frac{r}{\\pi}$ and then by $\\frac{s}{\\pi}$ to get $(r-2)s = \\frac{59}{58} (s-2)r$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 55, "metadata": { "path_label": "let_p_1_be_a_regular_r_mbox_gon_and_p_2_be_a_regular__2023-02-08T03-27-07.652437|chosen" } }, { "text": "Expanding and rearranging, I get $58rs - 116s = 59rs - 118r$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 55, "metadata": { "path_label": "let_p_1_be_a_regular_r_mbox_gon_and_p_2_be_a_regular__2023-02-08T03-27-07.652437|chosen" } }, { "text": "I notice that the numerator and denominator are both sums of powers of $\\sin x$ and $\\cos x$, which makes me think of using the identity $\\sin^2 x + \\cos^2 x = 1$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "If I square both sides of this identity, I get $\\sin^4 x + \\cos^4 x + 2\\sin^2 x \\cos^2 x = 1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "This means I can replace the $\\sin^4 x + \\cos^4 x$ term in the denominator with $1 - 2\\sin^2 x \\cos^2 x$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "Similarly, if I cube both sides of the identity, I get $\\sin^6 x + \\cos^6 x + 3\\sin^2 x \\cos^2 x (\\sin^2 x + \\cos^2 x) = 1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "This means I can replace the $\\sin^6 x + \\cos^6 x$ term in the numerator with $1 - 3\\sin^2 x \\cos^2 x$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "So the fraction becomes \\[\\frac{1 - 3\\sin^2 x \\cos^2 x + 1}{1 - 2\\sin^2 x \\cos^2 x + 1}\\] which simplifies to \\[\\frac{2 - 3\\sin^2 x \\cos^2 x}{2 - 2\\sin^2 x \\cos^2 x}\\] or \\[\\frac{1 - \\frac{3}{2}\\sin^2 x \\cos^2 x}{1 - \\sin^2 x \\cos^2 x}\\].", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "Now I want to find the minimum value of this fraction over all real values of $x$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "To do that, I need to maximize the value of $\\sin^2 x \\cos^2 x$, since it appears with a negative coefficient in both the numerator and the denominator.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "I recall that $\\sin^2 x \\cos^2 x = \\frac{1}{4}\\sin^2 2x$, so I can rewrite the fraction as \\[\\frac{1 - \\frac{3}{8}\\sin^2 2x}{1 - \\frac{1}{4}\\sin^2 2x}\\].", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "Now I see that the fraction is a function of $2x$, so I can let $y = 2x$ and consider the fraction as a function of $y$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "I want to maximize $\\sin^2 y$, which happens when $\\sin y = \\pm 1$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "This means that $y = \\pm \\frac{\\pi}{2}, \\pm \\frac{3\\pi}{2}, \\dots$ or equivalently, $x = \\pm \\frac{\\pi}{4}, \\pm \\frac{3\\pi}{4}, \\dots$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "When $\\sin^2 y = 1$, the fraction becomes \\[\\frac{1 - \\frac{3}{8}}{1 - \\frac{1}{4}} = \\frac{5}{8} \\cdot \\frac{4}{3} = \\frac{5}{6}\\].", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "This is the minimum value of the fraction, since any other value of $y$ would make $\\sin^2 y < 1$ and increase the fraction.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "Therefore, the minimum value of the original fraction is $\\frac{5}{6}$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 56, "metadata": { "path_label": "find_the_minimum_value_of_frac_sin_6_x_cos_6_x_1__2023-02-07T23-49-13.562700|chosen" } }, { "text": "To find the number of lattice points, I need to know which integers satisfy the equation of the circle.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "The equation of the circle is $x^2 + y^2 = 25$, since the radius is 5 and the center is at the origin.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "I can try different values of $x$ and see if they have a corresponding integer value of $y$ that satisfies the equation.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = 0$, then $y = \\pm 5$, so that gives me two lattice points: $(0,5)$ and $(0,-5)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = \\pm 1$, then $y^2 = 25 - 1 = 24$, which is not a perfect square, so there are no integer values of $y$ that work.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = \\pm 2$, then $y^2 = 25 - 4 = 21$, which is also not a perfect square, so there are no integer values of $y$ that work.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = \\pm 3$, then $y^2 = 25 - 9 = 16$, which is a perfect square, so $y = \\pm 4$. That gives me four lattice points: $(3,4)$, $(3,-4)$, $(-3,4)$, and $(-3,-4)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = \\pm 4$, then $y^2 = 25 - 16 = 9$, which is a perfect square, so $y = \\pm 3$. That gives me four more lattice points: $(4,3)$, $(4,-3)$, $(-4,3)$, and $(-4,-3)$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "If $x = \\pm 5$, then $y = 0$, so that gives me two more lattice points: $(5,0)$ and $(-5,0)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "I have exhausted all the possible values of $x$ from $-5$ to $5$, and I have found a total of 12 lattice points that lie within or on the circle.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 57, "metadata": { "path_label": "a_circle_of_radius_5_with_its_center_at_0_0_is_drawn_on__2023-02-07T22-50-26.230179|chosen" } }, { "text": "To find the probability, I need to count the number of favorable outcomes and divide by the total number of possible outcomes.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 58, "metadata": { "path_label": "john_rolls_a_pair_of_standard_6_sided_dice_what_is_the_prob__2023-02-03T23-25-49.179517|chosen" } }, { "text": "The total number of possible outcomes is 36, since there are 6 choices for each die.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 58, "metadata": { "path_label": "john_rolls_a_pair_of_standard_6_sided_dice_what_is_the_prob__2023-02-03T23-25-49.179517|chosen" } }, { "text": "The favorable outcomes are those where the two numbers have no common factors other than 1.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 58, "metadata": { "path_label": "john_rolls_a_pair_of_standard_6_sided_dice_what_is_the_prob__2023-02-03T23-25-49.179517|chosen" } }, { "text": "I can list them as pairs: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5), (4, 1), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (6, 1), (6, 5).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 58, "metadata": { "path_label": "john_rolls_a_pair_of_standard_6_sided_dice_what_is_the_prob__2023-02-03T23-25-49.179517|chosen" } }, { "text": "This problem involves the inverse sine function, which is the function that undoes the sine function.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 59, "metadata": { "path_label": "solve_arcsin_sin_x_frac_x_2_enter_all_the_solut__2023-02-03T22-09-18.841005|chosen" } }, { "text": "For example, \\[\\arcsin (\\sin (\\frac{\\pi}{4})) = \\frac{\\pi}{4},\\] because the sine of $\\pi/4$ is $\\sqrt{2}/2$, and the inverse sine of $\\sqrt{2}/2$ is $\\pi/4$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 59, "metadata": { "path_label": "solve_arcsin_sin_x_frac_x_2_enter_all_the_solut__2023-02-03T22-09-18.841005|chosen" } }, { "text": "However, the inverse sine function is not a true inverse of the sine function, because the sine function is not one-to-one.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 59, "metadata": { "path_label": "solve_arcsin_sin_x_frac_x_2_enter_all_the_solut__2023-02-03T22-09-18.841005|chosen" } }, { "text": "This means that there are infinitely many values of $x$ that have the same sine, such as \\[\\sin (\\frac{\\pi}{4}) = \\sin (\\frac{3\\pi}{4}) = \\sin (\\frac{9\\pi}{4}) = \\sin (\\frac{11\\pi}{4}) = \\cdots.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 59, "metadata": { "path_label": "solve_arcsin_sin_x_frac_x_2_enter_all_the_solut__2023-02-03T22-09-18.841005|chosen" } }, { "text": "I notice that $n^2-3n+2$ can be factored as $(n-2)(n-1)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "So, unless one of the factors is $1$ and the other is prime, the expression will not be prime.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "That means I only need to check the cases where $n-2=1$ or $n-1=1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "If $n-2=1$, then $n=3$ and $n^2-3n+2=2$, which is prime.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "If $n-1=1$, then $n=2$ and $n^2-3n+2=0$, which is not prime.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "So, there is only one positive integer $n$ that makes $n^2-3n+2$ prime, and that is $n=3$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 60, "metadata": { "path_label": "for_how_many_positive_integers_n_is_n_2_3n_2_a_prime_num__2023-02-03T00-58-14.607679|chosen" } }, { "text": "To answer this question, I need to think of pairs of single-digit factors that multiply to 12.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "I can list them out: 1 and 12, 2 and 6, 3 and 4, 4 and 3, 6 and 2, and 12 and 1.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "However, not all of these pairs can be the digits of a two-digit number.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "For example, 12 and 1 would make either 121 or 112, which are both three-digit numbers.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "So I need to eliminate the pairs that have a digit greater than 9.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "That leaves me with 2 and 6, 3 and 4, 4 and 3, and 6 and 2.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "These pairs can form four different two-digit numbers: 26, 34, 43, and 62.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "Therefore, there are four two-digit positive integers whose product of the digits is 12.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 61, "metadata": { "path_label": "for_how_many_two_digit_positive_integers_does_the_product_of__2023-02-02T10-57-56.317261|chosen" } }, { "text": "I notice that this expression is a quadratic function in two variables, so it might have a minimum value at some point where the partial derivatives are zero.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "To find the partial derivatives, I treat one variable as constant and differentiate with respect to the other. For example, the partial derivative with respect to $x$ is \\[\\frac{\\partial}{\\partial x}(2x^2 + 2xy + 4y + 5y^2 - x) = 4x + 2y - 1.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "Similarly, the partial derivative with respect to $y$ is \\[\\frac{\\partial}{\\partial y}(2x^2 + 2xy + 4y + 5y^2 - x) = 2x + 4 + 10y.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "To find the point where both partial derivatives are zero, I need to solve the system of equations \\[\\begin{cases} 4x + 2y - 1 = 0 \\\\ 2x + 4 + 10y = 0 \\end{cases}\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "I can use substitution or elimination to solve this system. I choose to eliminate $y$ by multiplying the first equation by $5$ and the second equation by $-1$, and then adding them.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "This gives me \\[20x - 5 - 2x - 4 = 0\\] or \\[18x - 9 = 0\\] or \\[x = \\frac{1}{2}.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "To find the corresponding value of $y$, I can plug in $x = \\frac{1}{2}$ into either equation. I choose the first one.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 62, "metadata": { "path_label": "find_the_minimum_value_of_2x_2_2xy_4y_5y_2_x_over__2023-02-02T10-27-00.883545|chosen" } }, { "text": "I notice that the problem is asking about the winning score, not the winning team or the order of the runners. So I can ignore the names and colors of the teams and just focus on the possible scores.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "I also notice that the score of a team is the sum of the positions of its 5 runners, and that the positions are integers from 1 to 10.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "So the problem is really asking how many different sums of 5 distinct integers from 1 to 10 are possible, such that the sum is smaller than the sum of the remaining 5 integers.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "One way to approach this is to list all the possible sums and count them, but that seems tedious and prone to error.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "Another way is to look for patterns or symmetry in the sums.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "I notice that the smallest possible sum is 1 + 2 + 3 + 4 + 5 = 15, and the largest possible sum is 6 + 7 + 8 + 9 + 10 = 40.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "I also notice that for any sum x between 15 and 40, there is a corresponding sum 55 - x that is the complement of x.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "For example, if x = 15, then 55 - x = 40. If x = 20, then 55 - x = 35.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "This means that there are pairs of sums that add up to 55, and only one of them can be a winning score, depending on which team has it.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "For example, if one team has a sum of 15 and the other has a sum of 40, then the team with 15 wins. But if one team has a sum of 20 and the other has a sum of 35, then the team with 20 wins.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "So I can divide the possible sums into two groups: those that are smaller than 55/2 = 27.5, and those that are larger.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "The winning scores are the ones that are smaller than 27.5, because they are paired with larger sums that are losing scores.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "So how many sums are smaller than 27.5?", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "Well, I know that 15 is the smallest sum, and 27 is the largest sum that is smaller than 27.5.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "I also know that the sums increase by 1 when I replace the smallest integer in the sum with the next larger integer.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "For example, if I replace 5 with 6 in the sum 1 + 2 + 3 + 4 + 5, I get 1 + 2 + 3 + 4 + 6, which has a sum of 16.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "So the sums that are smaller than 27.5 are 15, 16, 17, ..., 27.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "To count how many there are, I can subtract the endpoints and add 1: 27 - 15 + 1 = 13.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "So there are 13 possible winning scores.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 63, "metadata": { "path_label": "in_a_certain_cross_country_meet_between_2_teams_of_5_runners__2023-02-02T08-24-02.518058|chosen" } }, { "text": "This is a tricky problem, because the sequence is not arithmetic or geometric, but it has a recursive formula.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 64, "metadata": { "path_label": "a_sequence_of_integers_a_1_a_2_a_3_ldots_is_chosen_so__2023-02-10T05-33-14.678556|chosen" } }, { "text": "I wonder if I can find a pattern or a formula for the sequence or its sum.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 64, "metadata": { "path_label": "a_sequence_of_integers_a_1_a_2_a_3_ldots_is_chosen_so__2023-02-10T05-33-14.678556|chosen" } }, { "text": "Let me start by writing down some terms of the sequence, using an arbitrary $a_1$ and $a_2.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 64, "metadata": { "path_label": "a_sequence_of_integers_a_1_a_2_a_3_ldots_is_chosen_so__2023-02-10T05-33-14.678556|chosen" } }, { "text": "I notice that both numbers are of the form $2^n-1$, which makes me think of the Euclidean algorithm for finding the greatest common divisor (gcd) of two numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 65, "metadata": { "path_label": "what_is_the_greatest_common_divisor_of_2_1998_1_and_2__2023-02-10T05-24-28.131689|chosen" } }, { "text": "The Euclidean algorithm says that if $a$ and $b$ are positive integers, then the gcd of $a$ and $b$ is the same as the gcd of $b$ and $a \\mod b$, where $\\mod$ means the remainder after division.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 65, "metadata": { "path_label": "what_is_the_greatest_common_divisor_of_2_1998_1_and_2__2023-02-10T05-24-28.131689|chosen" } }, { "text": "So I can apply this algorithm to $a = 2^{1998}-1$ and $b = 2^{1989}-1$ and see what happens.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 65, "metadata": { "path_label": "what_is_the_greatest_common_divisor_of_2_1998_1_and_2__2023-02-10T05-24-28.131689|chosen" } }, { "text": "To find $a \\mod b$, I need to divide $a$ by $b$ and find the remainder.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 65, "metadata": { "path_label": "what_is_the_greatest_common_divisor_of_2_1998_1_and_2__2023-02-10T05-24-28.131689|chosen" } }, { "text": "But dividing $a$ by $b$ is not easy, since both numbers are huge.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 65, "metadata": { "path_label": "what_is_the_greatest_common_divisor_of_2_1998_1_and_2__2023-02-10T05-24-28.131689|chosen" } }, { "text": "I notice that the summand is a rational function of n, and I wonder if I can use partial fractions to simplify it.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "I write $\\frac1{n^2-4}=\\frac{A}{n-2}+\\frac{B}{n+2}$ for some constants A and B, and I multiply both sides by $n^2-4$ to get $1=A(n+2)+B(n-2)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "I want this equation to hold for all n, so I can choose any two values of n and get a system of equations for A and B.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "For example, if I let n=2, I get $1=4A$, so $A=\\frac14$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "Similarly, if I let n=-2, I get $1=-4B$, so $B=-\\frac14$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "Therefore, $\\frac1{n^2-4}=\\frac14\\left(\\frac1{n-2}-\\frac1{n+2}\\right)$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "Now I can rewrite the summation as $1000\\sum_{n=3}^{10000}\\frac1{n^2-4}=250\\sum_{n=3}^{10000}\\left(\\frac1{n-2}-\\frac1{n+2}\\right)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "I notice that this sum is telescoping, meaning that most of the terms cancel out with each other.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "For example, the first term is $\\frac1{1}-\\frac1{5}$, the second term is $\\frac1{2}-\\frac1{6}$, the third term is $\\frac1{3}-\\frac1{7}$, and so on.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "The only terms that do not cancel are the first few and the last few.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "Specifically, the terms that do not cancel are $\\frac1{1}+\\frac1{2}+\\frac1{3}+\\frac1{4}-\\frac1{9999}-\\frac1{10000}-\\frac1{10001}-\\frac1{10002}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "So the summation is equal to $250\\left(\\frac1{1}+\\frac1{2}+\\frac1{3}+\\frac1{4}-\\frac1{9999}-\\frac1{10000}-\\frac1{10001}-\\frac1{10002}\\right)$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "I can use a calculator or some approximation to find the value of this expression.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 66, "metadata": { "path_label": "find_the_integer_that_is_closest_to_1000_sum__n_3_10000__2023-02-10T02-19-43.835227|chosen" } }, { "text": "I recognize that this sum is related to the binomial expansion of $(1+i)^{99}$, where $i$ is the imaginary unit.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "To see why, I recall that $(1+i)^{99}=\\sum_{k=0}^{99}\\binom{99}{k}i^k$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "If I group the terms by the parity of $k$, I get $(1+i)^{99}=\\sum_{k=0}^{49}\\binom{99}{2k}i^{2k}+\\sum_{k=0}^{49}\\binom{99}{2k+1}i^{2k+1}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "Since $i^2=-1$ and $i^3=-i$, I can simplify this expression to $(1+i)^{99}=\\sum_{k=0}^{49}\\binom{99}{2k}(-1)^k+\\sum_{k=0}^{49}\\binom{99}{2k+1}(-1)^k i$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "Comparing this with the sum $S$, I see that $S$ is the real part of $(1+i)^{99}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "To find the real part of $(1+i)^{99}$, I can write $(1+i)^{99}$ in polar form, using the fact that $1+i=\\sqrt{2}e^{i\\pi/4}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "Then $(1+i)^{99}=(\\sqrt{2}e^{i\\pi/4})^{99}=\\sqrt{2}^{99}e^{i99\\pi/4}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "The real part of this expression is $\\sqrt{2}^{99}\\cos(99\\pi/4)$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "To simplify this, I use the periodicity of the cosine function and the fact that $\\cos(\\pi/4)=\\sqrt{2}/2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 67, "metadata": { "path_label": "what_is_the_value_of_the_sum_s_sum__k_0_49_1_k_binom__2023-02-06T04-00-34.418432|chosen" } }, { "text": "This problem looks like it involves some kind of ratio or similarity theorem.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 68, "metadata": { "path_label": "in_triangle_abc_a_b_and_c_are_on_the_sides_bc__2023-02-06T03-38-02.558200|chosen" } }, { "text": "I recall that there is a theorem that says that if $AA'$, $BB'$, and $CC'$ are concurrent cevians (segments from a vertex to the opposite side) in a triangle, then $\\frac{BA'}{A'C}\\cdot \\frac{CB'}{B'A}\\cdot \\frac{AC'}{C'B}=1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 68, "metadata": { "path_label": "in_triangle_abc_a_b_and_c_are_on_the_sides_bc__2023-02-06T03-38-02.558200|chosen" } }, { "text": "This is sometimes called Ceva's theorem, and it can be proved using areas of triangles.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 68, "metadata": { "path_label": "in_triangle_abc_a_b_and_c_are_on_the_sides_bc__2023-02-06T03-38-02.558200|chosen" } }, { "text": "I wonder if I can use this theorem to relate the given expression to the one I want to find.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 68, "metadata": { "path_label": "in_triangle_abc_a_b_and_c_are_on_the_sides_bc__2023-02-06T03-38-02.558200|chosen" } }, { "text": "To find the rational roots of $g(x)$, I can use the rational root theorem, which says that any rational root of the form $p/q$, where $p$ and $q$ are integers, must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "The constant term is $-4$, so its possible factors are $\\pm 1, \\pm 2, \\pm 4$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "The leading coefficient is $1$, so its only factor is $\\pm 1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "This means that the possible rational roots are $\\pm 1, \\pm 2, \\pm 4$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "I can check each of these values by plugging them into $g(x)$ and seeing if I get zero.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=1$, I get $g(1)=1-9+16-4=4$, which is not zero.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=-1$, I get $g(-1)=-1-9-16-4=-30$, which is not zero.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=2$, I get $g(2)=8-36+32-4=0$, which is zero.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=-2$, I get $g(-2)=-8-36-32-4=-80$, which is not zero.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=4$, I get $g(4)=64-144+64-4=-20$, which is not zero.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "If I try $x=-4$, I get $g(-4)=-64-144-64-4=-276$, which is not zero.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "So the only rational root of $g(x)$ is $x=2$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "The sum of the rational roots of $g(x)$ is therefore 2.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 69, "metadata": { "path_label": "find_the_sum_of_the_rational_roots_of_g_x_x_3_9x_2_16x_4__2023-02-05T10-18-21.301633|chosen" } }, { "text": "I want to find a number that is both of the form $1 + 6n$ and $4 + 7m$, where $n$ and $m$ are positive integers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "I can rewrite this as $6n - 7m = 3$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "This is a linear diophantine equation, and I can use the Euclidean algorithm to find a solution.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "I start by dividing $6$ by $7$ and getting a quotient of $0$ and a remainder of $6$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "Then I divide $7$ by $6$ and get a quotient of $1$ and a remainder of $1$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "This means that $7 - 1 \\cdot 6 = 1$, which is a solution to $6x - 7y = 1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "To get a solution to $6n - 7m = 3$, I can multiply both sides by $3$ and get $3(7 - 1 \\cdot 6) = 3 \\cdot 1$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 70, "metadata": { "path_label": "an_arithmetic_sequence_with_first_term_1_has_a_common_diff__2023-02-05T10-12-06.793554|chosen" } }, { "text": "I notice that the expression has the form \\[(a+b)^n+(a-b)^n\\]where \\[a=x, b=y+z, n=2006\\].", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I recall that the binomial theorem tells me how to expand \\[(a+b)^n\\]as a sum of terms of the form \\[\\binom{n}{k}a^{n-k}b^k\\]where \\[k=0,1,2,\\dots,n\\].", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I also recall that the binomial coefficients \\[\\binom{n}{k}\\]are symmetric, meaning that \\[\\binom{n}{k}=\\binom{n}{n-k}\\].", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I wonder what happens when I expand \\[(a-b)^n\\]using the same theorem.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I see that I get a similar sum of terms, but with alternating signs: \\[(a-b)^n=\\sum_{k=0}^n (-1)^k \\binom{n}{k}a^{n-k}b^k\\].", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I realize that when I add \\[(a+b)^n+(a-b)^n\\], some of the terms will cancel out, and some will double up.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I think about which terms will cancel out and which will double up.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I observe that the terms with odd values of \\[k\\]will cancel out, because they have opposite signs in the two expansions.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I infer that the terms with even values of \\[k\\]will double up, because they have the same sign in the two expansions.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I deduce that the simplified expression will have only the terms with even values of \\[k\\], and each term will have a coefficient of \\[2\\binom{n}{k}\\].", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I count how many terms there are with even values of \\[k\\].", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I note that \\[k\\]can be any even integer between \\[0\\]and \\[n\\], inclusive.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I estimate that there are about half as many even integers as there are integers between \\[0\\]and \\[n\\].", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I round up to the nearest integer, because \\[n\\]is even and \\[0\\]is included.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I get that there are \\[\\frac{n}{2}+1\\]terms with even values of \\[k\\].", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "I plug in \\[n=2006\\]and get that there are \\[1003+1=1004\\]terms in the simplified expression.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 71, "metadata": { "path_label": "the_expression_x_y_z_2006_x_y_z_2006_is_simplifie__2023-02-05T01-52-43.648919|chosen" } }, { "text": "To find the least common multiple of two numbers, I need to factor them into prime factors and then multiply the highest power of each factor that appears in either number.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 72, "metadata": { "path_label": "compute_the_least_common_multiple_of_9999_and_100_001__2023-02-02T05-10-42.796593|chosen" } }, { "text": "Factoring $9999$ and $100{,}001$ seems daunting, but I notice that they are both close to multiples of $10{,}000$, which is $2^4 \\cdot 5^4$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 72, "metadata": { "path_label": "compute_the_least_common_multiple_of_9999_and_100_001__2023-02-02T05-10-42.796593|chosen" } }, { "text": "Maybe I can use the difference of squares formula to factor them more easily.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 72, "metadata": { "path_label": "compute_the_least_common_multiple_of_9999_and_100_001__2023-02-02T05-10-42.796593|chosen" } }, { "text": "For example, $9999 = 10{,}000 - 1 = (2^4 \\cdot 5^4) - 1^4 = (2^2 \\cdot 5^2 + 1)(2^2 \\cdot 5^2 - 1)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 72, "metadata": { "path_label": "compute_the_least_common_multiple_of_9999_and_100_001__2023-02-02T05-10-42.796593|chosen" } }, { "text": "I notice that the expression for N has a pattern of alternating signs in pairs, starting with a plus sign.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "I also notice that the terms are perfect squares of consecutive integers, starting from 100 and going down to 1.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "This makes me think of using the difference of squares formula, which says that \\[a^2 - b^2 = (a + b)(a - b).\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "I can try to apply this formula to each pair of terms in N, and see if I can simplify the expression.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "For example, the first pair of terms is \\[100^2 + 99^2 - 98^2 - 97^2 = (100^2 - 97^2) + (99^2 - 98^2).\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "Using the difference of squares formula, I can write this as \\[(100 + 97)(100 - 97) + (99 + 98)(99 - 98).\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "This simplifies to \\[197 \\cdot 3 + 197 \\cdot 1 = 197 \\cdot 4.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "Similarly, the next pair of terms is \\[96^2 + 95^2 - 94^2 - 93^2 = (96^2 - 93^2) + (95^2 - 94^2),\\] which simplifies to \\[189 \\cdot 3 + 189 \\cdot 1 = 189 \\cdot 4.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "I notice that each pair of terms simplifies to a multiple of 4, and the multiplier decreases by 8 for each pair.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "This suggests that I can write N as a sum of multiples of 4, with the multiplier starting from 197 and decreasing by 8 until it reaches 5.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "That is, \\[N = 4(197 + 189 + 181 + \\cdots + 13 + 5).\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "Now, I need to find a way to add up the terms in the parentheses.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "I notice that they form an arithmetic sequence, with a common difference of -8.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "To find the sum of an arithmetic sequence, I can use the formula \\[S_n = \\frac{n}{2}(a_1 + a_n),\\] where n is the number of terms, a_1 is the first term, and a_n is the last term.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "In this case, a_1 = 197, a_n = 5, and the common difference is -8, so I can find n by solving \\[a_n = a_1 + (n - 1)d,\\] which gives \\[5 = 197 + (n - 1)(-8).\\]", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 73, "metadata": { "path_label": "compute_the_value_of_n_100_2_99_2_98_2_97_2_96_2__2023-02-02T03-40-23.383406|chosen" } }, { "text": "To find the perfect squares less than 1000, I need to think of the integers whose squares are less than 1000.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 74, "metadata": { "path_label": "how_many_perfect_squares_less_than_1000_have_a_ones_digit_of__2023-02-01T23-57-37.964832|chosen" } }, { "text": "The largest such integer is 31, since 31^2 = 961 and 32^2 = 1024.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 74, "metadata": { "path_label": "how_many_perfect_squares_less_than_1000_have_a_ones_digit_of__2023-02-01T23-57-37.964832|chosen" } }, { "text": "So I only need to consider the integers from 1 to 31 and check their squares.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 74, "metadata": { "path_label": "how_many_perfect_squares_less_than_1000_have_a_ones_digit_of__2023-02-01T23-57-37.964832|chosen" } }, { "text": "This problem involves counting the number of ways to choose marbles from two bags with different numbers of marbles and different ranges of numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "A common strategy for counting problems is to break them down into simpler cases and add up the results.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "In this problem, I can consider each possible choice that Mathew makes from his bag and then see how many ways I can match that choice with two marbles from my bag.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "Mathew has $12$ possible choices, from $1$ to $12.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "If Mathew chooses $1,$ there is no way I can match that with two marbles from my bag, since the smallest sum I can get is $1 + 2 = 3.$ So this case has zero ways.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "If Mathew chooses $2,$ there is also no way I can match that with two marbles from my bag, since the smallest sum I can get is still $3.$ So this case also has zero ways.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 75, "metadata": { "path_label": "i_have_a_bag_with_6_marbles_numbered_from_1_to_6_math__2023-02-01T23-40-32.416844|chosen" } }, { "text": "To find the values of $x$ that make $f(f(x)) = 3$, I need to first find the values of $f(x)$ that make $f(y) = 3$, where $y = f(x)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "That means I need to solve the equation $f(y) = 3$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "I notice that $f(y)$ has two different expressions depending on whether $y \\ge 3$ or $y < 3$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to consider both cases and find the values of $y$ that satisfy the equation in each case.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y \\ge 3$, then $f(y) = 2y - 5$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to solve $2y - 5 = 3$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Adding $5$ to both sides, I get $2y = 8$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Dividing by $2$, I get $y = 4$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So one value of $y$ that makes $f(y) = 3$ is $y = 4$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y < 3$, then $f(y) = -y + 5$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to solve $-y + 5 = 3$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Subtracting $5$ from both sides, I get $-y = -2$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Multiplying by $-1$, I get $y = 2$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So another value of $y$ that makes $f(y) = 3$ is $y = 2$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Now I have two values of $y$ that make $f(y) = 3$: $y = 4$ and $y = 2$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "But I need to find the values of $x$ that make $y = f(x)$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to plug in $y = 4$ and $y = 2$ into the definition of $f(x)$ and solve for $x$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Again, I need to consider both cases depending on whether $x \\ge 3$ or $x < 3$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $x \\ge 3$, then $f(x) = 2x - 5$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to solve $2x - 5 = y$, where $y = 4$ or $y = 2$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y = 4$, then $2x - 5 = 4$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Adding $5$ to both sides, I get $2x = 9$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Dividing by $2$, I get $x = 4.5$.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So one value of $x$ that makes $f(f(x)) = 3$ is $x = 4.5$.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y = 2$, then $2x - 5 = 2$.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Adding $5$ to both sides, I get $2x = 7$.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Dividing by $2$, I get $x = 3.5$.", "role": "entry", "step_index": 26, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So another value of $x$ that makes $f(f(x)) = 3$ is $x = 3.5$.", "role": "entry", "step_index": 27, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $x < 3$, then $f(x) = -x + 5$.", "role": "entry", "step_index": 28, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I need to solve $-x + 5 = y$, where $y = 4$ or $y = 2$.", "role": "entry", "step_index": 29, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y = 4$, then $-x + 5 = 4$.", "role": "entry", "step_index": 30, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Subtracting $5$ from both sides, I get $-x = -1$.", "role": "entry", "step_index": 31, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Multiplying by $-1$, I get $x = 1$.", "role": "entry", "step_index": 32, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So another value of $x$ that makes $f(f(x)) = 3$ is $x = 1$.", "role": "entry", "step_index": 33, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "If $y = 2$, then $-x + 5 = 2$.", "role": "entry", "step_index": 34, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Subtracting $5$ from both sides, I get $-x = -3$.", "role": "entry", "step_index": 35, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Multiplying by $-1$, I get $x = 3$.", "role": "entry", "step_index": 36, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "But wait, this value of $x$ does not satisfy the condition that $x < 3$.", "role": "entry", "step_index": 37, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So I can ignore this value of $x$.", "role": "entry", "step_index": 38, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "Therefore, I have found three values of $x$ that make $f(f(x)) = 3$: $x = 4.5$, $x = 3.5$, and $x = 1$.", "role": "entry", "step_index": 39, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "So the answer is 3.", "role": "entry", "step_index": 40, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 76, "metadata": { "path_label": "if_f_x_begin_cases_2x_5_quad_text_if_x_ge_3__2023-02-01T23-38-52.643498|chosen" } }, { "text": "To find the greatest common divisor (gcd) of two integers, I need to factor them into their prime factors and then look for the common ones.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 77, "metadata": { "path_label": "when_the_greatest_common_divisor_and_least_common_multiple_o__2023-02-01T22-40-07.885767|chosen" } }, { "text": "The least common multiple (lcm) of two integers is the product of all their prime factors, with the highest power of each factor.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 77, "metadata": { "path_label": "when_the_greatest_common_divisor_and_least_common_multiple_o__2023-02-01T22-40-07.885767|chosen" } }, { "text": "So the product of the gcd and the lcm is just the product of all the prime factors of both integers, with no repeated factors.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 77, "metadata": { "path_label": "when_the_greatest_common_divisor_and_least_common_multiple_o__2023-02-01T22-40-07.885767|chosen" } }, { "text": "That means that 200 must be the product of all the prime factors of both integers, with no repeated factors.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 77, "metadata": { "path_label": "when_the_greatest_common_divisor_and_least_common_multiple_o__2023-02-01T22-40-07.885767|chosen" } }, { "text": "I can factor 200 into its prime factors as 2^3 * 5^2.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 77, "metadata": { "path_label": "when_the_greatest_common_divisor_and_least_common_multiple_o__2023-02-01T22-40-07.885767|chosen" } }, { "text": "I notice that the problem involves the floor function, which rounds any real number down to the nearest integer, and the fractional part function, which gives the decimal part of any real number.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "I also notice that the problem says that the three numbers form an arithmetic sequence, which means that there is a common difference between consecutive terms.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "I can use this fact to write an equation involving the common difference. Let $d$ be the common difference. Then I have $\\lfloor x\\rfloor - \\{x\\} = d$ and $x - \\lfloor x\\rfloor = d.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "I can simplify the second equation by using the definition of $\\{x\\}.$ I get $x - \\lfloor x\\rfloor = \\{x\\} = d.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "Now I have two equations involving $d,$ so I can set them equal to each other and solve for $x.$ I get $\\lfloor x\\rfloor - \\{x\\} = \\{x\\}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "I can rearrange this equation to get $2\\{x\\} = \\lfloor x\\rfloor.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "This means that the fractional part of $x$ is exactly half of the integer part of $x.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 78, "metadata": { "path_label": "find_x_given_that_x_is_nonzero_and_the_numbers_x__2023-02-01T22-33-51.077784|chosen" } }, { "text": "This problem looks like it involves the distance formula, since each term under the square root is the square of the difference between two coordinates.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 79, "metadata": { "path_label": "find_the_minimum_value_of_sqrt_x_2_1_x_2_sqrt__2023-02-01T22-20-54.246542|chosen" } }, { "text": "Maybe I can interpret the expression as the sum of the distances from some point $(x, 0)$ to two other fixed points.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 79, "metadata": { "path_label": "find_the_minimum_value_of_sqrt_x_2_1_x_2_sqrt__2023-02-01T22-20-54.246542|chosen" } }, { "text": "Let me try to find those points by equating the terms under the square roots to the squares of some distances.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 79, "metadata": { "path_label": "find_the_minimum_value_of_sqrt_x_2_1_x_2_sqrt__2023-02-01T22-20-54.246542|chosen" } }, { "text": "For the first term, I have $x^2 + (1 - x)^2 = d_1^2,$ where $d_1$ is the distance from $(x, 0)$ to some point $(a, b).$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 79, "metadata": { "path_label": "find_the_minimum_value_of_sqrt_x_2_1_x_2_sqrt__2023-02-01T22-20-54.246542|chosen" } }, { "text": "Expanding and simplifying, I get $x^2 + 1 - 2x + x^2 = d_1^2,$ or $2x^2 - 2x + 1 = d_1^2.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 79, "metadata": { "path_label": "find_the_minimum_value_of_sqrt_x_2_1_x_2_sqrt__2023-02-01T22-20-54.246542|chosen" } }, { "text": "This problem is asking me to find all the values of $\\alpha$ that make the given rational function reducible, meaning it can be factored into two linear factors in the numerator and the denominator.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "To do that, I need to look at the numerator and the denominator separately and see when they have a common factor.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "I notice that the denominator is already factored: $x^2 + 7x - 44 = (x + 11)(x - 4)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "So, the denominator has two roots: $x = -11$ and $x = 4$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "These are the only possible values of $x$ that can make the rational function undefined, so they are also the only possible values of $x$ that can make the rational function reducible.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "Therefore, I need to find the values of $\\alpha$ that make the numerator zero when $x = -11$ or $x = 4$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "I can do this by plugging in these values of $x$ into the numerator and setting it equal to zero.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 80, "metadata": { "path_label": "let_s_be_the_set_of_all_real_numbers_alpha_such_that_th__2023-02-01T19-36-31.437770|chosen" } }, { "text": "To solve this problem, I need to find the value of $k$ that makes the system have a nonzero solution.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "I notice that the coefficient of $y$ is the same in the first two equations, so I can try to eliminate $y$ by subtracting them.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "If I do that, I get $-2x + 5z = 0,$ or $x = \\frac{5}{2} z.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "Now I can substitute this expression for $x$ into any of the original equations to get rid of one more variable.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "I choose the first equation, since it has the simplest coefficients.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "Substituting $x = \\frac{5}{2} z,$ I get $\\frac{5}{2} z + ky + 3z = 0,$ or $(\\frac{5}{2} + 3)z + ky = 0.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "Simplifying, I get $\\frac{11}{2} z + ky = 0.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "Now I have an equation that relates $y$ and $z.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "To find $k,$ I need to use the fact that $x,$ $y,$ and $z$ are all nonzero.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "That means I cannot have $z = 0,$ since that would make $x = 0$ as well.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "So I can divide both sides of the equation by $z,$ and get $k = -\\frac{11}{2} \\frac{z}{y}.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "This is the value of $k$ that makes the system have a nonzero solution.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "To find $\\frac{xz}{y^2},$ I can use the expressions I found earlier for $x$ and $k$ in terms of $z$ and $y.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "Substituting, I get $\\frac{xz}{y^2} = \\frac{\\frac{5}{2} z \\cdot z}{y^2} = \\frac{5}{2} \\frac{z^2}{y^2} = \\frac{5}{2} \\frac{z}{y} \\cdot \\frac{z}{y}.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "But I also know that $k = -\\frac{11}{2} \\frac{z}{y},$ so I can write $\\frac{xz}{y^2} = -\\frac{5}{11} k.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 81, "metadata": { "path_label": "for_a_certain_value_of_k_the_system_begin_align_x_ky__2023-02-01T12-29-51.230745|chosen" } }, { "text": "I know that compound interest means that the interest earned in each period is added to the principal, and then the next period's interest is calculated based on the new principal.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "So, for Jose's investment, I can use the formula $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate per period, and $t$ is the number of periods.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "In Jose's case, the interest rate per period is $4$ percent divided by $1$, since there is only one compounding period per year. That's $0.04$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "The number of periods is $2$, since he invested for $2$ years.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "So, plugging these values into the formula, I get $A = 50,\\!000(1 + 0.04)^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "I can simplify this by multiplying out the brackets: $A = 50,\\!000(1.04)^2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 82, "metadata": { "path_label": "jose_invested_50_000_for_2_years_at_an_annual_intere__2023-02-01T12-05-12.229016|chosen" } }, { "text": "I notice that each term in the product has the form \\[\\dfrac{n^3-1}{n^3+1},\\] where n is a natural number from 2 to 6.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 83, "metadata": { "path_label": "compute_dfrac_2_3_1_2_3_1_cdot_dfrac_3_3_1_3_3_1_cdot__2023-02-01T11-32-04.021143|chosen" } }, { "text": "I also recall that \\[n^3-1=(n-1)(n^2+n+1)\\] and \\[n^3+1=(n+1)(n^2-n+1)\\] by using the difference and sum of cubes formulas.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 83, "metadata": { "path_label": "compute_dfrac_2_3_1_2_3_1_cdot_dfrac_3_3_1_3_3_1_cdot__2023-02-01T11-32-04.021143|chosen" } }, { "text": "So, I can rewrite each term as \\[\\dfrac{(n-1)(n^2+n+1)}{(n+1)(n^2-n+1)}.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 83, "metadata": { "path_label": "compute_dfrac_2_3_1_2_3_1_cdot_dfrac_3_3_1_3_3_1_cdot__2023-02-01T11-32-04.021143|chosen" } }, { "text": "Now, I wonder if I can simplify the product by canceling out some factors.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 83, "metadata": { "path_label": "compute_dfrac_2_3_1_2_3_1_cdot_dfrac_3_3_1_3_3_1_cdot__2023-02-01T11-32-04.021143|chosen" } }, { "text": "I want to find the number of times the graphs of the two functions $y = \\sin x$ and $y = \\left( \\frac{1}{2} \\right)^x$ intersect on the given interval.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "One way to do this is to sketch the graphs and count the intersections visually, but that might be tedious and prone to error.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "Another way is to use some algebra and calculus to find the exact points of intersection, but that might be even more tedious and difficult.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "I will try a third way, which is to use some intuition and estimation to find an approximate answer.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "I notice that the function $y = \\left( \\frac{1}{2} \\right)^x$ is always positive and decreasing, while the function $y = \\sin x$ oscillates between -1 and 1 with a period of $2 \\pi$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "This means that the two functions can only intersect when $0 < \\sin x < 1$, which happens on the intervals $(2 \\pi k, 2 \\pi k + \\pi)$ for some integer $k$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "There are 50 such intervals on the given domain, since $2 \\pi \\cdot 50 = 100 \\pi$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "Now, on each interval, I want to estimate how many times the two functions intersect.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "I notice that the function $y = \\left( \\frac{1}{2} \\right)^x$ is very close to 0 when $x$ is large, and it approaches 1 when $x$ approaches 0.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 84, "metadata": { "path_label": "find_the_number_of_solutions_to_sin_x_left_frac_1_2__2023-02-01T10-18-22.421790|chosen" } }, { "text": "I want to solve for $x$, so I need to isolate it on one side of the equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "I could try to expand the products on both sides and simplify, but that might be messy and tedious.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "I notice that the equation has a quadratic form, so I could try to use the quadratic formula to find the solutions.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "To do that, I need to bring everything to one side and write it as $ax^2+bx+c=0$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "So I subtract $x(2x+7)$ from both sides and get $(4x+11)(2x-8)-x(2x+7)=0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "Now I need to expand the products and combine like terms.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 85, "metadata": { "path_label": "what_is_the_largest_value_of_x_such_that_4x_11_2x_8_x__2023-02-01T03-35-26.400308|chosen" } }, { "text": "To find the shortest distance, I need to minimize the square of the distance, since squaring preserves the order of positive numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "The square of the distance between $(6,12)$ and a point $(x,y)$ on the parabola is $(x-6)^2 + (y-12)^2$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "To minimize this expression, I can use calculus and find the critical points where the derivative is zero or undefined.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "I can also use the fact that $x = \\frac{y^2}{2}$ to substitute for $x$ and simplify the expression.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "So I have $(\\frac{y^2}{2} - 6)^2 + (y-12)^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "Expanding the first term, I get $\\frac{y^4}{4} - 6y^2 + 36 + (y-12)^2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "Taking the derivative with respect to $y$, I get $y^3 - 12y + 2y - 24 = y^3 - 10y - 24$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "To find the critical points, I need to solve $y^3 - 10y - 24 = 0$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "This is a cubic equation, which can be hard to solve in general.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "But I notice that $y = 4$ is a solution, since $4^3 - 10 \\cdot 4 - 24 = 0$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "So I can factor out $(y-4)$ and use polynomial division to find the other factors.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "I get $(y-4)(y^2 + 4y + 6) = 0$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "The quadratic factor has no real solutions, since its discriminant is negative.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "So the only real critical point is $y = 4$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "To check if this is a minimum, a maximum, or neither, I can use the second derivative test.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "The second derivative is $3y^2 - 10$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "Plugging in $y = 4$, I get $3 \\cdot 4^2 - 10 = 38$, which is positive.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "This means that the critical point is a local minimum.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "Since there is only one critical point, it must also be the global minimum.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "So the shortest distance occurs when $y = 4$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "To find the corresponding value of $x$, I can use the equation of the parabola.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "I get $x = \\frac{4^2}{2} = 8$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "So the point on the parabola that is closest to $(6,12)$ is $(8,4)$. [* from the grader: You should also find the value of the shortest distance, not just the point where it occurs.]", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 86, "metadata": { "path_label": "find_the_shortest_distance_between_the_point_6_12_and_th__2023-02-01T03-25-50.092754|chosen" } }, { "text": "I want to find the volume of $S,$ which means I need to integrate some function over the region defined by the inequalities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "I notice that the inequalities involve absolute values of $x,y,$ and $z,$ which means that $S$ is symmetric with respect to the coordinate planes.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "That is, if $(x,y,z)$ is in $S,$ then so are $(-x,y,z),$ $(x,-y,z),$ and $(x,y,-z),$ and any other combination of changing signs.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "This suggests that I can focus on one octant of $S,$ say the one where $x,y,$ and $z$ are all nonnegative, and then multiply the volume by $8$ to get the total volume.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "So I will assume that $x,y,z \\ge 0$ and rewrite the inequalities as $x + y \\le 1,$ $x + z \\le 1,$ and $y + z \\le 1.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "Now I need to find the limits of integration for each variable.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 87, "metadata": { "path_label": "the_solid_s_consists_of_the_set_of_all_points_x_y_z_su__2023-02-02T23-16-44.586612|chosen" } }, { "text": "I notice that the eight-digit integers of this form are multiples of the four-digit integers that are repeated.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "So, if I divide any such eight-digit integer by the corresponding four-digit integer, I get a quotient of 10,001.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "This means that 10,001 is a common divisor of all eight-digit integers of this form.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "But is it the greatest common divisor?", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "I wonder if 10,001 has any non-trivial factors.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "I try some small prime factors, such as 2, 3, 5, and 7, and find that none of them divide 10,001 evenly.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "I also notice that 10,001 is not a perfect square, since it is between 100^2 = 10,000 and 101^2 = 10,201.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 88, "metadata": { "path_label": "an_eight_digit_integer_is_formed_by_repeating_a_positive_fou__2023-01-31T22-30-09.176260|chosen" } }, { "text": "This problem involves finding the maximum value of a fraction, so I want to think about how the numerator and denominator relate to each other.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "I notice that the numerator is a sum of products of adjacent variables, and the denominator is a sum of squares of all the variables.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "This makes me think of the Cauchy-Schwarz inequality, which says that for any real numbers $a_1, a_2, \\dots, a_n$ and $b_1, b_2, \\dots, b_n,$ we have \\[(a_1^2 + a_2^2 + \\cdots + a_n^2)(b_1^2 + b_2^2 + \\cdots + b_n^2) \\geq (a_1b_1 + a_2b_2 + \\cdots + a_nb_n)^2.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "I wonder if I can apply this inequality to the given fraction, by choosing appropriate values for the $a_i$ and $b_i.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "I see that if I let $a_1 = w, a_2 = x, a_3 = y, a_4 = z$ and $b_1 = x, b_2 = y, b_3 = z, b_4 = 0,$ then the inequality becomes \\[(w^2 + x^2 + y^2 + z^2)(x^2 + y^2 + z^2) \\geq (wx + xy + yz)^2,\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "which is exactly what I need to bound the fraction from above.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "If I divide both sides by $(w^2 + x^2 + y^2 + z^2)^2,$ I get \\[\\frac{x^2 + y^2 + z^2}{w^2 + x^2 + y^2 + z^2} \\geq \\frac{(wx + xy + yz)^2}{(w^2 + x^2 + y^2 + z^2)^2},\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "and taking the square root of both sides, I get \\[\\sqrt{\\frac{x^2 + y^2 + z^2}{w^2 + x^2 + y^2 + z^2}} \\geq \\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2},\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 89, "metadata": { "path_label": "let_w_x_y_and_z_be_positive_real_numbers_find__2023-01-31T08-53-33.474511|chosen" } }, { "text": "To find the number of polynomials of the form $x^2 + mx + n$ with positive integer roots and $n = 28$, I need to think of all the ways I can factor 28 into two positive integers, since those will be the roots of the polynomial.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 90, "metadata": { "path_label": "how_many_polynomials_of_the_form_x_2_mx_n_are_there_wi__2023-01-31T05-31-38.641098|chosen" } }, { "text": "One way to do this is to list all the factors of 28 and pair them up: 1 and 28, 2 and 14, 4 and 7.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 90, "metadata": { "path_label": "how_many_polynomials_of_the_form_x_2_mx_n_are_there_wi__2023-01-31T05-31-38.641098|chosen" } }, { "text": "These are the only pairs of positive integers that multiply to give 28, so there are no other ways to factor 28. [*. So, the possible polynomials are $x^2 + 29x + 28$, $x^2 + 16x + 28$, and $x^2 + 11x + 28$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 90, "metadata": { "path_label": "how_many_polynomials_of_the_form_x_2_mx_n_are_there_wi__2023-01-31T05-31-38.641098|chosen" } }, { "text": "These are the only three polynomials that meet the given criteria, so the answer is 3.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 90, "metadata": { "path_label": "how_many_polynomials_of_the_form_x_2_mx_n_are_there_wi__2023-01-31T05-31-38.641098|chosen" } }, { "text": "I need to find the area of the region where $P$ is closer to the origin than to $(3,1)$, and divide it by the area of the whole rectangle, which is $2\\times 1=2$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "To find the boundary of this region, I can use the distance formula and set it equal for the origin and $(3,1)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "That is, $\\sqrt{x^2+y^2}=\\sqrt{(x-3)^2+(y-1)^2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "Squaring both sides and expanding, I get $x^2+y^2=x^2-6x+9+y^2-2y+1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "Simplifying, I get $6x+2y=10$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "This is a line that passes through $(\\frac{5}{3},0)$ and $(0,5)$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "I can draw this line on the rectangle and see that it divides it into two regions: one where $P$ is closer to the origin, and one where $P$ is closer to $(3,1)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 91, "metadata": { "path_label": "a_point_p_is_randomly_selected_from_the_rectangular_region__2023-01-31T05-18-31.867854|chosen" } }, { "text": "This is a parametric equation for a curve in the $xy$-plane, where $t$ is the parameter.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "To find where the curve crosses itself, I need to find two distinct values of $t$, say $t_1$ and $t_2$, such that $(x(t_1),y(t_1)) = (x(t_2),y(t_2))$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "That is, I need to solve the system of equations \\begin{align*} t_1^2 - 2 &= t_2^2 - 2, \\\\ t_1^3 - 9t_1 + 5 &= t_2^3 - 9t_2 + 5. \\end{align*}", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "The first equation simplifies to $t_1^2 = t_2^2$, which implies that $t_1 = \\pm t_2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "I can plug this into the second equation and get two cases: either $t_1 = t_2$, which means the curve is tangent to itself at that point, or $t_1 = -t_2$, which means the curve is symmetric about the origin.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "I want to find the latter case, where the curve actually crosses itself.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "So I assume that $t_1 = -t_2$ and substitute it into the second equation.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 92, "metadata": { "path_label": "as_t_takes_on_all_real_values_the_set_of_points_x_y_d__2023-01-30T17-27-17.082846|chosen" } }, { "text": "I notice that the given values of the polynomial are all multiples of 17, and they increase by 17 as the input increases by 1. This suggests that $p(x) - 17x$ might be a simpler polynomial that has some nice properties.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 93, "metadata": { "path_label": "let_p_x_be_a_monic_polynomial_of_degree_4_such_that_p_1__2023-01-30T16-03-08.697462|chosen" } }, { "text": "In fact, I can see that $p(x) - 17x$ must have 1, 2, and 3 as roots, since $p(1) - 17(1) = 0,$ $p(2) - 17(2) = 0,$ and $p(3) - 17(3) = 0.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 93, "metadata": { "path_label": "let_p_x_be_a_monic_polynomial_of_degree_4_such_that_p_1__2023-01-30T16-03-08.697462|chosen" } }, { "text": "Since $p(x) - 17x$ is also a monic polynomial of degree 4, it must have the form $p(x) - 17x = (x-1)(x-2)(x-3)(x-r),$ where $r$ is some unknown root.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 93, "metadata": { "path_label": "let_p_x_be_a_monic_polynomial_of_degree_4_such_that_p_1__2023-01-30T16-03-08.697462|chosen" } }, { "text": "To find $r,$ I can plug in any other value of $x$ and use the fact that $p(x) - 17x$ is zero. For example, let me try $x = 0.$ Then I get $p(0) - 17(0) = (-1)(-2)(-3)(-r),$ which simplifies to $p(0) = 6r.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 93, "metadata": { "path_label": "let_p_x_be_a_monic_polynomial_of_degree_4_such_that_p_1__2023-01-30T16-03-08.697462|chosen" } }, { "text": "I notice that the equation $a^{-1} \\equiv a \\pmod{24}$ is equivalent to saying that $a^2 \\equiv 1 \\pmod{24}$, since I can multiply both sides by $a$ without changing the congruence.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "So I want to find all the values of $a$ between 1 and 23 such that $a^2$ leaves a remainder of 1 when divided by 24.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "I can try some small values of $a$ and see if they work.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "For example, $a = 1$ works, since $1^2 = 1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "$a = 2$ does not work, since $2^2 = 4$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "$a = 3$ does not work, since $3^2 = 9$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "$a = 4$ does not work, since $4^2 = 16$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 94, "metadata": { "path_label": "for_how_many_integers_a_satisfying_1_le_a_le_23_is_it__2023-01-30T08-00-47.753612|chosen" } }, { "text": "I notice that the numerator and denominator of the fraction have a common factor of $(x - 1)^5$, so I can simplify the fraction by dividing both by that factor.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "This gives me \\[\\frac{(x - 1)^7 + 3(x - 1)^6 + (x - 1)^5 + 1}{(x - 1)^5} = (x - 1)^2 + 3(x - 1) + 1 + \\frac{1}{(x - 1)^5}.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "Now I want to find the minimum value of this expression for $x > 1.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "I could try to use calculus and find the critical points where the derivative is zero or undefined, but that might be tedious and messy.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "Instead, I notice that the expression is a sum of four terms, each of which is either increasing or decreasing as $x$ increases.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "Specifically, $(x - 1)^2$ is increasing, $3(x - 1)$ is increasing, $1$ is constant, and $\\frac{1}{(x - 1)^5}$ is decreasing as $x$ increases.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 95, "metadata": { "path_label": "find_the_minimum_value_of_frac_x_1_7_3_x_1_6__2023-01-24T16-32-29.296341|chosen" } }, { "text": "I need to find the fraction of the unit interval that satisfies the given conditions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "I can use geometry to visualize the problem.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "I can draw a square with vertices at (0,0), (1,0), (1,1), and (0,1), where the x-axis represents the green point and the y-axis represents the purple point.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "The diagonal line y = x divides the square into two equal halves, where the points below the line have y < x, and the points above the line have y > x.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "The points below the line do not satisfy the condition that the purple point is greater than the green point, so I can ignore that half of the square.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "The points above the line do satisfy the condition that the purple point is greater than the green point, but I need to check the second condition, that the purple point is less than twice the green point.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "That means I need to find the line y = 2x and see where it intersects the square.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "The line y = 2x passes through the origin and the point (1/2, 1), so it cuts off a triangle from the upper half of the square.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "The points in that triangle do not satisfy the second condition, because they have y > 2x.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 96, "metadata": { "path_label": "a_point_is_chosen_at_random_on_the_number_line_between_0_and__2023-01-24T15-32-43.295956|chosen" } }, { "text": "I need to find the smallest $t$ such that $\\cos t = \\cos \\frac{t \\pi}{180}.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 97, "metadata": { "path_label": "let_t_be_the_smallest_positive_real_number_for_which_cos__2023-01-24T07-37-58.809630|chosen" } }, { "text": "I can use the fact that $\\cos x = \\cos y$ if and only if $x = \\pm y + 2k \\pi$ for some integer $k.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 97, "metadata": { "path_label": "let_t_be_the_smallest_positive_real_number_for_which_cos__2023-01-24T07-37-58.809630|chosen" } }, { "text": "So I need to find the smallest $t$ such that $t = \\pm \\frac{t \\pi}{180} + 2k \\pi$ for some integer $k.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 97, "metadata": { "path_label": "let_t_be_the_smallest_positive_real_number_for_which_cos__2023-01-24T07-37-58.809630|chosen" } }, { "text": "I can simplify this equation by dividing both sides by $\\pi.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 97, "metadata": { "path_label": "let_t_be_the_smallest_positive_real_number_for_which_cos__2023-01-24T07-37-58.809630|chosen" } }, { "text": "To find the probability of an event, I need to multiply the probabilities of each individual outcome that makes up the event, and then add the probabilities of all the different ways the event can happen.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "So first, I need to identify the individual outcomes and their probabilities.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "The probability of spinning an arm is $\\frac{1}{2}$, since there are two equally likely options: arm or leg.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "The probability of spinning a red color is $\\frac{1}{4}$, since there are four equally likely colors: red, blue, green, and yellow.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "The probability of spinning an arm and a red color is $\\frac{1}{2} \\times \\frac{1}{4} = \\frac{1}{8}$, by the multiplication rule.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "Similarly, the probability of spinning a leg and a red color is $\\frac{1}{2} \\times \\frac{1}{4} = \\frac{1}{8}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "The probability of spinning an arm and a non-red color is $\\frac{1}{2} \\times \\frac{3}{4} = \\frac{3}{8}$, since there are three non-red colors out of four.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "And the probability of spinning a leg and a non-red color is $\\frac{1}{2} \\times \\frac{3}{4} = \\frac{3}{8}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "Now, I need to figure out how many different ways the event can happen.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "The event is that there are exactly two moves to a red spot, and the body part selected to move is an arm exactly $3$ times.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 98, "metadata": { "path_label": "in_the_game_of_twister_a_spinner_randomly_selects_either_an__2023-01-24T06-56-14.201402|chosen" } }, { "text": "This problem involves addition of three-digit numbers in base $6$, which is similar to base $10$ addition but with different digits and place values.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "I can use the standard algorithm for adding numbers in base $6$, starting from the rightmost digits and carrying over when the sum exceeds $5$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "First, I add the units digits: $C + A + B = 0$ (in base $6$). This means that $C + A + B$ must be a multiple of $6$, since $0$ is the only digit that is congruent to $0$ modulo $6$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "I can use trial and error to find values of $A,B$ and $C$ that satisfy this equation. For example, if $A = 1$, then $B + C = 5$ or $11$ or $17$ or $23$ or $29$ or $35$ (in base $10$). The only pair of distinct digits in base $6$ that add up to these numbers are $B = 2$ and $C = 3$ or vice versa.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "If I try these values, I get $\\overline{123}_6 + \\overline{231}_6 + \\overline{312}_6 = \\overline{1110}_6$, which is indeed true.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "Alternatively, I could have noticed that $A = 1$ implies that $C + A + B = 6$, since $A0$ is a multiple of $6$. Then I could have used the fact that $B + C = 6 - A = 5$ (in base $10$) to find $B$ and $C$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "In any case, I have found one possible solution for $A,B$ and $C$. There might be other solutions, but the problem only asks for $B + C$ in base $6$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "To convert $B + C = 5$ from base $10$ to base $6$, I can use repeated division by $6$ and write the remainders as digits. I get $5 = 6 \\cdot 0 + 5$, so $B + C = 5$ in base $6$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 99, "metadata": { "path_label": "if_a_b_and_c_are_non_zero_distinct_digits_in_base_6_su__2023-01-24T06-18-12.768285|chosen" } }, { "text": "To find the volume of the sphere, I need its radius.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 100, "metadata": { "path_label": "a_right_circular_cone_sits_on_a_table_pointing_up_the_cros__2023-01-24T03-41-25.183566|chosen" } }, { "text": "I can draw a right triangle inside the cone, with one leg as the radius of the cone's base, the other leg as the height of the cone, and the hypotenuse as the slant height of the cone.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 100, "metadata": { "path_label": "a_right_circular_cone_sits_on_a_table_pointing_up_the_cros__2023-01-24T03-41-25.183566|chosen" } }, { "text": "The radius of the base is half of the diameter, so it is $6\\sqrt{3}$ inches.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 100, "metadata": { "path_label": "a_right_circular_cone_sits_on_a_table_pointing_up_the_cros__2023-01-24T03-41-25.183566|chosen" } }, { "text": "To find the domain of $f_n,$ I need to consider the restrictions on the argument of the square root function.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 101, "metadata": { "path_label": "let_f__1_x_sqrt_1_x_and_for_integers_n_geq_2_let__2023-01-23T23-56-33.453706|chosen" } }, { "text": "The argument of $f_1$ is $1-x,$ so $f_1$ is only defined for $x \\leq 1.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 101, "metadata": { "path_label": "let_f__1_x_sqrt_1_x_and_for_integers_n_geq_2_let__2023-01-23T23-56-33.453706|chosen" } }, { "text": "The argument of $f_2$ is $\\sqrt{4-x},$ so $f_2$ is only defined for $x \\leq 4$ and $\\sqrt{4-x} \\leq 1.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 101, "metadata": { "path_label": "let_f__1_x_sqrt_1_x_and_for_integers_n_geq_2_let__2023-01-23T23-56-33.453706|chosen" } }, { "text": "The latter inequality implies that $x \\geq 3,$ so the domain of $f_2$ is $[3,4].$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 101, "metadata": { "path_label": "let_f__1_x_sqrt_1_x_and_for_integers_n_geq_2_let__2023-01-23T23-56-33.453706|chosen" } }, { "text": "I notice that this problem is asking me to find a congruence relation modulo 260, which means I am working with the remainders when dividing by 260.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "I also notice that 55 and 260 are not relatively prime, since they have a common factor of 5.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "This means that there might not be a unique solution to the congruence, but there could be multiple solutions that differ by multiples of 260/5 = 52.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "To simplify the congruence, I can divide both sides by 5, as long as I also divide the modulus by 5.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "This gives me $$11n\\equiv 33\\pmod{52}~.$$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Now I need to find the inverse of 11 modulo 52, which is a number $a$ such that $$11a\\equiv 1\\pmod{52}~.$$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "One way to do this is to use the extended Euclidean algorithm, which finds integers $x$ and $y$ such that $$11x+52y=1~.$$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Then $x$ will be the inverse of 11 modulo 52.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "To apply the algorithm, I start by writing 52 as a multiple of 11 plus a remainder: $$52=4\\cdot 11+8~.$$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Then I write 11 as a multiple of 8 plus a remainder: $$11=1\\cdot 8+3~.$$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Then I write 8 as a multiple of 3 plus a remainder: $$8=2\\cdot 3+2~.$$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Then I write 3 as a multiple of 2 plus a remainder: $$3=1\\cdot 2+1~.$$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Then I write 2 as a multiple of 1 plus a remainder: $$2=2\\cdot 1+0~.$$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "I stop here because I have reached a remainder of zero.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Now I work backwards from the last equation and express 1 as a linear combination of 11 and 52.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "From the last equation, I see that 1=3-1\\cdot 2.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Substituting 2=8-2\\cdot 3, I get 1=3-(8-2\\cdot 3)=3\\cdot 3-8.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Substituting 8=52-4\\cdot 11, I get 1=3\\cdot 3-(52-4\\cdot 11)=3\\cdot 3+4\\cdot 11-52.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Substituting 3=11-1\\cdot 8, I get 1=(11-1\\cdot 8)\\cdot 3+4\\cdot 11-52=11\\cdot 3-3\\cdot 8+4\\cdot 11-52=11\\cdot 7-3\\cdot (52-4\\cdot 11)-52=11\\cdot 7-3\\cdot 52+12\\cdot 11-52=11\\cdot 19-4\\cdot 52.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "Therefore, I have found that $$11\\cdot 19+52\\cdot (-4)=1~.$$", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "This means that the inverse of 11 modulo 52 is 19.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "To solve the original congruence, I can multiply both sides by 19, which gives me $$n\\equiv 33\\cdot 19\\pmod{52}~.$$", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 102, "metadata": { "path_label": "what_is_the_largest_three_digit_integer_n_that_satisfies__2023-01-23T23-51-24.680223|chosen" } }, { "text": "I need to find out how many people drank coffee and milk that morning.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Let me start by assigning some variables to the amounts of coffee and milk in each person's cup.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Let C be the total amount of coffee in ounces, and M be the total amount of milk in ounces.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Then the total amount of liquid in ounces is C + M.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Since each person drank 8 ounces, I can divide the total by 8 to get the number of people: (C + M) / 8.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Now I need to use the information about Angela's cup.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "She drank a quarter of the total milk, so her cup had M / 4 ounces of milk.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "She also drank a sixth of the total coffee, so her cup had C / 6 ounces of coffee.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Adding these together, I get that Angela's cup had (M / 4) + (C / 6) ounces of liquid.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Since this is also 8 ounces, I can set up an equation: (M / 4) + (C / 6) = 8.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "I want to solve for (C + M) / 8, so I need to eliminate the fractions.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "To do that, I can multiply both sides by the least common multiple of 4 and 6, which is 12: 3M + 2C = 96.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Now I can solve for C in terms of M: C = (96 - 3M) / 2.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Plugging this into (C + M) / 8, I get: ((96 - 3M) / 2 + M) / 8.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 103, "metadata": { "path_label": "one_morning_each_member_of_angela_s_family_drank_an_8_ounce__2023-01-23T23-41-08.970764|chosen" } }, { "text": "Synthetic division is a way to divide a polynomial by a linear factor of the form $x-a$ by using only the coefficients of the polynomial and the value of $a$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "To divide by $x^2-3x+2$, I need to factor it into two linear factors: $(x-1)(x-2)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "This means I can use synthetic division twice, once for each factor.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "First, I will divide by $x-2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "To do that, I write down the coefficients of the dividend: $1, -3, 4, 11, -9$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Then I write down the value of $a$ for $x-a$, which is $2$ in this case, and draw a vertical bar and a horizontal bar to separate the coefficients and the result.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Next, I bring down the first coefficient, which is $1$, and write it under the horizontal bar.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Then I multiply it by $a$, which is $2$, and write the product above the second coefficient.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "I get $1 \\cdot 2 = 2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Then I add the second coefficient and the product, and write the sum under the horizontal bar.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "I get $-3 + 2 = -1$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "I repeat this process for the remaining coefficients: multiply by $a$, write the product above, and add to the coefficient below.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "I get $-1 \\cdot 2 = -2$, then $4 + (-2) = 2$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Then $2 \\cdot 2 = 4$, then $11 + 4 = 15$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "Finally, $15 \\cdot 2 = 30$, then $-9 + 30 = 21$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "The last number I get under the horizontal bar is the remainder of the division.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "So the remainder is $21$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "But this is not the final answer, because I still need to divide by $x-1$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "To do that, I use the result of the first division as the new dividend, and $a = 1$ as the new divisor.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 104, "metadata": { "path_label": "use_synthetic_division_to_divide_x_4_3x_3_4x_2_11x_9_by_x__2023-01-23T23-27-22.527209|chosen" } }, { "text": "I know that complementary angles are angles that add up to 90 degrees, so I can write an equation: $A + B = 90$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "I also know that $A$ is a multiple of $B$, so I can write another equation: $A = kB$, where $k$ is some positive integer.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "I can substitute $A = kB$ into the first equation and get: $kB + B = 90$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "I can factor out $B$ and get: $B(k + 1) = 90$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "This means that $B$ must be a factor of 90, and $k + 1$ must be the other factor.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "For each factor of 90, I can find the corresponding value of $k + 1$ by dividing 90 by the factor.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "For example, if $B = 1$, then $k + 1 = 90$, and if $B = 2$, then $k + 1 = 45$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "The value of $k + 1$ must also be a positive integer, so I can eliminate any fractions or decimals.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "For example, if $B = 4$, then $k + 1 = 22.5$, which is not an integer, so I can ignore this possibility.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "The only values of $B$ that give integer values of $k + 1$ are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "There are 12 such values of $B$, and for each one, I can find the value of $A$ by multiplying $B$ by $k$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "For example, if $B = 1$, then $k = 89$, and $A = 89$, and if $B = 2$, then $k = 44$, and $A = 88$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "The values of $A$ are 89, 88, 84, 80, 72, 63, 60, 45, 36, 30, 18, and 0.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "However, I have to remember that $A$ and $B$ are both positive, so I cannot include 0 as a possible value of $A$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "Therefore, there are 11 possible measures for angle $A$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 105, "metadata": { "path_label": "the_measures_of_angles_a_and_b_are_both_positive_intege__2023-01-23T23-21-55.665622|chosen" } }, { "text": "I want to find the sum of 2009 fractions of the form $\\frac{2}{n(n+2)}$, where $n$ ranges from 1 to 2009.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "This looks like a series problem, so I wonder if there is a way to simplify or rewrite the fractions to make them easier to add up.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "One thing I notice is that the denominator of each fraction is a product of two consecutive integers, $n$ and $n+2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "This reminds me of a technique called partial fraction decomposition, where I can write a fraction with a product in the denominator as a sum of two fractions with linear denominators.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "For example, $\\frac{2}{n(n+2)} = \\frac{A}{n} + \\frac{B}{n+2}$ for some constants $A$ and $B$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "To find $A$ and $B$, I can multiply both sides by the common denominator $n(n+2)$ and get $2 = A(n+2) + Bn$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "This must hold for any value of $n$, so I can plug in some convenient values and solve for $A$ and $B$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "For example, if I let $n=0$, I get $2 = 2A$, so $A=1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "If I let $n=-2$, I get $2 = -2B$, so $B=-1$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "Therefore, I can write $\\frac{2}{n(n+2)} = \\frac{1}{n} - \\frac{1}{n+2}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "Now, I have a sum of 2009 fractions of this form, so I can write it as $\\sum_{n=1}^{2009} \\frac{2}{n(n+2)} = \\sum_{n=1}^{2009} \\frac{1}{n} - \\frac{1}{n+2}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "I can split this sum into two sums and write it as $\\sum_{n=1}^{2009} \\frac{1}{n} - \\sum_{n=1}^{2009} \\frac{1}{n+2}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "I notice that the second sum has the same terms as the first sum, but shifted by 2.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "This means that most of the terms will cancel out when I subtract them, except for the first two terms of the first sum and the last two terms of the second sum.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "That is, $\\sum_{n=1}^{2009} \\frac{1}{n} - \\sum_{n=1}^{2009} \\frac{1}{n+2} = \\frac{1}{1} + \\frac{1}{2} - \\frac{1}{2010} - \\frac{1}{2011}$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "Now, I just need to simplify this expression and round it to the nearest thousandth.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 106, "metadata": { "path_label": "what_is_the_sum_of_the_2009_fractions_of_the_form_frac_2__2023-01-23T16-21-42.174193|chosen" } }, { "text": "I know that the area of a parallelogram is given by the magnitude of the cross product of its adjacent sides.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "So I want to find $|(\\mathbf{p} + 2 \\mathbf{q}) \\times (2 \\mathbf{p} + \\mathbf{q})|$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "To do that, I need to use the distributive property of the cross product and simplify the expression.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "I get $(\\mathbf{p} + 2 \\mathbf{q}) \\times (2 \\mathbf{p} + \\mathbf{q}) = 2 \\mathbf{p} \\times \\mathbf{p} + \\mathbf{p} \\times \\mathbf{q} + 4 \\mathbf{q} \\times \\mathbf{p} + 2 \\mathbf{q} \\times \\mathbf{q}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "Now I recall that the cross product of two parallel vectors is zero, and that the cross product of two unit vectors is another unit vector whose magnitude is the sine of the angle between them.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "So I can simplify further and get $(\\mathbf{p} + 2 \\mathbf{q}) \\times (2 \\mathbf{p} + \\mathbf{q}) = 0 + \\mathbf{p} \\times \\mathbf{q} - 4 \\mathbf{p} \\times \\mathbf{q} + 0 = -3 \\mathbf{p} \\times \\mathbf{q}.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "Therefore, the area of the parallelogram is $|-3 \\mathbf{p} \\times \\mathbf{q}| = 3 |\\mathbf{p} \\times \\math bf{q}| = 3 \\sin 30^\\circ = 3/2.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 107, "metadata": { "path_label": "let_mathbf_p_and_mathbf_q_be_two_three_dimensional_u__2023-01-23T11-07-06.415979|chosen" } }, { "text": "I want to find the smallest positive integer $n$ such that $n > 100$, $n \\equiv 6 \\pmod{13}$, and $n \\equiv 5 \\pmod{14}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "I know that the Chinese remainder theorem guarantees a unique solution for $n$ modulo $13 \\cdot 14 = 182$, so I can use that as a starting point.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "I can write $n = 182k + r$, where $r$ is the remainder when dividing $n$ by $182$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "Then I have to find $r$ such that $r \\equiv 6 \\pmod{13}$ and $r \\equiv 5 \\pmod{14}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "I can use the extended Euclidean algorithm to find integers $a$ and $b$ such that $13a + 14b = 1$, since $13$ and $14$ are coprime.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "I start by dividing $14$ by $13$ and get a quotient of $1$ and a remainder of $1$. So $14 = 13 \\cdot 1 + 1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "Then I divide $13$ by $1$ and get a quotient of $13$ and a remainder of $0$. So $13 = 1 \\cdot 13 + 0$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "Working backwards, I can express $1$ as a linear combination of $13$ and $14$ by substituting $1 = 14 - 13$ into the second equation.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "I get $1 = 14 - (1 \\cdot 13)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "So $a = -1$ and $b = 1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 108, "metadata": { "path_label": "in_a_solar_system_of_n_planets_zorn_the_world_conqueror_c__2023-01-23T08-25-31.476077|chosen" } }, { "text": "To solve this problem, I need to find a value of $x$ that makes the area and perimeter of the rectangle equal.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "I recall that the area of a rectangle is length times width, and the perimeter is 2 times the sum of length and width.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "So I can write two expressions for the area and perimeter, and set them equal to each other.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "The area is $3x(x + 5)$, and the perimeter is $2(3x + x + 5)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "Simplifying the expressions, I get $3x^2 + 15x = 8x + 10$. [* I want to bring everything to one side and factor the resulting quadratic equation.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "Subtracting $8x + 10$ from both sides, I get $3x^2 + 7x - 10 = 0$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "To factor this, I look for two numbers that multiply to $-30$ (the product of the leading and constant coefficients) and add to $7$ (the middle coefficient).", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 109, "metadata": { "path_label": "given_that_a_rectangle_with_length_3x_inches_and_width_x__2023-01-23T03-59-32.722321|chosen" } }, { "text": "To find the largest possible 4-digit palindrome in base 3, I need to use the largest possible digit in base 3, which is 2.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "So the number will look like 2??2 in base 3, where the two middle digits are the same.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "The largest possible value for the middle digits is also 2, so the number is 2222 in base 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "Now I need to convert this number to another base $b>2$ and see if it is still a palindrome.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I can use the place value method to convert from base 3 to base $b$. The place values of the digits in base 3 are $3^3, 3^2, 3^1, 3^0$, so the number is $2\\times 3^3 + 2\\times 3^2 + 2\\times 3^1 + 2\\times 3^0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "To simplify, I can factor out a 2 and get $2(3^3 + 3^2 + 3^1 + 3^0)$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I can also use the formula for the sum of a geometric series to get $2\\left(\\frac{3^4-1}{3-1}\\right) = 2\\left(\\frac{80}{2}\\right) = 80$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "So the number is 80 in base 10.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "To convert from base 10 to base $b$, I need to divide 80 by $b$ repeatedly and write down the remainders until I get 0.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "The remainders will be the digits of the number in base $b$ from right to left.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "For example, if $b=4$, I get 80 divided by 4 is 20 with a remainder of 0, then 20 divided by 4 is 5 with a remainder of 0, then 5 divided by 4 is 1 with a remainder of 1, then 1 divided by 4 is 0 with a remainder of 1.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "So the number in base 4 is 1100, which is not a palindrome.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I need to try a different value of $b$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I notice that 80 is a multiple of 8, so I can try $b=8$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I get 80 divided by 8 is 10 with a remainder of 0, then 10 divided by 8 is 1 with a remainder of 2, then 1 divided by 8 is 0 with a remainder of 1.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 110, "metadata": { "path_label": "a_palindrome_is_a_number_that_reads_the_same_forward_and_bac__2023-01-23T03-53-56.534468|chosen" } }, { "text": "I see that this is a rational equation, so I want to get rid of the fractions by multiplying both sides by the common denominator.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 111, "metadata": { "path_label": "compute_the_value_of_k_such_that_the_equation_frac_x__2023-01-23T03-44-09.040088|chosen" } }, { "text": "That is, I multiply both sides by $(kx - 1)$, which gives me \\[x + 2 = (kx - 1)x\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 111, "metadata": { "path_label": "compute_the_value_of_k_such_that_the_equation_frac_x__2023-01-23T03-44-09.040088|chosen" } }, { "text": "I can expand the right-hand side and simplify the equation to get \\[x + 2 = kx^2 - x\\]or \\[kx^2 - 2x - 2 = 0\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 111, "metadata": { "path_label": "compute_the_value_of_k_such_that_the_equation_frac_x__2023-01-23T03-44-09.040088|chosen" } }, { "text": "This problem involves inverse trigonometric functions and the angle addition formula for tangent.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 112, "metadata": { "path_label": "find_x_if_3_arctan_frac_1_4_arctan_frac_1_20__2023-01-22T07-27-23.175147|chosen" } }, { "text": "I recall that $\\arctan a + \\arctan b = \\arctan \\frac{a+b}{1-ab}$, as long as $ab < 1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 112, "metadata": { "path_label": "find_x_if_3_arctan_frac_1_4_arctan_frac_1_20__2023-01-22T07-27-23.175147|chosen" } }, { "text": "I can use this formula to simplify the left-hand side of the equation, by combining the first two terms and then the result with the third term.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 112, "metadata": { "path_label": "find_x_if_3_arctan_frac_1_4_arctan_frac_1_20__2023-01-22T07-27-23.175147|chosen" } }, { "text": "To find the number of positive real solutions, I can use the Descartes' rule of signs.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 113, "metadata": { "path_label": "how_many_positive_real_solutions_are_there_to_x_10_7x_9_1__2023-01-30T04-26-12.401506|chosen" } }, { "text": "This rule says that the number of positive real solutions is either equal to the number of sign changes in the coefficients of the polynomial, or less than that by a multiple of 2.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 113, "metadata": { "path_label": "how_many_positive_real_solutions_are_there_to_x_10_7x_9_1__2023-01-30T04-26-12.401506|chosen" } }, { "text": "The coefficients of the polynomial are 1, 7, 14, 1729, -1379, and 0.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 113, "metadata": { "path_label": "how_many_positive_real_solutions_are_there_to_x_10_7x_9_1__2023-01-30T04-26-12.401506|chosen" } }, { "text": "I see that there is one sign change, from 1729 to -1379.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 113, "metadata": { "path_label": "how_many_positive_real_solutions_are_there_to_x_10_7x_9_1__2023-01-30T04-26-12.401506|chosen" } }, { "text": "I notice that the equation is quadratic in $x,$ and it has complex coefficients.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "I wonder if I can use the discriminant of the quadratic formula to determine when the equation has real roots.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is $b^2 - 4ac.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots, only complex ones.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "In this problem, $a = 3 - i, b = a + 4i, c = -115 + 5i.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "So the discriminant is $(a + 4i)^2 - 4(3 - i)(-115 + 5i).$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "I want to simplify this expression and see what conditions on $a$ make it nonnegative.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "To simplify $(a + 4i)^2,$ I can use the formula $(x + yi)^2 = x^2 - y^2 + 2xyi.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "So $(a + 4i)^2 = a^2 - 16 + 8ai.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "To simplify $-4(3 - i)(-115 + 5i),$ I can use the distributive property and the fact that $i^2 = -1.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 114, "metadata": { "path_label": "find_all_real_values_of_a_for_which_the_quadratic_equation__2023-01-30T04-17-33.593849|chosen" } }, { "text": "I need to find the number of cubes that the internal diagonal intersects.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "The internal diagonal is the hypotenuse of a right triangle whose legs are the face diagonals of the solid.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "To find the length of the internal diagonal, I can use the Pythagorean theorem twice: first on one face, then on the right triangle formed by two face diagonals and the internal diagonal.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "For example, the diagonal of the $150\\times 324$ face is $\\sqrt{150^2+324^2}$ by the Pythagorean theorem.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "Then, the internal diagonal is $\\sqrt{(\\sqrt{150^2+324^2})^2+375^2}$ by applying the Pythagorean theorem again.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "I can simplify this expression by squaring both sides and taking the square root: the internal diagonal is $\\sqrt{150^2+324^2+375^2}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 115, "metadata": { "path_label": "a_150_times_324_times_375_rectangular_solid_is_made_by_glu__2023-01-30T04-12-06.806962|chosen" } }, { "text": "To approach this problem, I need to think of all the possible positive perfect squares that are less than or equal to 81, since any larger perfect square would make the sum exceed 81.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 116, "metadata": { "path_label": "in_how_many_ways_can_81_be_written_as_the_sum_of_three_posit__2023-01-29T05-49-10.688347|chosen" } }, { "text": "The possible positive perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, and 81.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 116, "metadata": { "path_label": "in_how_many_ways_can_81_be_written_as_the_sum_of_three_posit__2023-01-29T05-49-10.688347|chosen" } }, { "text": "Now, I need to find three of these numbers that add up to 81.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 116, "metadata": { "path_label": "in_how_many_ways_can_81_be_written_as_the_sum_of_three_posit__2023-01-29T05-49-10.688347|chosen" } }, { "text": "One way to do this is to start with the largest possible perfect square, 81, and see if there are two other perfect squares that add up to 0. There is only one way to do this, and that is to use 0 and 0, but 0 is not a positive integer, so this is not a valid way.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 116, "metadata": { "path_label": "in_how_many_ways_can_81_be_written_as_the_sum_of_three_posit__2023-01-29T05-49-10.688347|chosen" } }, { "text": "I notice that the modulus of a complex number is related to its real and imaginary parts by $|z| = \\sqrt{x^2 + y^2}$, where $z = x + yi$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "Therefore, the condition $|z| = 1$ implies that $x^2 + y^2 = 1$, which is the equation of a circle with radius 1 centered at the origin.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "The other condition, \\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1,\\] looks more complicated, but I can simplify it by using the fact that $z \\overline{z} = |z|^2 = 1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "Multiplying both sides of the equation by $z \\overline{z}$, I get \\[|z^2 + \\overline{z}^2| = z \\overline{z}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "Now I can write $z^2$ and $\\overline{z}^2$ in terms of $x$ and $y$ using the identity $(x + yi)^2 = x^2 - y^2 + 2xyi$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "I get \\[|x^2 - y^2 + 2xyi + x^2 - y^2 - 2xyi| = x^2 + y^2.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "Simplifying the left-hand side, I get \\[|2x^2 - 2y^2| = x^2 + y^2.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "Since both sides are nonnegative, I can drop the absolute value signs and rearrange the terms to get \\[x^2 - y^2 = \\pm \\frac{1}{2}.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 117, "metadata": { "path_label": "find_the_number_of_complex_numbers_z_satisfying_z_1__2023-01-29T05-40-57.682251|chosen" } }, { "text": "I need to find the values of $k$ that make the quadratic equation have two distinct integer solutions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 118, "metadata": { "path_label": "what_is_the_sum_of_all_values_of_k_such_that_the_equation__2023-01-29T05-30-01.998950|chosen" } }, { "text": "That means the equation has to factor into two linear terms with integer coefficients, like $(2x-a)(x-b)=0$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 118, "metadata": { "path_label": "what_is_the_sum_of_all_values_of_k_such_that_the_equation__2023-01-29T05-30-01.998950|chosen" } }, { "text": "Then I can use the zero product property to get $x=a/2$ or $x=b$ as the solutions.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 118, "metadata": { "path_label": "what_is_the_sum_of_all_values_of_k_such_that_the_equation__2023-01-29T05-30-01.998950|chosen" } }, { "text": "To find $k$, I can expand the factors and compare the coefficients of $x$ and the constant term.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 118, "metadata": { "path_label": "what_is_the_sum_of_all_values_of_k_such_that_the_equation__2023-01-29T05-30-01.998950|chosen" } }, { "text": "I will use a geometric approach to this problem.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 119, "metadata": { "path_label": "three_points_are_chosen_randomly_and_independently_on_a_circ__2023-01-29T04-48-59.258191|chosen" } }, { "text": "I will fix one point on the circle and call it A.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 119, "metadata": { "path_label": "three_points_are_chosen_randomly_and_independently_on_a_circ__2023-01-29T04-48-59.258191|chosen" } }, { "text": "Then I will choose a second point B on the circle, and consider the arc AB.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 119, "metadata": { "path_label": "three_points_are_chosen_randomly_and_independently_on_a_circ__2023-01-29T04-48-59.258191|chosen" } }, { "text": "The length of the arc AB is proportional to the central angle subtended by it, so I can use angles to measure distances on the circle.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 119, "metadata": { "path_label": "three_points_are_chosen_randomly_and_independently_on_a_circ__2023-01-29T04-48-59.258191|chosen" } }, { "text": "I want the distance between A and B to be less than the radius of the circle, which means that the angle AOB, where O is the center of the circle, must be less than 60 degrees.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 119, "metadata": { "path_label": "three_points_are_chosen_randomly_and_independently_on_a_circ__2023-01-29T04-48-59.258191|chosen" } }, { "text": "To solve this inequality, I need to find the values of $x$ that make the left-hand side smaller than the right-hand side.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "One way to do that is to simplify the inequality by expanding the brackets and collecting like terms.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "I get $2x^2+16x+30<3x^2+33x+72$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "Then I can subtract $2x^2+16x+30$ from both sides to get $00$, so the product is positive in this interval.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "Similarly, if I pick $x=0$ in the third interval, I get $(0+14)(0+3)=14\\times 3=42>0$, so the product is positive in this interval as well.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "However, if I pick $x=-5$ in the second interval, I get $(-5+14)(-5+3)=9\\times -2=-18<0$, so the product is negative in this interval.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "Therefore, the values of $x$ that satisfy the inequality are those that lie in the first and third intervals, excluding the boundary points.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "In interval notation, the answer is $(-\\infty,-14)\\cup (-3,\\infty)$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 120, "metadata": { "path_label": "find_all_x_that_satisfy_the_inequality_2x_10_x_3_3x_9__2023-01-28T18-56-08.824707|chosen" } }, { "text": "I need to find the sum of the integers from $-25$ to $n$, which is an arithmetic series with a common difference of $1$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "The first term is $-25$ and the last term is $n$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "To find the number of terms in the series, I can use the formula $n = a_1 + (k - 1)d$, where $a_1$ is the first term, $d$ is the common difference, and $k$ is the number of terms.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Plugging in the values, I get $n = -25 + (k - 1)1$, which simplifies to $n = k - 26$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Therefore, the number of terms is $k = n + 26$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "To find the sum of the series, I can use the formula $S_k = \\frac{k}{2}(a_1 + a_k)$, where $a_k$ is the last term.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Plugging in the values, I get $S_k = \\frac{n + 26}{2}(-25 + n)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "This is the sum of the integers from $-25$ to $n$, and I want it to be at least $26$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "So I need to solve the inequality $\\frac{n + 26}{2}(-25 + n) \\geq 26$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Multiplying both sides by $2$, I get $(n + 26)(-25 + n) \\geq 52$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Expanding the brackets, I get $-25n + n^2 - 650 + 26n \\geq 52$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "Simplifying, I get $n^2 + n - 702 \\geq 0$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "I can try to factor this quadratic expression by looking for two numbers that multiply to $-702$ and add to $1$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "One pair that works is $27$ and $-26$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "So I can write $n^2 + n - 702 = (n + 27)(n - 26)$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "To solve the inequality, I need to consider the sign of each factor.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "When $n < -27$, both factors are negative, so the product is positive.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "When $-27 < n < 26$, one factor is negative and the other is positive, so the product is negative.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "When $n > 26$, both factors are positive, so the product is positive.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "So the inequality is satisfied when $n \\leq -27$ or $n \\geq 26$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "But I want the smallest integer $n$ that works, so I choose $n = 26$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "This means that the sum of the integers from $-25$ to $26$ is exactly $26$, and adding any larger integer would make the sum greater than $26$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 121, "metadata": { "path_label": "find_the_smallest_integer_n_for_which_the_sum_of_the_integ__2023-01-27T23-36-45.138375|chosen" } }, { "text": "This problem is asking me to find the number of pennies that Betty originally had, which I'll call $p$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 122, "metadata": { "path_label": "betty_has_between_40_and_50_pennies_that_she_wants_to_ar__2023-01-27T23-34-12.166487|chosen" } }, { "text": "I know that $p$ is a multiple of three different pairs of factors, since she can arrange the pennies in three different rectangles.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 122, "metadata": { "path_label": "betty_has_between_40_and_50_pennies_that_she_wants_to_ar__2023-01-27T23-34-12.166487|chosen" } }, { "text": "I also know that $p + 2$ is a prime number, since she can only arrange the pennies in a straight line.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 122, "metadata": { "path_label": "betty_has_between_40_and_50_pennies_that_she_wants_to_ar__2023-01-27T23-34-12.166487|chosen" } }, { "text": "I can list some prime numbers that are two more than multiples of three: $5, 11, 17, 23, 29, 41, 47, 53, 59, \\dots$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 122, "metadata": { "path_label": "betty_has_between_40_and_50_pennies_that_she_wants_to_ar__2023-01-27T23-34-12.166487|chosen" } }, { "text": "I need to find the value of the $ 50$th term in an arithmetic sequence of $ 200$ numbers that sum to $ 10{,}000$ and are between $ 10$ and $ 100$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I know that the formula for the sum of an arithmetic series is $ S_n = \\frac{n}{2}(a_1 + a_n)$, where $ S_n$ is the sum of the first $ n$ terms, $ a_1$ is the first term, and $ a_n$ is the $ n$th term.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "In this case, $ S_{200} = 10{,}000$, so I can plug that in and get $ 10{,}000 = \\frac{200}{2}(a_1 + a_{200})$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Simplifying, I get $ 100 = a_1 + a_{200}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "This means that the first and last terms of the sequence add up to $ 100$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I also know that the formula for the $ n$th term of an arithmetic sequence is $ a_n = a_1 + (n - 1)d$, where $ d$ is the common difference between consecutive terms.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I can use this formula to find $ a_{50}$ in terms of $ a_1$ and $ d$. I get $ a_{50} = a_1 + 49d$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I want to find the least and greatest possible values of $ a_{50}$, given that $ 10 \\leq a_n \\leq 100$ for all $ n$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "To find the least possible value of $ a_{50}$, I need to minimize $ a_1$ and $ d$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "The smallest possible value of $ a_1$ is $ 10$, since that is the lower bound for the terms of the sequence.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "If $ a_1 = 10$, then $ a_{200} = 90$, since they add up to $ 100$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "This means that $ d = \\frac{a_{200} - a_1}{199} = \\frac{80}{199}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Therefore, the least possible value of $ a_{50}$ is $ a_1 + 49d = 10 + \\frac{3920}{199}$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "To find the greatest possible value of $ a_{50}$, I need to maximize $ a_1$ and $ d$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "The largest possible value of $ a_1$ is $ 100$, since that is the upper bound for the terms of the sequence.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "If $ a_1 = 100$, then $ a_{200} = 0$, since they add up to $ 100$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "However, this is not possible, since $ 0$ is not between $ 10$ and $ 100$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Therefore, the largest possible value of $ a_1$ is actually $ 90$, since that allows $ a_{200}$ to be $ 10$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "If $ a_1 = 90$, then $ a_{200} = 10$, and $ d = \\frac{a_{200} - a_1}{199} = -\\frac{80}{199}$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Therefore, the greatest possible value of $ a_{50}$ is $ a_1 + 49d = 90 - \\frac{3920}{199}$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Now, to find the value of $ G - L$, I need to subtract the least and greatest possible values of $ a_{50}$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I get $ G - L = 90 - \\frac{3920}{199} - (10 + \\frac{3920}{199})$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "Simplifying, I get $ G - L = 80 - 2\\frac{3920}{199}$.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 123, "metadata": { "path_label": "an_arithmetic_sequence_consists_of_200_numbers_that_are_e__2023-01-27T04-07-18.804318|chosen" } }, { "text": "I notice that both equations are fractions of polynomials, and they are equal to zero when the numerator is zero and the denominator is not zero.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "So I can focus on finding the roots of the numerators and avoiding the roots of the denominators.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "For the first equation, the numerator is a product of three linear factors, so it has three roots: $-a, -b, -12$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "The denominator is a perfect square, so it has a double root at $-3$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "Therefore, the first equation has exactly $3$ distinct roots when $-a, -b, -12$ are all different from $-3$ and each other.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "This means that $a, b, 12$ are three distinct numbers, none of them equal to $3$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "For the second equation, the numerator is also a product of three linear factors, so it has three roots: $-2a, -3, -6$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "The denominator is a product of two linear factors, so it has two roots: $-b, -12$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 124, "metadata": { "path_label": "let_a_and_b_be_constants_suppose_that_the_equation_f__2023-01-26T06-47-03.799387|chosen" } }, { "text": "I notice that the expression inside the absolute value is a quadratic function of x, and I know that quadratic functions are symmetric about their axis of symmetry, which is the line x = -b/2a, where a and b are the coefficients of x^2 and x respectively.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 125, "metadata": { "path_label": "find_min__y_in_mathbb_r_max__0_le_x_le_1_x_2_x__2023-01-25T22-19-39.435920|chosen" } }, { "text": "In this case, the axis of symmetry is x = y/2, since a = 1 and b = -y.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 125, "metadata": { "path_label": "find_min__y_in_mathbb_r_max__0_le_x_le_1_x_2_x__2023-01-25T22-19-39.435920|chosen" } }, { "text": "I also know that quadratic functions have a maximum or minimum value at their vertex, which is the point where x = -b/2a and y = f(-b/2a), where f is the quadratic function.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 125, "metadata": { "path_label": "find_min__y_in_mathbb_r_max__0_le_x_le_1_x_2_x__2023-01-25T22-19-39.435920|chosen" } }, { "text": "Therefore, I want to find the value of y that minimizes the maximum value of |x^2 - xy| on the interval [0, 1].", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 125, "metadata": { "path_label": "find_min__y_in_mathbb_r_max__0_le_x_le_1_x_2_x__2023-01-25T22-19-39.435920|chosen" } }, { "text": "I can think of two possible cases: either the maximum value of |x^2 - xy| occurs at the vertex, or it occurs at one of the endpoints of the interval.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 125, "metadata": { "path_label": "find_min__y_in_mathbb_r_max__0_le_x_le_1_x_2_x__2023-01-25T22-19-39.435920|chosen" } }, { "text": "To solve this problem, I need to find a two-digit integer that satisfies a certain equation involving its digits.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "Let me call the tens digit of $n$ as $a$ and the ones digit as $b$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "Then $n = 10a + b$ and switching its digits gives $10b + a$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "The equation I need to solve is $10b + a + 3 = 2(10a + b)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "I can simplify this equation by expanding the parentheses and moving everything to one side: $8b - 19a + 3 = 0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "I can see that this equation is linear in $a$ and $b$, so I can try to find integer solutions by using modular arithmetic.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "If I take both sides of the equation modulo 8, I get $-3a + 3 \\equiv 0 \\pmod{8}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "I can divide both sides by -3, which is coprime to 8, and get $a \\equiv 1 \\pmod{8}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "This means that $a$ must be of the form $8k + 1$, where $k$ is an integer.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "Since $a$ is a digit, it can only take values from 0 to 9, so the only possible values for $a$ are 1 and 9.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "If I plug in $a = 1$ into the original equation, I get $8b - 19 + 3 = 0$, or $b = 2$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "This gives me one possible value for $n$, which is $n = 10a + b = 12$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "If I plug in $a = 9$ into the original equation, I get $8b - 171 + 3 = 0$, or $b = 21$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "This is not a valid digit, so there is no other solution for $n$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "Therefore, the smallest two-digit integer that satisfies the given condition is 12.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 126, "metadata": { "path_label": "what_is_the_smallest_two_digit_integer_n_such_that_switchi__2023-01-25T22-15-13.354390|chosen" } }, { "text": "I need to find the volume of the tetrahedron, which is one-third of the product of the base area and the height.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "I can choose any face as the base, but I need to find the height perpendicular to that face.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "I notice that the given information involves two faces that share the edge $AB$, so I will try to use one of them as the base.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "Let me use face $ABD$ as the base, since it has a smaller area.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "To find the height, I need to find the distance from the vertex $C$ to the plane containing $ABD$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "I recall that the distance from a point to a plane can be found by dividing the absolute value of the dot product of a normal vector to the plane and the position vector of the point by the magnitude of the normal vector.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "To find a normal vector to the plane $ABD$, I can use the cross product of two vectors in the plane, such as $\\vec{AB}$ and $\\vec{AD}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "I also need to find the position vector of $C$ relative to any point in the plane $ABD$, such as $A$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 127, "metadata": { "path_label": "in_tetrahedron_abcd_edge_ab_has_length_3_cm_the_area_o__2023-01-25T22-06-47.566905|chosen" } }, { "text": "To find the maximum value of the expression, I need to use the properties of complex numbers and their absolute values.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "I know that $|z|^2 = z \\overline{z},$ where $\\overline{z}$ is the complex conjugate of $z.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "I also know that $|z_1 z_2| = |z_1| |z_2|$ for any complex numbers $z_1$ and $z_2.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "So, I can rewrite the expression as \\[|(z - 1)^2 (z + 1)| = |(z - 1)^2| |z + 1| = |(z - 1) \\overline{(z - 1)}| |z + 1|.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "Now, I can expand the first factor and simplify it using the fact that $|z| = \\sqrt{2}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "I get \\[|(z - 1) \\overline{(z - 1)}| = |z^2 - 2z + 1| = |\\overline{z}^2 - 2\\overline{z} + 1| = |(\\overline{z} - 1)^2|.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "So, the expression becomes \\[|(z - 1)^2 (z + 1)| = |(\\overline{z} - 1)^2| |z + 1|.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 128, "metadata": { "path_label": "let_z_be_a_complex_number_with_z_sqrt_2_find_the__2023-01-25T06-38-10.461014|chosen" } }, { "text": "To find $f^{-1}(x)$, I need to switch $x$ and $y$ and then solve for $y$ in terms of $x$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "So, let $y = f(x)$ and then swap them to get \\[x = \\frac{2y + 3}{ky - 2}.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "To solve for $y$, I need to clear the fraction by multiplying both sides by $ky - 2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "This gives me \\[x(ky - 2) = 2y + 3.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "Now, I want to isolate $y$ on one side, so I will expand the left side and then move all the terms with $y$ to one side and all the constants to the other side.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "This gives me \\[kxy - 2x - 2y - 3 = 0.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "I can factor out $y$ from the terms that have $y$ in them and get \\[y(kx - 2) - (2x + 3) = 0.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "Then, I can add $2x + 3$ to both sides and get \\[y(kx - 2) = 2x + 3.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "Finally, I can divide both sides by $kx - 2$ and get \\[y = \\frac{2x + 3}{kx - 2}.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "This means that $f^{-1}(x) = \\frac{2x + 3}{kx - 2}$, which is the same as $f(x)$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 129, "metadata": { "path_label": "let_f_x_frac_2x_3_kx_2_find_all_real_numbers__2023-01-25T04-37-33.226962|chosen" } }, { "text": "I notice that this equation is linear in both $m$ and $n,$ so I can try to isolate one of them and use the fact that they are positive integers to narrow down the possible values.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 130, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-25T02-43-08.473155|chosen" } }, { "text": "For example, I can solve for $m$ in terms of $n$ by adding $3m$ to both sides and dividing by $6n:$ $$m = \\frac{27 - 2n}{6n + 3}$$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 130, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-25T02-43-08.473155|chosen" } }, { "text": "Now, since $m$ is a positive integer, the numerator and the denominator of this fraction must both be positive integers, and the numerator must be divisible by the denominator.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 130, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-25T02-43-08.473155|chosen" } }, { "text": "I also notice that the denominator is always odd, so the numerator must be odd as well.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 130, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-25T02-43-08.473155|chosen" } }, { "text": "This is a tricky problem, because it involves a function that is not given explicitly, but only defined by a functional equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "A good way to approach such problems is to look for some special values of $x$ and $y$ that simplify the equation or give useful information about $f(x).$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "One obvious choice is to let $y = x,$ since then the left-hand side becomes zero and we get $f(1) = 0.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "Another useful choice is to let $y = 1,$ since then the right-hand side becomes $f(x)$ and we get \\[xf(1) - f(x) = f(x),\\] or \\[f(x) = \\frac{xf(1)}{2}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "But we already know that $f(1) = 0,$ so this implies that $f(x) = 0$ for all $x.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "This means that $f(100) = 0$ as well, and that is the only possible value.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 131, "metadata": { "path_label": "the_function_f_x_takes_positive_real_numbers_to_real_numb__2023-01-22T06-01-28.694159|chosen" } }, { "text": "I know that the congruence $a\\equiv b\\pmod m$ means that $m$ divides $a-b$, or that $a$ and $b$ have the same remainder when divided by $m$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "So, I want to find the smallest $n$ such that $5^n$ and $n^5$ have the same remainder when divided by 3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "I also know that the remainders of powers of 5 when divided by 3 follow a cycle of 2, 1, 2, 1, ... depending on whether the exponent is even or odd.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "For example, $5^0\\equiv 1\\pmod 3$, $5^1\\equiv 2\\pmod 3$, $5^2\\equiv 1\\pmod 3$, and so on.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "This means that $5^n$ will have a remainder of 2 if $n$ is odd, and a remainder of 1 if $n$ is even.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "Now, what about the remainders of powers of $n$ when divided by 3?", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "Well, that depends on the value of $n$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "If $n\\equiv 0\\pmod 3$, then $n^5\\equiv 0\\pmod 3$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "If $n\\equiv 1\\pmod 3$, then $n^5\\equiv 1\\pmod 3$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "If $n\\equiv 2\\pmod 3$, then $n^5\\equiv 2\\pmod 3$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "So, the only way that $5^n\\equiv n^5\\pmod 3$ is if $n\\equiv 1\\pmod 3$ and $n$ is even, or if $n\\equiv 2\\pmod 3$ and $n$ is odd.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 132, "metadata": { "path_label": "determine_the_smallest_positive_integer_n_such_that_5_n_e__2023-01-22T03-41-05.647189|chosen" } }, { "text": "I notice that the fly's path forms a triangle on the circular ceiling, and the center of the circle is the midpoint of the first part of the journey.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 133, "metadata": { "path_label": "a_fly_is_on_the_edge_of_a_ceiling_of_a_circular_room_with_a__2023-01-21T19-06-28.120403|chosen" } }, { "text": "That means the first part of the journey is a diameter of the circle, so it has length 2 times the radius, or 2 times 58 feet, which is 116 feet.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 133, "metadata": { "path_label": "a_fly_is_on_the_edge_of_a_ceiling_of_a_circular_room_with_a__2023-01-21T19-06-28.120403|chosen" } }, { "text": "The second part of the journey is also a chord of the circle, but not a diameter, so I don't know its length yet. [*: The third part of the journey is given as 80 feet, so I can use that to find the angle of the arc that it subtends at the center of the circle.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 133, "metadata": { "path_label": "a_fly_is_on_the_edge_of_a_ceiling_of_a_circular_room_with_a__2023-01-21T19-06-28.120403|chosen" } }, { "text": "The formula for the arc length of a circle is s = r theta, where s is the arc length, r is the radius, and theta is the central angle in radians.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 133, "metadata": { "path_label": "a_fly_is_on_the_edge_of_a_ceiling_of_a_circular_room_with_a__2023-01-21T19-06-28.120403|chosen" } }, { "text": "I need to find the values of $n$ that satisfy two conditions: $n<1000$ and $\\lfloor \\log_{2} n \\rfloor$ is a positive even integer.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "I can start by listing some examples of $n$ that meet the second condition.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "If $\\lfloor \\log_{2} n \\rfloor = 2$, then $n$ must be between $4$ and $7$, inclusive. That's four values of $n$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "If $\\lfloor \\log_{2} n \\rfloor = 4$, then $n$ must be between $16$ and $31$, inclusive. That's sixteen values of $n$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "If $\\lfloor \\log_{2} n \\rfloor = 6$, then $n$ must be between $64$ and $127$, inclusive. That's sixty-four values of $n$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "If $\\lfloor \\log_{2} n \\rfloor = 8$, then $n$ must be between $256$ and $511$, inclusive. That's two hundred fifty-six values of $n$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "If $\\lfloor \\log_{2} n \\rfloor = 10$, then $n$ must be between $1024$ and $2047$, inclusive. But wait, that's too big for the first condition. So I can stop here.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "The total number of values of $n$ that meet both conditions is $4 + 16 + 64 + 256 = 340$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 134, "metadata": { "path_label": "for_each_real_number_x_let_lfloor_x_rfloor_denote_the__2023-01-21T08-05-57.549597|chosen" } }, { "text": "This is a tricky problem, because it involves a function of both $x$ and $\\frac{1}{x}$, which are reciprocals of each other.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 135, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_2_f_left_frac_1_x__2023-01-20T23-13-27.544291|chosen" } }, { "text": "One way to approach this problem is to try to find a way to eliminate one of the variables, by using the fact that $x$ and $\\frac{1}{x}$ are reciprocals.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 135, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_2_f_left_frac_1_x__2023-01-20T23-13-27.544291|chosen" } }, { "text": "For example, what if we plug in $x = \\frac{1}{x}$ into the given equation? Then we get \\[f \\left( \\frac{1}{x} \\right) - 2 f(x) = 4^{\\frac{1}{x}},\\]which is another equation involving $f(x)$ and $f \\left( \\frac{1}{x} \\right)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 135, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_2_f_left_frac_1_x__2023-01-20T23-13-27.544291|chosen" } }, { "text": "Now we have two equations and two unknowns, so we can try to solve for $f(x)$ by eliminating $f \\left( \\frac{1}{x} \\right)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 135, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_2_f_left_frac_1_x__2023-01-20T23-13-27.544291|chosen" } }, { "text": "To find the probability of an event, I need to divide the number of favorable outcomes by the number of possible outcomes.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "First, I will count the number of ways Ryan can arrange the lamps in a row.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "This is a permutation problem, since the order of the lamps matters.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "I can use the formula for permutations of n objects, where r of them are identical of one type, and s of them are identical of another type: n!/(r!s!)", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "In this case, n = 6, r = 3, and s = 3, so the number of ways to arrange the lamps is 6!/(3!3!) = 20.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "Next, I will count the number of ways Ryan can turn 3 lamps on.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "This is a combination problem, since the order of the lamps being on or off does not matter.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "I can use the formula for combinations of n objects, where r of them are chosen: nCr = n!/(r!(n-r)!)", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "In this case, n = 6 and r = 3, so the number of ways to turn 3 lamps on is 6C3 = 6!/(3!(6-3)!) = 20.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "Now, I will count the number of favorable outcomes, that is, the number of ways the leftmost lamp is blue and off, and the rightmost lamp is red and on.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "To do this, I will fix the leftmost and rightmost lamps and count the ways to arrange and turn on the remaining 4 lamps.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "The leftmost lamp must be blue and off, so I have 2 blue lamps left.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "The rightmost lamp must be red and on, so I have 2 red lamps left and 2 lamps on.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "The remaining 4 lamps can be arranged in 4!/(2!2!) = 6 ways, and turned on in 4C2 = 6 ways.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "Therefore, the number of favorable outcomes is 6 * 6 = 36.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "Finally, I will find the probability by dividing the number of favorable outcomes by the number of possible outcomes.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "The probability is 36/(20 * 20) = 9/100.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 136, "metadata": { "path_label": "ryan_has_3_red_lava_lamps_and_3_blue_lava_lamps_he_arranges__2023-01-20T22-30-05.890034|chosen" } }, { "text": "This problem involves finding the integer solutions of a quadratic equation and checking if the constant term is a prime number.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 137, "metadata": { "path_label": "find_the_product_of_all_positive_integral_values_of_n_such__2023-01-20T16-26-08.347681|chosen" } }, { "text": "A natural way to approach this problem is to try to factor the quadratic expression on the left-hand side of the equation.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 137, "metadata": { "path_label": "find_the_product_of_all_positive_integral_values_of_n_such__2023-01-20T16-26-08.347681|chosen" } }, { "text": "To do this, I will look for two numbers that add up to -35 and multiply to 306.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 137, "metadata": { "path_label": "find_the_product_of_all_positive_integral_values_of_n_such__2023-01-20T16-26-08.347681|chosen" } }, { "text": "One possible pair of numbers is -17 and -18, since -17 + -18 = -35 and -17 * -18 = 306.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 137, "metadata": { "path_label": "find_the_product_of_all_positive_integral_values_of_n_such__2023-01-20T16-26-08.347681|chosen" } }, { "text": "I notice that the problem involves the vertex and the $y$-intercept of a parabola, which are related to the coefficients $a$, $b$, and $c$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I recall that the vertex of a parabola in the form $y=ax^2+bx+c$ has $x$-coordinate $-\\frac{b}{2a}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "Since the vertex is $(p,p)$, I can substitute $p$ for $x$ and $y$ and get $p=ap^2+bp+c$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I also recall that the $y$-intercept of a parabola is the value of $y$ when $x=0$, which is $c$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "Since the $y$-intercept is $(0,-p)$, I can substitute $-p$ for $y$ and $0$ for $x$ and get $-p=c$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "Now I have two equations involving $a$, $b$, and $c$: $p=ap^2+bp+c$ and $-p=c$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I can eliminate $c$ by plugging in $-p$ for $c$ in the first equation and get $p=ap^2+bp-p$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I can simplify this equation by subtracting $p$ from both sides and get $0=ap^2+bp-2p$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I can factor out $p$ from the right-hand side and get $0=p(ap+b-2)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "Since $p\\ne 0$, I can divide both sides by $p$ and get $0=ap+b-2$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I can isolate $b$ by subtracting $ap$ and adding $2$ to both sides and get $b=2-ap$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "This is the expression for $b$ in terms of $a$ and $p$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 138, "metadata": { "path_label": "the_parabola_y_ax_2_bx_c_has_vertex_p_p_and_y_interc__2023-01-19T07-07-31.145306|chosen" } }, { "text": "I recognize that the inverse sine function, $\\sin^{-1}(x)$, returns the angle between $-\\frac{\\pi}{2}$ and $\\frac{\\pi}{2}$ whose sine is $x$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 139, "metadata": { "path_label": "compute_sin_1_sin_3_sin_1_sin_4_sin_1__2023-01-19T06-21-25.671011|chosen" } }, { "text": "So, for example, $\\sin^{-1} (\\sin 3)$ is not equal to $3$, because $3$ is not in that interval.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 139, "metadata": { "path_label": "compute_sin_1_sin_3_sin_1_sin_4_sin_1__2023-01-19T06-21-25.671011|chosen" } }, { "text": "Instead, I need to find an angle between $-\\frac{\\pi}{2}$ and $\\frac{\\pi}{2}$ that has the same sine as $3$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 139, "metadata": { "path_label": "compute_sin_1_sin_3_sin_1_sin_4_sin_1__2023-01-19T06-21-25.671011|chosen" } }, { "text": "I recall that the sine function is periodic with period $2\\pi$, and it is symmetric about the origin.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 139, "metadata": { "path_label": "compute_sin_1_sin_3_sin_1_sin_4_sin_1__2023-01-19T06-21-25.671011|chosen" } }, { "text": "An arithmetic progression is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 140, "metadata": { "path_label": "determine_the_number_of_pairs_a_b_of_real_numbers_such_t__2023-01-18T22-48-36.186686|chosen" } }, { "text": "Let me call this constant $d$, so that $a = 10 + d$, $b = 10 + 2d$, and $ab = 10 + 3d$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 140, "metadata": { "path_label": "determine_the_number_of_pairs_a_b_of_real_numbers_such_t__2023-01-18T22-48-36.186686|chosen" } }, { "text": "I want to find all values of $d$ that make this equation true.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 140, "metadata": { "path_label": "determine_the_number_of_pairs_a_b_of_real_numbers_such_t__2023-01-18T22-48-36.186686|chosen" } }, { "text": "Multiplying out $ab$, I get $ab = (10 + d)(10 + 2d) = 100 + 30d + 2d^2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 140, "metadata": { "path_label": "determine_the_number_of_pairs_a_b_of_real_numbers_such_t__2023-01-18T22-48-36.186686|chosen" } }, { "text": "I need to find the values of $a$ and $b$ that make the modulus of $a \\omega + b$ equal to 1.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I can start by finding the exact value of $\\omega.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "Since $z^3 = 1,$ I can write $z = e^{i \\theta}$ where $\\theta$ is an angle in radians.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "Then, $z^3 = e^{3i \\theta} = 1.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "This implies that $3 \\theta$ is a multiple of $2 \\pi.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "The smallest positive values of $\\theta$ that satisfy this are $\\frac{2 \\pi}{3}$ and $\\frac{4 \\pi}{3}.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "These correspond to the complex numbers $\\frac{-1 + i \\sqrt{3}}{2}$ and $\\frac{-1 - i \\sqrt{3}}{2},$ respectively.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I can choose either one as $\\omega,$ since they are both nonreal roots of $z^3 = 1.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I will pick $\\omega = \\frac{-1 + i \\sqrt{3}}{2}$ for convenience.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "Now, I need to find the integers $a$ and $b$ that make $|a \\omega + b| = 1.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I can use the fact that $|a \\omega + b|^2 = (a \\omega + b)(a \\overline{\\omega} + b),$ where $\\overline{\\omega}$ is the complex conjugate of $\\omega.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "Expanding this, I get $|a \\omega + b|^2 = a^2 \\omega \\overline{\\omega} + ab (\\omega + \\overline{\\omega}) + b^2.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I can simplify this by noting that $\\omega \\overline{\\omega} = 1,$ since $|\\omega| = 1,$ and that $\\omega + \\overline{\\omega} = -1,$ since they are both on the unit circle and have a negative real part.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "So, $|a \\omega + b|^2 = a^2 - ab + b^2.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "Since I want this to be equal to 1, I have the equation $a^2 - ab + b^2 = 1.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "This is a quadratic equation in $a,$ which I can solve by using the quadratic formula.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I get $a = \\frac{b \\pm \\sqrt{b^2 - 4(b^2 - 1)}}{2}.$", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "The expression under the square root simplifies to $4 - 3b^2.$", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "For this to be nonnegative, $b$ has to be either 0, 1, or -1.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "If $b = 0,$ then $a = \\pm 1.$", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 141, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2023-01-27T23-20-23.567975|chosen" } }, { "text": "I notice that this problem is asking about the minimum value of a quadratic expression, which might have something to do with completing the square or using the vertex form.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 142, "metadata": { "path_label": "find_the_minimum_value_of_x_2_y_2_where_x_and_y_are__2023-01-27T23-03-30.430421|chosen" } }, { "text": "I also notice that the constraint $x^2 + y^2 = 4xy$ is a symmetric equation in $x$ and $y,$ which means that swapping their values does not change the equation.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 142, "metadata": { "path_label": "find_the_minimum_value_of_x_2_y_2_where_x_and_y_are__2023-01-27T23-03-30.430421|chosen" } }, { "text": "This makes me think of using polar coordinates, where $x = r \\cos \\theta$ and $y = r \\sin \\theta,$ to simplify the constraint.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 142, "metadata": { "path_label": "find_the_minimum_value_of_x_2_y_2_where_x_and_y_are__2023-01-27T23-03-30.430421|chosen" } }, { "text": "If I substitute these expressions into the constraint, I get $r^2 \\cos^2 \\theta + r^2 \\sin^2 \\theta = 4r \\cos \\theta r \\sin \\theta.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 142, "metadata": { "path_label": "find_the_minimum_value_of_x_2_y_2_where_x_and_y_are__2023-01-27T23-03-30.430421|chosen" } }, { "text": "This simplifies to $r^2 = 4r^2 \\cos \\theta \\sin \\theta,$ or $r^2 (1 - 4 \\cos \\theta \\sin \\theta) = 0.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 142, "metadata": { "path_label": "find_the_minimum_value_of_x_2_y_2_where_x_and_y_are__2023-01-27T23-03-30.430421|chosen" } }, { "text": "I notice that the problem involves tangents and secants of a circle, which often leads to using the power of a point theorem.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 143, "metadata": { "path_label": "let_p_be_a_point_outside_of_circle_o_a_segment_is_drawn__2023-01-27T22-49-41.267887|chosen" } }, { "text": "This theorem states that if a line drawn from a point $P$ intersects a circle at two points $A$ and $B,$ then $PA \\cdot PB = PT^2,$ where $T$ is the point of tangency.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 143, "metadata": { "path_label": "let_p_be_a_point_outside_of_circle_o_a_segment_is_drawn__2023-01-27T22-49-41.267887|chosen" } }, { "text": "In this problem, I have $PA = 3$ and $PT = AB - PA.$ I want to find $PB.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 143, "metadata": { "path_label": "let_p_be_a_point_outside_of_circle_o_a_segment_is_drawn__2023-01-27T22-49-41.267887|chosen" } }, { "text": "So I can plug these values into the power of a point formula and get $3 \\cdot PB = (AB - PA)^2.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 143, "metadata": { "path_label": "let_p_be_a_point_outside_of_circle_o_a_segment_is_drawn__2023-01-27T22-49-41.267887|chosen" } }, { "text": "I can simplify this equation by expanding the square and canceling out the $PA$ terms. I get $3 \\cdot PB = AB^2 - 2 \\cdot AB \\cdot PA + PA^2.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 143, "metadata": { "path_label": "let_p_be_a_point_outside_of_circle_o_a_segment_is_drawn__2023-01-27T22-49-41.267887|chosen" } }, { "text": "This is a problem about number theory and modular arithmetic.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 144, "metadata": { "path_label": "the_positive_integers_n_and_n_2_both_end_in_the_same_seq__2023-01-27T05-41-59.083880|chosen" } }, { "text": "The fact that $N$ and $N^2$ both end in the same four digits means that they have the same remainder when divided by $10^4$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 144, "metadata": { "path_label": "the_positive_integers_n_and_n_2_both_end_in_the_same_seq__2023-01-27T05-41-59.083880|chosen" } }, { "text": "In other words, $N \\equiv N^2 \\pmod{10^4}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 144, "metadata": { "path_label": "the_positive_integers_n_and_n_2_both_end_in_the_same_seq__2023-01-27T05-41-59.083880|chosen" } }, { "text": "I want to count how many numbers are divisible by 3 or 4, but not both, since that would make them divisible by 12.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "One way to do this is to use the inclusion-exclusion principle, which says that the number of elements in the union of two sets is equal to the sum of the numbers in each set minus the number of elements in their intersection.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "So, I can write the number of numbers divisible by 3 or 4 as N(3 or 4) = N(3) + N(4) - N(3 and 4).", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "Now, I need to find each of these terms.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "N(3) is the number of multiples of 3 between 1 and 2005. To find this, I can divide 2005 by 3 and round down to get the largest multiple of 3 that is at most 2005. That is 2005 / 3 = 668. So, N(3) = 668.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "Similarly, N(4) is the number of multiples of 4 between 1 and 2005. To find this, I can divide 2005 by 4 and round down to get the largest multiple of 4 that is at most 2005. That is 2005 / 4 = 501. So, N(4) = 501.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "N(3 and 4) is the number of multiples of both 3 and 4, or equivalently, the number of multiples of 12 between 1 and 2005. To find this, I can divide 2005 by 12 and round down to get the largest multiple of 12 that is at most 2005. That is 2005 / 12 = 167. So, N(3 and 4) = 167.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "Therefore, N(3 or 4) = 668 + 501 - 167 = 1002.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "However, this is not the final answer, because I want to exclude the numbers that are divisible by 12. So, I need to subtract N(3 and 4) from N(3 or 4) to get the final answer. That is, the answer is 1002 - 167 = 835.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 145, "metadata": { "path_label": "how_many_numbers_between_1_and_2005_are_integer_multiples_of__2023-01-27T05-40-09.403415|chosen" } }, { "text": "An isosceles triangle is a triangle with two congruent sides.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "Let x be the length of the congruent sides and y be the length of the base.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "Then x + x + y = 23, or 2x + y = 23.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "I want to find all the integer solutions to this equation that also satisfy the triangle inequality, which states that the sum of any two sides of a triangle must be greater than the third side.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "This means that x + y > x, or y > 0, and 2x > y, or y < 2x.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "So I have three conditions: y > 0, y < 2x, and 2x + y = 23.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "I can try different values of x and see if they produce valid values of y.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 1, then y = 21, which violates the triangle inequality.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 2, then y = 19, which also violates the triangle inequality.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 3, then y = 17, which also violates the triangle inequality.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 4, then y = 15, which also violates the triangle inequality.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 5, then y = 13, which also violates the triangle inequality.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 6, then y = 11, which satisfies the triangle inequality.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 7, then y = 9, which satisfies the triangle inequality.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 8, then y = 7, which satisfies the triangle inequality.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 9, then y = 5, which satisfies the triangle inequality.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 10, then y = 3, which satisfies the triangle inequality.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x = 11, then y = 1, which satisfies the triangle inequality.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "If x > 11, then y < 0, which violates the condition that y > 0.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "So the only possible values of x are 6, 7, 8, 9, 10, and 11.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "For each value of x, there is a unique value of y that makes 2x + y = 23.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "Therefore, there are six different isosceles triangles with integer side lengths and perimeter 23.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 146, "metadata": { "path_label": "how_many_different_isosceles_triangles_have_integer_side_len__2023-01-17T22-47-49.164888|chosen" } }, { "text": "I need to find the number of ways to seat four couples around a circular table, such that each husband is opposite his wife, and rotations and reflections do not matter.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 147, "metadata": { "path_label": "george_is_planning_a_dinner_party_for_three_other_couples_h__2023-01-17T09-42-16.816745|chosen" } }, { "text": "I can start by fixing one of the couples at a certain pair of opposite seats. This will eliminate the rotations, since I can always rotate the table so that this couple is at the same position.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 147, "metadata": { "path_label": "george_is_planning_a_dinner_party_for_three_other_couples_h__2023-01-17T09-42-16.816745|chosen" } }, { "text": "Now, I have three remaining couples to seat around the remaining six seats.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 147, "metadata": { "path_label": "george_is_planning_a_dinner_party_for_three_other_couples_h__2023-01-17T09-42-16.816745|chosen" } }, { "text": "I can choose any of the three couples to sit in the next pair of opposite seats, and then there are two couples left for the last two pairs of opposite seats.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 147, "metadata": { "path_label": "george_is_planning_a_dinner_party_for_three_other_couples_h__2023-01-17T09-42-16.816745|chosen" } }, { "text": "I need to find a number that is divisible by 9, so it must be a multiple of 9.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "I also need to make sure it has two even and two odd digits, so I can't use any multiples of 9 that have all even or all odd digits.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The smallest four-digit number is 1000, but that is not divisible by 9.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1000 is 1008, but that has three even digits and one odd digit.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1008 is 1017, but that has three odd digits and one even digit.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1017 is 1026, but that has three even digits and one odd digit.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1026 is 1035, but that has three odd digits and one even digit.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1035 is 1044, but that has three even digits and one odd digit.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1044 is 1053, but that has three odd digits and one even digit.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1053 is 1062, but that has three even digits and one odd digit.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1062 is 1071, but that has three odd digits and one even digit.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "The next multiple of 9 after 1071 is 1080, but that has three even digits and one odd digit.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 148, "metadata": { "path_label": "what_is_the_smallest_positive_four_digit_number_divisible_by__2023-01-17T03-08-52.087037|chosen" } }, { "text": "This problem involves finding the minimum value of a function of two variables, so I might want to use calculus or some algebraic techniques.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "I notice that the expression $x^6 + y^6 - 54xy$ is symmetric in $x$ and $y,$ so it might be helpful to consider the case where $x = y.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "If $x = y,$ then the expression becomes $2x^6 - 54x^2.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "This is a one-variable function, so I can find its critical points by taking the derivative and setting it equal to zero.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "The derivative of $2x^6 - 54x^2$ is $12x^5 - 108x.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "Setting this equal to zero, I get $12x^5 - 108x = 0.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "I can factor out $12x$ from both terms and get $12x(x^4 - 9) = 0.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "This means that either $x = 0$ or $x^4 - 9 = 0.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "The latter equation implies that $x^4 = 9,$ so $x = \\pm \\sqrt[4]{9}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "These are the three possible values of $x$ that make the derivative zero, so they are the critical points of the function.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "To find out which one is the minimum, I need to check the second derivative of the function.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 149, "metadata": { "path_label": "find_the_minimum_value_of_x_6_y_6_54xy_over_all_real_n__2023-01-17T02-04-03.881259|chosen" } }, { "text": "I notice that this expression is the sum of two perfect squares, which might make it easier to factor.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I recall the identity $(a^2+b^2)=(a+b)^2-2ab$, which can be used to rewrite the sum of two squares as a difference of two squares.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I apply this identity with $a=25$ and $b=72$, and get $(25^2+72^2)=(25+72)^2-2\\cdot 25\\cdot 72$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I simplify this expression and get $(25^2+72^2)=97^2-3600$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "Now I can use the difference of two squares formula to factor this expression further.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I write $(97^2-3600)=(97+60)(97-60)$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I simplify these factors and get $(97+60)(97-60)=157\\cdot 37$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "I recognize that both 157 and 37 are prime numbers, so I have factored the expression completely.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "The largest prime factor is 157, so that is the answer.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 150, "metadata": { "path_label": "find_the_largest_prime_divisor_of_25_2_72_2__2023-01-17T02-00-00.885226|chosen" } }, { "text": "To find the volume of a parallelepiped, I need to use the scalar triple product of its three generating vectors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "The scalar triple product is the determinant of the matrix formed by the vectors as its columns, or its rows.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "So I can write the volume as $V = \\det \\begin{pmatrix} 2 & 1 & 1 \\\\ 3 & k & 2 \\\\ 4 & 2 & k \\end{pmatrix}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "I can use any method to find the determinant, but I'll use cofactor expansion along the first row, since it has two ones.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "That gives me $V = 2 \\det \\begin{pmatrix} k & 2 \\\\ 2 & k \\end{pmatrix} - \\det \\begin{pmatrix} 3 & 2 \\\\ 4 & k \\end{pmatrix} + \\det \\begin{pmatrix} 3 & k \\\\ 4 & 2 \\end{pmatrix}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "The determinants of the 2x2 matrices are easy to find: they are just the products of the diagonal entries minus the products of the off-diagonal entries.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "So I can write $V = 2(k^2 - 4) - (3k - 8) + (6 - 4k).$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 151, "metadata": { "path_label": "the_volume_of_the_parallelepiped_generated_by_begin_pmatri__2023-01-17T01-56-51.656192|chosen" } }, { "text": "I notice that the function behaves differently depending on whether the input is less than 10 or greater than or equal to 10.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 152, "metadata": { "path_label": "the_function_f_is_defined_on_positive_integers_as_follows__2023-01-16T23-49-50.257223|chosen" } }, { "text": "If the input is less than 10, the function simply adds 10 to it, so the maximum value in that case would be 19, when the input is 9.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 152, "metadata": { "path_label": "the_function_f_is_defined_on_positive_integers_as_follows__2023-01-16T23-49-50.257223|chosen" } }, { "text": "If the input is greater than or equal to 10, the function calls itself on a smaller input, subtracting 5 each time.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 152, "metadata": { "path_label": "the_function_f_is_defined_on_positive_integers_as_follows__2023-01-16T23-49-50.257223|chosen" } }, { "text": "This looks like a rational equation, so I want to clear the denominators by multiplying both sides by the common denominator of the fractions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "To find the common denominator, I need to factor the quadratic expressions in each denominator.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "I notice that all the quadratics have the same coefficient of x^2 and x, which is -10, so I can use a shortcut to factor them.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "I look at the constant term of each quadratic and try to find two factors that add up to -10.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "For the first quadratic, x^2-10x-29, I see that -29 can be factored as -1 and 29, and -1+29=28, which is not -10.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "So I try the other pair of factors, 1 and -29, and 1+(-29)=-28, which is also not -10. [*.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 153, "metadata": { "path_label": "find_the_positive_solution_to_frac_1_x_2_10x_29_frac1_x__2023-01-16T20-24-36.419882|chosen" } }, { "text": "I need to count the number of ways to assign five people to five seats, subject to some constraints.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "One way to do this is to start from the driver's seat, since it has the most restrictions.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "Only Mr. Lopez or Mrs. Lopez can sit there, so I have two choices for the driver's seat.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "Next, I look at the other front seat. The youngest child cannot sit there, so I have three choices for the other front seat, as long as I don't pick the same person as the driver.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "Now I have three seats left in the back row, and three people left to assign them to.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "The order matters, since the seats are distinct, so I can use the permutation formula to find the number of ways to arrange three people in three seats.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "That formula is 3! = 3 * 2 * 1 = 6.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "So for each of the two choices I made for the front seats, I have six choices for the back seats.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "That means the total number of seating arrangements is 2 * 3 * 6 = 36.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 154, "metadata": { "path_label": "mr_and_mrs_lopez_have_three_children_when_they_get_into__2023-01-16T06-16-59.353088|chosen" } }, { "text": "I want to find all the values of $n$ such that 3 is the largest common factor that divides both 15 and $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "I know that 15 is divisible by 3, so 3 must also divide $n$ for this to be possible.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "That means $n$ must be a multiple of 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "So I can write $n = 3k$ for some integer $k$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "Now I need to check when 3 is the only common factor of 15 and $n$, and not any larger number.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "I know that the other factors of 15 are 1 and 5, so I need to avoid any $k$ that makes $n$ divisible by 5 as well.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "That means $k$ cannot be a multiple of 5, or else $n = 3k$ would be divisible by 15.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "So I have to exclude the multiples of 5 from the possible values of $k$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "I also know that $n$ has to be between 1 and 100, so $k$ has to be between 1 and 33, since $3 \\cdot 33 = 99$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "So I can count how many values of $k$ there are in that range that are not multiples of 5.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "I can do this by dividing 33 by 5 and rounding down to get the number of multiples of 5, and then subtracting that from 33.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "So 33 divided by 5 is 6 with a remainder of 3, so there are 6 multiples of 5 in that range.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "Subtracting 6 from 33 gives me 27.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "So there are 27 values of $k$ that satisfy the conditions, and hence 27 values of $n$ that have 3 as the greatest common divisor with 15.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 155, "metadata": { "path_label": "for_how_many_integers_n_between_1_and_100_is_the_greatest__2023-01-20T04-35-47.649164|chosen" } }, { "text": "I know that angle bisectors divide the opposite side in the same ratio as the adjacent sides, so I can use that to find some segments.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 156, "metadata": { "path_label": "in_triangle_abc_angle_bisectors_overline_ad_and_ove__2023-01-19T11-09-48.117243|chosen" } }, { "text": "For example, since $\\overline{AD}$ bisects $\\angle BAC,$ I can write $\\frac{BD}{DC} = \\frac{AB}{AC} = \\frac{7}{5}.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 156, "metadata": { "path_label": "in_triangle_abc_angle_bisectors_overline_ad_and_ove__2023-01-19T11-09-48.117243|chosen" } }, { "text": "Similarly, since $\\overline{BE}$ bisects $\\angle ABC,$ I can write $\\frac{AE}{EC} = \\frac{AB}{BC} = \\frac{7}{3}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 156, "metadata": { "path_label": "in_triangle_abc_angle_bisectors_overline_ad_and_ove__2023-01-19T11-09-48.117243|chosen" } }, { "text": "Now, I can use these ratios to express $BD, DC, AE,$ and $EC$ in terms of $AB, AC,$ and $BC.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 156, "metadata": { "path_label": "in_triangle_abc_angle_bisectors_overline_ad_and_ove__2023-01-19T11-09-48.117243|chosen" } }, { "text": "This problem involves finding the probability of an event that depends on two continuous random variables: the time when the train arrives and the time when John arrives.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "One way to approach this is to use a geometric representation of the possible outcomes.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "I will draw a coordinate plane where the horizontal axis is the time when the train arrives, labeled from 2:00 to 3:00, and the vertical axis is the time when John arrives, also labeled from 2:00 to 3:00.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "Each point on this plane represents a possible outcome, and the probability of each outcome is equal to the area of a small rectangle around it, divided by the total area of the square from 2:00 to 3:00.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The event that the train is there when John arrives is represented by the points that lie on or below the line y = x + 20, where 20 represents the 20 minutes that the train waits.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The probability of this event is the fraction of the square that is below this line.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "To find this fraction, I need to find the area of the triangular region that is above the line and subtract it from the area of the square.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The area of the square is 60 x 60 = 3600, since there are 60 minutes from 2:00 to 3:00.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The area of the triangle is half the product of its base and height.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The base of the triangle is 40, since it goes from 2:20 to 3:00.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The height of the triangle is also 40, since it goes from 2:00 to 2:40.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "So the area of the triangle is 40 x 40 / 2 = 800.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The area of the region below the line is 3600 - 800 = 2800.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "The probability of the event is 2800 / 3600 = 7/9.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 157, "metadata": { "path_label": "john_needs_to_catch_a_train_the_train_arrives_randomly_some__2023-01-19T10-29-00.033039|chosen" } }, { "text": "I want to find the smallest possible value of $a$, which is the sum of the three roots of the polynomial.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 158, "metadata": { "path_label": "the_polynomial_x_3_ax_2_bx_2010_has_three_positive_int__2023-01-19T09-36-10.049771|chosen" } }, { "text": "So I need to find three positive integers that multiply to 2010 and add up to as small a number as possible.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 158, "metadata": { "path_label": "the_polynomial_x_3_ax_2_bx_2010_has_three_positive_int__2023-01-19T09-36-10.049771|chosen" } }, { "text": "I can start by factoring 2010 into prime factors: $2010 = 2 \\times 3 \\times 5 \\times 67$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 158, "metadata": { "path_label": "the_polynomial_x_3_ax_2_bx_2010_has_three_positive_int__2023-01-19T09-36-10.049771|chosen" } }, { "text": "Now I can try to group these factors into three sets, each representing one root.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 158, "metadata": { "path_label": "the_polynomial_x_3_ax_2_bx_2010_has_three_positive_int__2023-01-19T09-36-10.049771|chosen" } }, { "text": "I know that if $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients, then those factors must have the form $(5x+a)(x+b)$, where $a$ and $b$ are integers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "Expanding $(5x+a)(x+b)$, I get $5x^2+(a+5b)x+ab$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "Comparing this with $5x^2+nx+48$, I can see that I need to have $a+5b=n$ and $ab=48$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "I want to find the largest possible value of $n$, so I want to maximize $a+5b$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "I also know that $a$ and $b$ must be factors of 48, since $ab=48$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "The factors of 48 are $\\pm 1, \\pm 2, \\pm 3, \\pm 4, \\pm 6, \\pm 8, \\pm 12, \\pm 16, \\pm 24, \\pm 48$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "I can try different pairs of factors and see which one gives me the largest value of $a+5b$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "For example, if $a=1$ and $b=48$, then $a+5b=241$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "If $a=2$ and $b=24$, then $a+5b=122$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "If $a=3$ and $b=16$, then $a+5b=83$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "If $a=4$ and $b=12$, then $a+5b=64$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "If $a=6$ and $b=8$, then $a+5b=46$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 159, "metadata": { "path_label": "find_the_largest_value_of_n_such_that_5x_2_nx_48_can_be__2023-01-16T05-35-51.781490|chosen" } }, { "text": "To find the distance between the two intersections, I need to first find the coordinates of the intersections.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "I can do this by substituting $x=y^4$ into the equation $x+y^2=1$ and solving for $y$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "This gives me $y^4+y^2-1=0$, which is a quadratic equation in terms of $y^2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "I can use the quadratic formula to find $y^2=\\frac{-1\\pm\\sqrt{5}}{2}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "Since $y^2$ cannot be negative, I only consider the positive root, which is $y^2=\\frac{-1+\\sqrt{5}}{2}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "Taking the square root of both sides, I get $y=\\pm\\sqrt{\\frac{-1+\\sqrt{5}}{2}}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "To find the corresponding values of $x$, I plug these values of $y$ into the equation $x=y^4$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "This gives me $x=\\left(\\frac{-1+\\sqrt{5}}{2}\\right)^2=\\frac{3-\\sqrt{5}}{4}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "So the coordinates of the intersections are $\\left(\\frac{3-\\sqrt{5}}{4},\\sqrt{\\frac{-1+\\sqrt{5}}{2}}\\right)$ and $\\left(\\frac{3-\\sqrt{5}}{4},-\\sqrt{\\frac{-1+\\sqrt{5}}{2}}\\right)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "Now I can use the distance formula to find the distance between these two points.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "The distance formula is $d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "In this case, $x_1=x_2=\\frac{3-\\sqrt{5}}{4}$, so the first term in the formula is zero.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "The second term is $(y_2-y_1)^2=\\left(\\sqrt{\\frac{-1+\\sqrt{5}}{2}}+\\sqrt{\\frac{-1+\\sqrt{5}}{2}}\\right)^2=4\\left(\\frac{-1+\\sqrt{5}}{2}\\right)$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "So the distance is $d=\\sqrt{4\\left(\\frac{-1+\\sqrt{5}}{2}\\right)}=2\\sqrt{\\frac{-1+\\sqrt{5}}{2}}$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "To write this in the form $\\sqrt{u+v\\sqrt5}$, I need to rationalize the denominator by multiplying the numerator and denominator by $\\sqrt{2}$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 160, "metadata": { "path_label": "the_distance_between_the_two_intersections_of_x_y_4_and_x__2023-01-15T05-44-31.011160|chosen" } }, { "text": "This problem involves a cubic equation and a determinant of a 3x3 matrix.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 161, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_x_3_px_q_0_expre__2023-01-14T11-31-39.592945|chosen" } }, { "text": "I wonder if there is a connection between the roots of the cubic and the determinant.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 161, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_x_3_px_q_0_expre__2023-01-14T11-31-39.592945|chosen" } }, { "text": "One way to find out is to expand the determinant using the cofactor expansion along the first row.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 161, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_x_3_px_q_0_expre__2023-01-14T11-31-39.592945|chosen" } }, { "text": "This gives me \\[\\begin{vmatrix} 1 + a & 1 & 1 \\\\ 1 & 1 + b & 1 \\\\ 1 & 1 & 1 + c \\end{vmatrix} = (1 + a) \\begin{vmatrix} 1 + b & 1 \\\\ 1 & 1 + c \\end{vmatrix} - \\begin{vmatrix} 1 & 1 \\\\ 1 & 1 + c \\end{vmatrix} + \\begin{vmatrix} 1 & 1 + b \\\\ 1 & 1 \\end{vmatrix}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 161, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_x_3_px_q_0_expre__2023-01-14T11-31-39.592945|chosen" } }, { "text": "Simplifying the 2x2 determinants, I get \\[(1 + a)((1 + b)(1 + c) - 1) - (1 + c - 1) + (1 - 1 - b).\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 161, "metadata": { "path_label": "let_a_b_c_be_the_roots_of_x_3_px_q_0_expre__2023-01-14T11-31-39.592945|chosen" } }, { "text": "I know that the equation of a hyperbola with foci on the $x$-axis and center at the origin is $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1,$ where $a$ and $b$ are the semi-major and semi-minor axes, respectively.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "Since the vertices of $\\mathcal{H}$ are at $(\\pm 3, 0),$ I can deduce that $a = 3.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "To find $b,$ I need to use the relationship $c^2 = a^2 + b^2,$ where $c$ is the distance from the center to each focus.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "Since the foci of $\\mathcal{H}$ are at $(\\pm 5, 0),$ I can deduce that $c = 5.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "Plugging these values into the equation, I get $5^2 = 3^2 + b^2,$ which implies that $b^2 = 16.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "Therefore, the equation of $\\mathcal{H}$ is $\\frac{x^2}{9} - \\frac{y^2}{16} = 1.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "The equation of $\\mathcal{C}$ is simply $x^2 + y^2 = 16,$ since it has center $(0,0)$ and radius $4.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "To find the points of intersection, I need to solve the system of equations $\\frac{x^2}{9} - \\frac{y^2}{16} = 1$ and $x^2 + y^2 = 16.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "I notice that both equations have a $y^2$ term, so I can try to eliminate that by multiplying the first equation by $16$ and the second by $9$ and then adding them together.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 162, "metadata": { "path_label": "let_mathcal_h_be_the_hyperbola_with_foci_at_pm_5_0__2023-01-14T04-31-38.435974|chosen" } }, { "text": "This problem involves complex numbers and their powers, so I will use the polar form to simplify the calculations.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Let $r = \\sqrt{a^2 + b^2}$ be the modulus of $a + bi$ and $a - bi$, and let $\\theta = \\tan^{-1}(b/a)$ be their argument.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Then, by De Moivre's theorem, I can write $(a + bi)^n = r^n(\\cos(n\\theta) + i\\sin(n\\theta))$ and $(a - bi)^n = r^n(\\cos(n\\theta) - i\\sin(n\\theta))$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Since these two expressions are equal, I can equate their real and imaginary parts separately.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "For the real parts, I get $\\cos(n\\theta) = \\cos(n\\theta)$, which is always true.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "For the imaginary parts, I get $\\sin(n\\theta) = -\\sin(n\\theta)$, which implies that $\\sin(n\\theta) = 0$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "This means that $n\\theta$ must be a multiple of $\\pi$, or equivalently, $\\theta$ must be a multiple of $\\pi/n$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Now, recall that $\\theta = \\tan^{-1}(b/a)$, so $\\tan(\\theta) = b/a$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Therefore, $b/a$ must be equal to $\\tan(k\\pi/n)$ for some integer $k$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "However, the problem asks for the smallest positive integer $n$ that satisfies the given condition, so I want to minimize $k$ as well.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "Since $\\tan(\\theta) = b/a$ is positive, I can assume that $k$ is positive as well.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "The smallest possible value of $k$ is $1$, since $k = 0$ would make $\\tan(k\\pi/n) = 0$ and $b = 0$, which is not allowed by the problem.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 163, "metadata": { "path_label": "if_n_is_the_smallest_positive_integer_for_which_there_exis__2023-01-14T04-27-54.042103|chosen" } }, { "text": "I notice that the equation involves the floor function, which rounds down any fractional part of a number.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "This means that the equation is equivalent to saying that $2002/n$ is between $k$ and $k+1$, or $k<\\frac{2002}{n}\\frac{2002}{k+1}$ and $n<\\frac{2002}{k}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "I want to find the smallest $k$ such that there is no integer $n$ in this interval.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "One way to do this is to look at the lengths of the intervals for different values of $k$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "The length of the interval is $\\frac{2002}{k}-\\frac{2002}{k+1}=\\frac{2002}{k(k+1)}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "As $k$ increases, this length decreases, so the intervals get narrower and it becomes more likely that there is no integer $n$ in them.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 164, "metadata": { "path_label": "find_the_least_positive_integer_k_for_which_the_equation__2023-01-14T03-13-35.227329|chosen" } }, { "text": "This is an infinite series, so I need to see if it converges or diverges.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "If it converges, I can find its sum by using a formula or a trick.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "To check for convergence, I can use the ratio test.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "The ratio test says that if the absolute value of the ratio of consecutive terms is less than 1, then the series converges.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "Let me find the ratio of consecutive terms.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "The nth term is \\(\\frac{n}{10^n}\\), and the (n+1)st term is \\(\\frac{n+1}{10^{n+1}}\\).", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "The ratio of these terms is \\(\\frac{n+1}{10^{n+1}} \\div \\frac{n}{10^n} = \\frac{n+1}{10n}\\).", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "The absolute value of this ratio is \\(\\frac{n+1}{10n}\\).", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "I want to see if this ratio is less than 1 for all n. [* to be continued]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 165, "metadata": { "path_label": "find_the_sum_0_1_0_02_0_003_0_0004_dots_frac_n__2023-01-13T03-23-25.299673|chosen" } }, { "text": "I want to find an integer x that is less than 100 and has a greatest common factor of 3 with 18.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 166, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2023-01-13T03-09-26.270847|chosen" } }, { "text": "One way to do this is to list the multiples of 3 that are less than 100 and see which ones have a common factor with 18 other than 3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 166, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2023-01-13T03-09-26.270847|chosen" } }, { "text": "The multiples of 3 that are less than 100 are 3, 6, 9, 12, 15, 18, 21, ..., 99.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 166, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2023-01-13T03-09-26.270847|chosen" } }, { "text": "This problem involves a lot of doubling and subtracting, so I want to find a way to simplify it.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 167, "metadata": { "path_label": "stacy_has_d_dollars_she_enters_a_mall_with_10_shops_and__2023-01-12T12-33-29.870932|chosen" } }, { "text": "I notice that $1024$ is a power of $2$, specifically $2^{10}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 167, "metadata": { "path_label": "stacy_has_d_dollars_she_enters_a_mall_with_10_shops_and__2023-01-12T12-33-29.870932|chosen" } }, { "text": "This means that every time Stacy plays the lottery and doubles her money, she is multiplying it by $2$, and every time she spends $1024$ dollars, she is subtracting $2^{10}$ from it.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 167, "metadata": { "path_label": "stacy_has_d_dollars_she_enters_a_mall_with_10_shops_and__2023-01-12T12-33-29.870932|chosen" } }, { "text": "I can rewrite this process using exponents: after playing the lottery and spending money in the first shop, Stacy has $2d - 2^{10}$ dollars left.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 167, "metadata": { "path_label": "stacy_has_d_dollars_she_enters_a_mall_with_10_shops_and__2023-01-12T12-33-29.870932|chosen" } }, { "text": "I want to find the angle at the center of the Earth that corresponds to the arc between Ajay and Billy.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "I can use the fact that the length of an arc is proportional to the central angle subtended by it, with a proportionality constant of the radius of the sphere.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "I can also use the fact that the circumference of a circle is $2\\pi r$, where $r$ is the radius.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "To find the length of the arc, I need to know the distance along the surface of the Earth from $A$ to $B$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "I can use the formula for the great-circle distance between two points on a sphere, which is $r\\Delta\\sigma$, where $\\Delta\\sigma$ is the angular distance in radians.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "The angular distance depends on the latitudes and longitudes of the points. I can use the formula $\\Delta\\sigma = \\cos^{-1}(\\sin\\phi_1\\sin\\phi_2 + \\cos\\phi_1\\cos\\phi_2\\cos(\\lambda_2 - \\lambda_1))$, where $\\phi_1$ and $\\phi_2$ are the latitudes and $\\lambda_1$ and $\\lambda_2$ are the longitudes of the points.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "Plugging in the given values, I get $\\Delta\\sigma = \\cos^{-1}(\\sin 0^\\circ\\sin 45^\\circ + \\cos 0^\\circ\\cos 45^\\circ\\cos(115^\\circ - (-110^\\circ)))$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "Simplifying, I get $\\Delta\\sigma = \\cos^{-1}(\\frac{1}{\\sqrt{2}}\\cos 225^\\circ)$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "Using the fact that $\\cos 225^\\circ = -\\frac{1}{\\sqrt{2}}$, I get $\\Delta\\sigma = \\cos^{-1}(-\\frac{1}{2})$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "Using the inverse cosine function, I get $\\Delta\\sigma \\approx 2.0944$ radians.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 168, "metadata": { "path_label": "ajay_is_standing_at_point_a_near_pontianak_indonesia_0__2023-01-12T10-01-37.872870|chosen" } }, { "text": "To be divisible by 12, a number must be divisible by both 3 and 4.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 169, "metadata": { "path_label": "the_five_digit_number_n_14_9ab_is_divisible_by_12_giv__2023-01-10T17-03-25.583596|chosen" } }, { "text": "A number is divisible by 3 if the sum of its digits is divisible by 3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 169, "metadata": { "path_label": "the_five_digit_number_n_14_9ab_is_divisible_by_12_giv__2023-01-10T17-03-25.583596|chosen" } }, { "text": "A number is divisible by 4 if its last two digits are divisible by 4.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 169, "metadata": { "path_label": "the_five_digit_number_n_14_9ab_is_divisible_by_12_giv__2023-01-10T17-03-25.583596|chosen" } }, { "text": "So, I need to find the smallest values of $A$ and $B$ that satisfy these conditions.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 169, "metadata": { "path_label": "the_five_digit_number_n_14_9ab_is_divisible_by_12_giv__2023-01-10T17-03-25.583596|chosen" } }, { "text": "This problem involves finding the maximum value of a difference of two square roots.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "I know that square roots are always nonnegative, so the maximum value of the difference will occur when the first square root is as large as possible and the second square root is as small as possible.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "I also know that square roots have zeros when their arguments are zero, and they are undefined when their arguments are negative.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "So I want to find the values of x that make the arguments of both square roots zero or positive.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "For the first square root, I have 8x - x^2, which I can factor as x(8 - x).", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "This expression is zero when x = 0 or x = 8, and it is positive when x is between 0 and 8.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 170, "metadata": { "path_label": "find_the_maximum_value_of_f_x_sqrt_8x_x_2_sqrt_1__2023-01-16T20-33-04.777236|chosen" } }, { "text": "I notice that the equation involves powers of 2 and 3, which are both prime numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "This means that any solution must have the same prime factorization as 55, which is $5 \\times 11$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "I also notice that the equation has the form $a^2-b^2$, which can be factored as $(a-b)(a+b)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "So I can rewrite the equation as $(2^x-3^y)(2^x+3^y)=55$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "Now I have two factors on the left side that multiply to 55.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "Since 55 is not very large, I can try to list all the pairs of factors that multiply to 55.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "They are $(1, 55)$, $(5, 11)$, $(11, 5)$, and $(55, 1)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "I can ignore the negative factors, since $2^x$ and $3^y$ are always positive for integer $x$ and $y$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "For each pair of factors, I can set them equal to $2^x-3^y$ and $2^2+3^y$, and see if I can find integer solutions for $x$ and $y$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "If I try $(1, 55)$, I get $2^x-3^y=1$ and $2^x+3^y=55$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "Adding these equations, I get $2^{x+1}=56$, which has no integer solutions, since 56 is not a power of 2.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "If I try $(5, 11)$, I get $2^x-3^y=5$ and $2^x+3^y=11$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "Adding these equations, I get $2^{x+1}=16$, which has the solution $x=3$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "Substituting this into the first equation, I get $8-3^y=5$, which has the solution $y=1$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "So $(x, y)=(3, 1)$ is one solution.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "If I try $(11, 5)$, I get $2^x-3^y=11$ and $2^x+3^y=5$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 171, "metadata": { "path_label": "the_number_of_solutions_of_2_2x_3_2y_55_in_which_x__2023-01-10T07-04-36.485055|chosen" } }, { "text": "I want to find the values of $x$ that make the numerator and denominator of the fraction have the same sign, or make the numerator zero.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "To do that, I need to factor the numerator and denominator as much as possible.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "I notice that the numerator has a common factor of $x$, so I can write it as \\[x(1-10x+25x^2)\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "I also notice that the expression in the parentheses is a perfect square, since it is equal to $(5x-1)^2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "So I can rewrite the numerator as \\[x(5x-1)^2\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "For the denominator, I recognize that it is a difference of cubes, which has a special factorization formula: \\[a^3-b^3 = (a-b)(a^2+ab+b^2)\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "In this case, $a=2$ and $b=x$, so I can write the denominator as \\[(2-x)(4+2x+x^2)\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "Now I have factored the fraction as \\[\\frac{x(5x-1)^2}{(2-x)(4+2x+x^2)}\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "I want this fraction to be nonnegative, which means either both factors in the numerator and denominator are positive, or both are negative, or the numerator is zero.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "To find the values of $x$ that satisfy these conditions, I need to consider the zeros of each factor and the intervals between them.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 172, "metadata": { "path_label": "for_what_values_of_x_is_frac_x_10x_2_25x_3_8_x_3_non__2023-01-10T06-38-15.777842|chosen" } }, { "text": "To solve this problem, I need to recall the condition for four points to be coplanar in terms of vectors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "I remember that four points $A,$ $B,$ $C,$ and $D$ are coplanar if and only if the vectors $\\overrightarrow{AB},$ $\\overrightarrow{AC},$ and $\\overrightarrow{AD}$ are linearly dependent.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "This means that there exist scalars $x,$ $y,$ and $z,$ not all zero, such that \\[x \\overrightarrow{AB} + y \\overrightarrow{AC} + z \\overrightarrow{AD} = \\mathbf{0}.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "I can rewrite this equation in terms of the position vectors $\\overrightarrow{OA},$ $\\overrightarrow{OB},$ $\\overrightarrow{OC},$ and $\\overrightarrow{OD}$ as follows: \\[x (\\overrightarrow{OB} - \\overrightarrow{OA}) + y (\\overrightarrow{OC} - \\overrightarrow{OA}) + z (\\overrightarrow{OD} - \\overrightarrow{OA}) = \\mathbf{0}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "Expanding and rearranging, I get \\[(-x - y - z) \\overrightarrow{OA} + x \\overrightarrow{OB} + y \\overrightarrow{OC} + z \\overrightarrow{OD} = \\mathbf{0}.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "Comparing this equation with the given equation, I notice that they have the same form, except for the coefficients of the vectors.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "This means that I can equate the coefficients of the corresponding vectors and get a system of four equations and four unknowns: \\[\\begin{cases} -x - y - z &= 3 \\\\ x &= -2 \\\\ y &= 5 \\\\ z &= k \\end{cases}\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "To solve this system, I can use substitution or elimination.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "I start by substituting $x = -2$ and $y = 5$ into the first equation and get \\[-(-2) - 5 - z = 3,\\] which simplifies to \\[-3 - z = 3.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "Solving for $z,$ I get \\[z = -6.\\]", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "Since $z = k,$ I can conclude that \\[k = -6.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 173, "metadata": { "path_label": "let_o_be_the_origin_there_exists_a_scalar_k_so_that_fo__2023-01-10T06-32-20.585568|chosen" } }, { "text": "I notice that the equation involves both x and \\sqrt{x}, so I wonder if I can simplify it by using a substitution.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 174, "metadata": { "path_label": "compute_the_sum_of_the_roots_of_the_equation_x_sqrt_x_6__2023-01-10T06-29-05.468648|chosen" } }, { "text": "Let me try letting y = \\sqrt{x}, then I can rewrite the equation as \\[y^3 - 6y^2 + 7y - 1 = 0.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 174, "metadata": { "path_label": "compute_the_sum_of_the_roots_of_the_equation_x_sqrt_x_6__2023-01-10T06-29-05.468648|chosen" } }, { "text": "This looks like a cubic equation, which might be hard to solve in general, but maybe it has some nice properties.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 174, "metadata": { "path_label": "compute_the_sum_of_the_roots_of_the_equation_x_sqrt_x_6__2023-01-10T06-29-05.468648|chosen" } }, { "text": "I want to find the sum of three arctan values, which are inverse trigonometric functions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "I recall that the arctan function gives the angle whose tangent is a given ratio of opposite to adjacent sides in a right triangle.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "So, for example, $\\arctan 1$ is the angle whose tangent is 1, which means it is the angle in a right triangle with equal sides.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "I also recall that $\\arctan 1$ is equal to $\\frac{\\pi}{4}$ radians, since it is one-fourth of a full revolution.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "Similarly, I can find $\\arctan \\frac{1}{2}$ and $\\arctan \\frac{1}{3}$ by thinking of right triangles with those ratios of sides.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "Alternatively, I can use the identity $\\arctan x + \\arctan y = \\arctan \\frac{x + y}{1 - xy}$, which holds for any $x$ and $y$ such that $xy < 1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "Using this identity, I can rewrite the sum as $\\arctan 1 + \\arctan \\frac{1}{2} + \\ar", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "Now, I can apply the identity again to the first two terms, and get $\\arctan \\frac{1 + \\frac{1}{2}}{1 - 1 \\cdot \\frac{1}{2}} + \\arctan \\frac{1}{3}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 175, "metadata": { "path_label": "compute_arctan_1_arctan_frac_1_2_arctan_frac_1__2023-01-09T08-41-45.211610|chosen" } }, { "text": "I want to find a point $P$ on the line $BC$ such that $AP$ is perpendicular to $BC.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "To do that, I need to find the equation of the line $BC$ and the vector $\\vec{AP}$ and then set their dot product equal to zero.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "The equation of the line $BC$ can be written in parametric form as $x = 0 + 2t, y = -1 - 2t, z = 3 - 4t,$ where $t$ is a parameter.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "The vector $\\vec{AP}$ can be written as $(x-1, y-8, z-4),$ where $(x,y,z)$ is the point $P$ on the line $BC.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "The dot product of $\\vec{AP}$ and $\\vec{BC}$ is $(x-1, y-8, z-4) \\cdot (2,-2,-4) = 2x - 2 - 2y + 16 - 4z + 16.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "Setting this equal to zero, I get $2x - 2y - 4z + 30 = 0.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "To find $x, y, z,$ I can plug in the expressions for $x, y, z$ in terms of $t$ into this equation.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 176, "metadata": { "path_label": "find_the_foot_of_the_perpendicular_from_the_point_a_1_8__2023-01-09T08-10-16.468961|chosen" } }, { "text": "I know that a repeating decimal can be converted to a fraction by using a system of equations.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 177, "metadata": { "path_label": "what_common_fraction_that_is_a_fraction_reduced_to_its_low__2023-01-09T07-21-08.955118|chosen" } }, { "text": "Let me first write $.3\\overline{25}$ as a sum of two decimals: $.3$ and $.0\\overline{25}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 177, "metadata": { "path_label": "what_common_fraction_that_is_a_fraction_reduced_to_its_low__2023-01-09T07-21-08.955118|chosen" } }, { "text": "I can find the fraction for $.3$ easily: it's just $\\frac{3}{10}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 177, "metadata": { "path_label": "what_common_fraction_that_is_a_fraction_reduced_to_its_low__2023-01-09T07-21-08.955118|chosen" } }, { "text": "To find the fraction for $.0\\overline{25}$, I need to use a trick.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 177, "metadata": { "path_label": "what_common_fraction_that_is_a_fraction_reduced_to_its_low__2023-01-09T07-21-08.955118|chosen" } }, { "text": "Let x be the decimal part of $.0\\overline{25}$, so x = $.0\\overline{25}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 177, "metadata": { "path_label": "what_common_fraction_that_is_a_fraction_reduced_to_its_low__2023-01-09T07-21-08.955118|chosen" } }, { "text": "I want to find the largest possible product of $x$ and $y$, given that they are positive integers and satisfy the equation $5x+3y=100$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "One way to approach this problem is to look at the possible values of $x$ and $y$ that make the equation true, and see which pair gives the largest product.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "To do this, I can use the fact that $5x+3y=100$ implies that $y$ is a multiple of $5$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "So, I can write $y=5k$ for some positive integer $k$, and then substitute it into the equation to get $5x+15k=100$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "Dividing both sides by $5$, I get $x+3k=20$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "Now, I can solve for $x$ in terms of $k$: $x=20-3k$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "This means that the product $xy$ is equal to $(20-3k)(5k)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "To find the maximum value of this product, I can use the distributive property to expand it as $100k-15k^2$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "This is a quadratic function of $k$, and I know that it has a maximum value when $k$ is equal to the average of the roots.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "The roots of this function are the values of $k$ that make it zero, which are $k=0$ and $k=\\frac{20}{3}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "The average of these roots is $\\frac{10}{3}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "However, $k$ has to be a positive integer, so I cannot use $\\frac{10}{3}$ as a possible value of $k$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "The closest integer to $\\frac{10}{3}$ that is positive is $k=3$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "So, I will try $k=3$ as a possible value of $k$ that maximizes the product.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "When $k=3$, I get $x=20-3k=20-9=11$ and $y=5k=5(3)=15$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "The product $xy$ is then $11(15)=165$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "This is the greatest possible value of $xy$, since any other value of $k$ that is an integer would either make $x$ negative or make the product smaller.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 178, "metadata": { "path_label": "if_x_and_y_are_positive_integers_such_that_5x_3y_100__2023-01-08T11-05-29.392026|chosen" } }, { "text": "I notice that the two graphs are reciprocals of each other, so they are symmetric about the line $y = x.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 179, "metadata": { "path_label": "find_the_area_of_the_region_bounded_by_the_graph_of_r_se__2023-01-08T06-34-30.265265|chosen" } }, { "text": "This means that the region bounded by them is a quadrilateral with two right angles and two equal sides.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 179, "metadata": { "path_label": "find_the_area_of_the_region_bounded_by_the_graph_of_r_se__2023-01-08T06-34-30.265265|chosen" } }, { "text": "I can use the formula for the area of a kite to find the area of this region.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 179, "metadata": { "path_label": "find_the_area_of_the_region_bounded_by_the_graph_of_r_se__2023-01-08T06-34-30.265265|chosen" } }, { "text": "The area of a kite is half the product of the lengths of its diagonals.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 179, "metadata": { "path_label": "find_the_area_of_the_region_bounded_by_the_graph_of_r_se__2023-01-08T06-34-30.265265|chosen" } }, { "text": "I need to find a plane that contains both the line $L$ and the point $(1,1,1).$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 180, "metadata": { "path_label": "let_line_l_be_the_intersection_of_the_planes_x_y_z__2023-01-08T06-00-36.918208|chosen" } }, { "text": "To do that, I need to know two things: a normal vector to the plane, and a point on the plane.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 180, "metadata": { "path_label": "let_line_l_be_the_intersection_of_the_planes_x_y_z__2023-01-08T06-00-36.918208|chosen" } }, { "text": "I can use the point $(1,1,1)$ as a point on the plane, since it is given.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 180, "metadata": { "path_label": "let_line_l_be_the_intersection_of_the_planes_x_y_z__2023-01-08T06-00-36.918208|chosen" } }, { "text": "I need to count how many numbers are divisible by 2 or 3, but not by 4.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 181, "metadata": { "path_label": "how_many_positive_integers_not_exceeding_100_are_multiples__2023-01-04T21-22-54.669829|chosen" } }, { "text": "One way to do this is to use the inclusion-exclusion principle: the number of elements in a union of two sets is equal to the sum of the numbers of elements in each set, minus the number of elements in their intersection.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 181, "metadata": { "path_label": "how_many_positive_integers_not_exceeding_100_are_multiples__2023-01-04T21-22-54.669829|chosen" } }, { "text": "In this case, the sets are the multiples of 2 and the multiples of 3, and their union is the multiples of 2 or 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 181, "metadata": { "path_label": "how_many_positive_integers_not_exceeding_100_are_multiples__2023-01-04T21-22-54.669829|chosen" } }, { "text": "The intersection of the sets is the multiples of 6, since 6 is the least common multiple of 2 and 3.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 181, "metadata": { "path_label": "how_many_positive_integers_not_exceeding_100_are_multiples__2023-01-04T21-22-54.669829|chosen" } }, { "text": "So, the number of multiples of 2 or 3 is the number of multiples of 2 plus the number of multiples of 3, minus the number of multiples of 6.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 181, "metadata": { "path_label": "how_many_positive_integers_not_exceeding_100_are_multiples__2023-01-04T21-22-54.669829|chosen" } }, { "text": "I notice that $F(p, q)$ is a linear function of $q$, so its maximum over the interval $[0, 1]$ occurs either at $q = 0$ or $q = 1$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "To decide which one, I can compare the values of $F(p, 0)$ and $F(p, 1)$ for any fixed $p$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "If I plug in $q = 0$, I get $F(p, 0) = 3p - 4(1-p) = 7p - 4$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "If I plug in $q = 1$, I get $F(p, 1) = -2p + 3(1-p) = -5p + 3$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "I can see that $F(p, 0) > F(p, 1)$ if and only if $7p - 4 > -5p + 3$, which simplifies to $p > \\frac{7}{12}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "Therefore, for $p \\le \\frac{7}{12}$, the maximum of $F(p, q)$ occurs at $q = 1$, and for $p > \\frac{7}{12}$, it occurs at $q = 0$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "This means that $G(p) = F(p, 1)$ for $p \\le \\frac{7}{12}$ and $G(p) = F(p, 0)$ for $p > \\frac{7}{12}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "Now I want to find the value of $p$ that minimizes $G(p)$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 182, "metadata": { "path_label": "if_0_le_p_le_1_and_0_le_q_le_1_define_f_p_q_by__2023-01-04T11-03-27.526548|chosen" } }, { "text": "I notice that both equations involve $x$ and $y$, so I might try to eliminate one of them by subtracting or adding the equations.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "If I subtract the second equation from the first, I get $\\sin y - 2008 \\cos y = 1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "This looks like a trigonometric identity, so I try to recall if there is a way to simplify it.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "I remember that $\\sin (A-B) = \\sin A \\cos B - \\cos A \\sin B$, so maybe I can rewrite the left-hand side as a sine of some angle.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "To do that, I need to find the coefficients $a$ and $b$ such that $a^2 + b^2 = 1$ and $a = 2008 b$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "This is a system of two equations in two variables, which I can solve by substitution.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "If I plug in $a = 2008 b$ into the first equation, I get $(2008 b)^2 + b^2 = 1$, which simplifies to $4032065 b^2 = 1$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 183, "metadata": { "path_label": "given_that_x_sin_y_2008_and_x_2008_cos_y_2007_where__2023-01-12T07-23-57.555785|chosen" } }, { "text": "I need to find a relationship between $n$ and the number of drummers, so I will start by comparing the areas of the two rectangles.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "The area of the first rectangle is $(n-2)(n+8)$, and the area of the second rectangle is $n(2n-3)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "The difference between these areas is the number of drummers, since they are not included in the second rectangle.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "So I can write an equation: $(n-2)(n+8) - n(2n-3) = d$, where $d$ is the number of drummers.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "I will expand and simplify this equation: $n^2 + 6n - 16 - 2n^2 + 3n = d$, or $-n^2 + 9n - 16 = d$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "I want to find the sum of all possible values of $n$, so I need to know the constraints on $n$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "One constraint is that $n$ must be a positive integer, since it is the number of members in one side of a rectangle.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "Another constraint is that $d$ must be at least 4, since there are at least 4 drummers.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "So I can write an inequality: $-n^2 + 9n - 16 \\geq 4$, or $n^2 - 9n + 20 \\leq 0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "I will factor this quadratic inequality: $(n-4)(n-5) \\leq 0$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "This means that $n$ must be between 4 and 5, inclusive, to satisfy the inequality.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "But since $n$ is an integer, the only possible values are 4 and 5.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "Therefore, the sum of all possible values of $n$ is 4 + 5 = 9.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 184, "metadata": { "path_label": "a_band_is_marching_in_a_rectangular_formation_with_dimension__2023-01-12T07-21-31.443136|chosen" } }, { "text": "I need to find a way to round the seven numbers $a_i$ to integers $A_i$ such that the sum of the $A_i$'s is still $19$ and the errors $|A_i - a_i|$ are as small as possible.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "One possible way to round the $a_i$'s is to use the nearest integer function, which rounds a number to the closest integer, and breaks ties by rounding up.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "For example, using the nearest integer function, I would round $2.56$ to $3,$ $2.61$ to $3,$ $2.65$ to $3,$ $2.71$ to $3,$ $2.79$ to $3,$ $2.82$ to $3,$ and $2.86$ to $3.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "However, this would not work, because the sum of the $A_i$'s would be $3 + 3 + 3 + 3 + 3 + 3 + 3 = 21,$ which is not $19.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "So I need to adjust some of the $A_i$'s to make the sum equal to $19.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "One way to do that is to round some of the $a_i$'s down instead of up, even if they are closer to the next integer.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "For example, I could round $2.56$ to $2,$ $2.61$ to $2,$ $2.65$ to $3,$ $2.71$ to $3,$ $2.79$ to $3,$ $2.82$ to $3,$ and $2.86$ to $3.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "This would give me $A_i$'s that sum to $2 + 2 + 3 + 3 + 3 + 3 + 3 = 19,$ as desired.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "The errors in this case would be $|2 - 2.56| = 0.56,$ $|2 - 2.61| = 0.61,$ $|3 - 2.65| = 0.35,$ $|3 - 2.71| = 0.29,$ $|3 - 2.79| = 0.21,$ $|3 - 2.82| = 0.18,$ and $|3 - 2.86| = 0.14.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "The maximum of these errors is $0.61,$ so this would be one possible value of $M.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "Is this the smallest possible value of $M,$ or can I do better?", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "I notice that the $a_i$'s are all between $2.5$ and $3,$ so the only possible values for the $A_i$'s are $2$ and $3.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "Therefore, to minimize the errors, I want to round as many $a_i$'s as possible to their nearest integer, and only round down the ones that are farther from $3.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "I also notice that the sum of the $a_i$'s is $19.00,$ so I need to round down exactly two of them to get the sum of the $A_i$'s to be $19.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "Which two $a_i$'s should I round down?", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "The ones that are farthest from $3,$ of course.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "Those are $a_1 = 2.56$ and $a_2 = 2.61.$", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "So I round $a_1$ to $2,$ $a_2$ to $2,$ and the rest to $3.$", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "This gives me $A_i$'s that sum to $2 + 2 + 3 + 3 + 3 + 3 + 3 = 19,$ as desired.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "The errors in this case are $|2 - 2.56| = 0.56,$ $|2 - 2.61| = 0.61,$ $|3 - 2.65| = 0.35,$ $|3 - 2.71| = 0.29,$ $|3 - 2.79| = 0.21,$ $|3 - 2.82| = 0.18,$ and $|3 - 2.86| = 0.14.$", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "The maximum of these errors is still $0.61,$ so this is the smallest possible value of $M.$", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 185, "metadata": { "path_label": "the_sum_of_the_following_seven_numbers_is_exactly_19_a_1__2023-01-11T23-47-45.891955|chosen" } }, { "text": "I need to count the number of different arrangements of balls and boxes, but I don't care about the order of the balls within a box, only the colors and the quantities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "One way to approach this problem is to consider how many ways I can distribute the green balls among the boxes, and then multiply that by how many ways I can distribute the red balls among the boxes.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "To distribute the green balls, I can either put both in the same box, or split them into different boxes.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "If I put both in the same box, there are 3 choices for which box to use.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "If I split them into different boxes, there are 3 choices for the first box, and then 2 choices for the second box, but I have to divide by 2 to avoid counting the same pair twice.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "So, the total number of ways to distribute the green balls is 3 + 3 * 2 / 2 = 6.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "Similarly, the total number of ways to distribute the red balls is also 6.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "Therefore, the total number of ways to put 4 balls in 3 boxes is 6 * 6 = 36.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 186, "metadata": { "path_label": "how_many_ways_are_there_to_put_4_balls_in_3_boxes_if_two_bal__2023-01-11T05-18-34.410230|chosen" } }, { "text": "Let the length and width of the rectangle be l and w, respectively.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 187, "metadata": { "path_label": "a_non_square_rectangle_has_integer_dimensions_the_number_o__2023-01-11T04-09-20.523355|chosen" } }, { "text": "Then the area is lw and the perimeter is 2(l+w).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 187, "metadata": { "path_label": "a_non_square_rectangle_has_integer_dimensions_the_number_o__2023-01-11T04-09-20.523355|chosen" } }, { "text": "The problem says that lw = 3(2l+2w), or equivalently, lw - 6l - 6w = 0.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 187, "metadata": { "path_label": "a_non_square_rectangle_has_integer_dimensions_the_number_o__2023-01-11T04-09-20.523355|chosen" } }, { "text": "I want to find the smallest positive integer values of l and w that satisfy this equation.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 187, "metadata": { "path_label": "a_non_square_rectangle_has_integer_dimensions_the_number_o__2023-01-11T04-09-20.523355|chosen" } }, { "text": "One way to approach this is to factor the equation as (l-6)(w-6) = 36.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 187, "metadata": { "path_label": "a_non_square_rectangle_has_integer_dimensions_the_number_o__2023-01-11T04-09-20.523355|chosen" } }, { "text": "I know that the distance from the center of the ellipse to each focus is called the focal length, and it is related to the major and minor axes of the ellipse by the formula $c^2 = a^2 - b^2,$ where $c$ is the focal length, $a$ is the semi-major axis, and $b$ is the semi-minor axis.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Since the ellipse is tangent to both axes, the center must be at the midpoint of the segment joining the foci. So the center is at $(\\frac{3+d}{2},7).$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "The focal length is then $c = \\frac{d-3}{2}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "The semi-major axis is the distance from the center to the vertex on the $x$-axis, which is $a = \\frac{3+d}{2}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "The semi-minor axis is the distance from the center to the vertex on the $y$-axis, which is $b = 7.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Plugging these values into the formula, I get $(\\frac{d-3}{2})^2 = (\\frac{3+d}{2})^2 - 7^2.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Expanding and simplifying, I get $d^2 - 6d + 9 = d^2 + 6d + 9 - 196.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Subtracting $d^2 + 9$ from both sides, I get $-12d = -196.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Dividing both sides by $-12,$ I get $d = \\frac{196}{12}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "Simplifying the fraction, I get $d = \\frac{49}{3}.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 188, "metadata": { "path_label": "an_ellipse_in_the_first_quadrant_is_tangent_to_both_the_x__2023-01-11T03-54-11.684832|chosen" } }, { "text": "I know that a number is divisible by 3 if the sum of its digits is divisible by 3.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "So I want to find the smallest four-digit number that is a palindrome and has a digit sum that is a multiple of 3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "The smallest four-digit palindrome is 1001, but the sum of its digits is 2, which is not divisible by 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "The next smallest palindrome is 1111, but the sum of its digits is 4, which is also not divisible by 3.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "The next one is 1221, but the sum of its digits is 6, which is divisible by 3.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "So 1221 is the smallest four-digit palindrome that is divisible by 3.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 189, "metadata": { "path_label": "palindromes_are_numbers_that_read_the_same_backwards_and_for__2023-01-11T03-24-24.319803|chosen" } }, { "text": "I know that the length of the major axis of an ellipse is twice the distance from the center to either vertex, where the vertices are the endpoints of the major axis.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "To find the center of the ellipse, I can use the fact that it is the midpoint of the segment joining the foci.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "The midpoint formula is $(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}),$ so the center of the ellipse is $(\\frac{2 + 2}{2}, \\frac{-3 + \\sqrt{5} - 3 - \\sqrt{5}}{2}) = (2, -3).$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "Since the ellipse is tangent to both axes, I can also use the fact that the distance from the center to either vertex is equal to the distance from the center to either axis.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "For example, the distance from the center to the vertex on the positive $x$-axis is the same as the distance from the center to the positive $x$-axis, which is $|2 - 0| = 2.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "Similarly, the distance from the center to the vertex on the positive $y$-axis is the same as the distance from the center to the positive $y$-axis, which is $|(-3) - 0| = 3.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "Therefore, the length of the major axis is twice the longer of these two distances, which is $2 \\cdot 3 = 6.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 190, "metadata": { "path_label": "a_certain_ellipse_is_tangent_to_both_the_x_axis_and_the_y__2023-01-11T02-49-55.574956|chosen" } }, { "text": "I need to find the number of permutations of six items, but with a constraint that one item must be after another.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "I can think of this as choosing a position for Dr. Jones, and then choosing a position for Dr. Smith among the remaining slots that are after Dr. Jones.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "For example, if Dr. Jones is in the first slot, then Dr. Smith can be in any of the five remaining slots. If Dr. Jones is in the second slot, then Dr. Smith can be in any of the four remaining slots, and so on.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "So the total number of ways to schedule Dr. Jones and Dr. Smith is 5 + 4 + 3 + 2 + 1, which is 15.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "Once I have scheduled Dr. Jones and Dr. Smith, I can schedule the other four lecturers in any order among the four remaining slots, which is 4! = 24 ways.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "So the total number of ways to schedule all six lecturers is 15 times 24, which is 360.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 191, "metadata": { "path_label": "a_math_conference_is_presenting_a_lecture_series_with_six_di__2023-01-04T10-38-40.656819|chosen" } }, { "text": "I want to find the midpoint of $\\overline{AB}$ and the distance between it and either $A$ or $B$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "The midpoint of $\\overline{AB}$ has coordinates $\\left(\\frac{t-4-2}{2},\\frac{-1+t+3}{2}\\right) = \\left(\\frac{t-6}{2},\\frac{t+2}{2}\\right)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "The distance between this midpoint and $A$ is $\\sqrt{\\left(\\frac{t-6}{2} - (t-4)\\right)^2 + \\left(\\frac{t+2}{2} - (-1)\\right)^2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "I want to simplify this expression and square it.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "I get $\\left(\\frac{t-6}{2} - (t-4)\\right)^2 + \\left(\\frac{t+2}{2} - (-1)\\right)^2 = \\left(-\\frac{t-2}{2}\\right)^2 + \\left(\\frac{t+4}{2}\\right)^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "Squaring and expanding the binomials, I get $\\frac{(t-2)^2}{4} + \\frac{(t+4)^2}{4}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "Combining like terms, I get $\\frac{t^2 - 4t + 4 + t^2 + 8t + 16}{4} = \\frac{2t^2 + 4t + 20}{4}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 192, "metadata": { "path_label": "let_a_b_be_the_points_on_the_coordinate_plane_with_coordin__2023-01-04T09-17-39.417410|chosen" } }, { "text": "I want to compare the number of times the digit 4 appears in the page numbers with the number of times the digit 8 appears.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "One way to do this is to count how many times each digit appears in each place value: ones, tens, and hundreds.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "For the ones place, I can use the fact that the page numbers repeat the same pattern of digits every 10 pages: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "So, in every 10 pages, there is one 4 and one 8 in the ones place.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "Since there are 488 pages, that means there are 48 full cycles of 10 pages, and then 8 extra pages at the end.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "So, in the ones place, there are 48 times 1, which is 48, 4's and 8's, plus one extra 4 and one extra 8 in the last 8 pages.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "That means there are 49 4's and 49 8's in the ones place, so they cancel out.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "For the tens place, I can use a similar idea, but the pattern of digits repeats every 100 pages: 10, 11, 12, ..., 19, 20, 21, ..., 29, ..., 90, 91, ..., 99, 100.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "In every 100 pages, there are 10 4's and 10 8's in the tens place, because each digit appears once for each ones digit from 0 to 9.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "Since there are 488 pages, that means there are 4 full cycles of 100 pages, and then 88 extra pages at the end.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 193, "metadata": { "path_label": "each_page_number_of_a_488_page_book_is_printed_one_time_in_t__2023-01-04T06-54-40.588694|chosen" } }, { "text": "I need to count the number of possible programs that satisfy the given requirements.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "Since English is required, I can ignore it for now and focus on the other three courses.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "I have five choices for the other courses: Algebra, Geometry, History, Art, and Latin.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "However, I must have at least one mathematics course, so I cannot choose all three courses from History, Art, and Latin.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "This means I have two cases to consider: either I choose one mathematics course and two non-mathematics courses, or I choose two mathematics courses and one non-mathematics course.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "In the first case, I have two choices for the mathematics course: Algebra or Geometry.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "For each of these choices, I have three choices for the first non-mathematics course: History, Art, or Latin.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "Then, for each of these choices, I have two choices for the second non-mathematics course: any of the remaining two courses.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 194, "metadata": { "path_label": "a_student_must_choose_a_program_of_four_courses_from_a_list__2023-01-03T09-45-04.752218|chosen" } }, { "text": "I notice that the addresses on the south side increase by 4 each time, and the addresses on the north side also increase by 4 each time.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "This means that the addresses on the south side are of the form 1 + 4k, where k is a nonnegative integer, and the addresses on the north side are of the form 3 + 4k, where k is a nonnegative integer.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "I also notice that the addresses on the south side are all odd, and the addresses on the north side are all odd.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "This implies that there are no houses with even addresses, and therefore no houses with two-digit addresses that are multiples of 10, such as 20, 30, etc.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "This also implies that there are no houses with three-digit addresses that are multiples of 100, such as 100, 200, etc.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "These observations simplify the problem, because I don't have to consider cases where the number of digits in the address changes.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "Now, I need to figure out how many digits are in each address.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "I can use the fact that there are 50 houses on each side of the street, and that the addresses increase by 4 each time, to find the largest address on each side.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "On the south side, the largest address is 1 + 4(49) = 197, which has three digits.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "On the north side, the largest address is 3 + 4(49) = 199, which also has three digits.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "This means that every address on the street has either one, two, or three digits.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "I can use the same formula to find the smallest address with two digits on each side.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "On the south side, the smallest address with two digits is 1 + 4k, where k is the smallest integer such that 1 + 4k >= 10. This implies that k >= 2.25, so the smallest integer k that works is 3. Therefore, the smallest address with two digits on the south side is 1 + 4(3) = 13.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "On the north side, the smallest address with two digits is 3 + 4k, where k is the smallest integer such that 3 + 4k >= 10. This implies that k >= 1.75, so the smallest integer k that works is 2. Therefore, the smallest address with two digits on the north side is 3 + 4(2) = 11.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "Now, I can count how many addresses have one digit, two digits, and three digits on each side.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "On the south side, the addresses with one digit are 1, 5, and 9, so there are 3 of them. The addresses with two digits are 13, 17, ..., 97, so there are (97 - 13) / 4 + 1 = 22 of them. The addresses with three digits are 101, 105, ..., 197, so there are (197 - 101) / 4 + 1 = 25 of them.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 195, "metadata": { "path_label": "a_street_has_50_houses_on_each_side_for_a_total_of_100_hous__2023-01-03T07-24-23.344793|chosen" } }, { "text": "To solve this equation, I want to get rid of the fractions by finding a common denominator.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "One way to do that is to multiply both sides by the product of all the denominators: (x + 9)(x + 7)(x + 10)(x + 6).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "This way, each fraction will cancel out with one of the factors, and I'll be left with a polynomial equation.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "So, after multiplying both sides by (x + 9)(x + 7)(x + 10)(x + 6), I get: (x + 7)(x + 10)(x + 6) + (x + 9)(x + 10)(x + 6) = (x + 9)(x + 7)(x + 6) + (x + 9)(x + 7)(x + 10).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "Now, I need to expand and simplify this equation.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "To do that, I'll use the distributive property and combine like terms.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "I'll start with the left side: (x + 7)(x + 10)(x + 6) + (x + 9)(x + 10)(x + 6).", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 196, "metadata": { "path_label": "solve_frac_1_x_9_frac_1_x_7_frac_1_x_10__2023-01-03T06-50-05.343770|chosen" } }, { "text": "I want to find the maximum value of a function of three variables, subject to a constraint.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "One way to do that is to use Lagrange multipliers, which is a technique from multivariable calculus.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "The idea is to find a scalar $\\lambda$ such that the gradient of the function is parallel to the gradient of the constraint.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "In other words, I want to solve the system of equations $\\nabla (x^3 y^2 z) = \\lambda \\nabla (x + y + z - 1).$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "The gradient of a function is a vector of its partial derivatives, so I need to compute those first.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "For the function $x^3 y^2 z,$ the partial derivatives are $3x^2 y^2 z,$ $2x^3 y z,$ and $x^3 y^2.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "For the constraint $x + y + z - 1,$ the partial derivatives are $1,$ $1,$ and $1.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "So the system of equations becomes \\begin{align*} 3x^2 y^2 z &= \\lambda \\\\ 2x^3 y z &= \\lambda \\\\ x^3 y^2 &= \\lambda \\\\ x + y + z &= 1 \\end{align*}", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "This looks like a hard system to solve, but I notice that the first three equations are all equal to $\\lambda,$ so I can try to eliminate that by dividing them.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "If I divide the first equation by the second, I get $\\frac{3x^2 y^2 z}{2x^3 y z} = \\frac{\\lambda}{\\lambda},$ which simplifies to $\\frac{3y}{2x} = 1.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "If I divide the second equation by the third, I get $\\frac{2x^3 y z}{x^3 y^2} = \\frac{\\lambda}{\\lambda},$ which simplifies to $\\frac{2z}{y} = 1.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "These two equations give me a relation between $x,$ $y,$ and $z,$ so I can try to substitute them into the fourth equation.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "If I solve for $x$ and $z$ in terms of $y,$ I get $x = \\frac{3y}{2}$ and $z = \\frac{y}{2}.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "Then the fourth equation becomes $\\frac{3y}{2} + y + \\frac{y}{2} = 1,$ which simplifies to $3y = 1.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "So $y = \\frac{1}{3},$ and then $x = \\frac{1}{2}$ and $z = \\frac{1}{6}.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "These are the values of $x,$ $y,$ and $z$ that make the gradient of the function parallel to the gradient of the constraint.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "But are they the values that maximize the function?", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "To check that, I need to plug them into the function and compare them to the values at the boundary of the constraint.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "The boundary of the constraint is where one or more of $x,$ $y,$ and $z$ are zero or one.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 197, "metadata": { "path_label": "let_x_y_and_z_be_positive_real_numbers_such_that_x__2023-01-03T06-31-32.028033|chosen" } }, { "text": "I want to find all the digits $n$ such that $\\underline{6}\\underline{n}$ is a multiple of $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "This means that $\\underline{6}\\underline{n}$ must be equal to $n$ times some integer $k$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "I can write this as $\\underline{6}\\underline{n} = kn$, or equivalently, $10 \\cdot 6 + n = kn$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "Subtracting $n$ from both sides, I get $60 = (k - 1)n$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "This means that $n$ must be a factor of 60, and also a digit from 0 to 9.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "Out of these, only 1, 2, 3, 4, 5, and 6 are digits.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "So there are 6 possible values for $n$ that make $\\underline{6}\\underline{n}$ divisible by $n$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 198, "metadata": { "path_label": "for_how_many_different_digits_n_is_the_two_digit_number__2023-01-03T06-07-58.939516|chosen" } }, { "text": "I need to find an integer $n$ such that $31n$ has the same remainder as $3$ when divided by $2293$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "This is equivalent to saying that $31n-3$ is divisible by $2293$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "In other words, I need to find a solution to the linear congruence \\[31n\\equiv 3\\pmod{2293}.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "One way to solve linear congruences is to use the extended Euclidean algorithm, which finds the greatest common divisor of two numbers and also expresses it as a linear combination of them.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "In this case, I need to find the greatest common divisor of $31$ and $2293$, and then see if I can multiply both sides of the equation by some factor to get $3$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "To apply the extended Euclidean algorithm, I start by writing the larger number as a multiple of the smaller one plus a remainder: \\[2293=31\\cdot73+30.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "Then I repeat this process with the smaller number and the remainder, until I get a remainder of zero: \\[31=30\\cdot1+1\\] \\[30=1\\cdot30+0.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "The last nonzero remainder is the greatest common divisor, so in this case it is $1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "This also means that $31$ and $2293$ are relatively prime, so there is a unique solution to the linear congruence modulo $2293$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "To find the solution, I work backwards from the equations I obtained, expressing $1$ as a linear combination of $31$ and $2293$: \\[1=31-30\\] \\[=31-(2293-31\\cdot73)\\] \\[=31\\cdot74-2293.\\]", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "This means that $31\\cdot74\\equiv 1\\pmod{2293}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "Therefore, to get $3$ instead of $1$, I can multiply both sides by $3$: \\[31\\cdot222\\equiv 3\\pmod{2293}.\\]", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "This gives me the value of $n$ that I was looking for: \\[n=222.\\]", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 199, "metadata": { "path_label": "notice_that_31_cdot37_1147_find_some_integer_n_with__2023-01-03T05-55-38.683581|chosen" } }, { "text": "I need to find the maximum distance between two consecutive exits, so I want to minimize the distances between the other exits.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "The problem says that each exit is at least 6 km from the next one, so I can assume that the distance between any two exits that are not 47 and 48 is 6 km.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "That means that the distance from exit 41 to exit 46 is 6 * (46 - 41) = 6 * 5 = 30 km.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "Similarly, the distance from exit 49 to exit 50 is 6 km.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "Therefore, the distance from exit 46 to exit 49, which includes the distance from exit 47 to exit 48, is 100 - 30 - 6 = 64 km.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "Now, I want to maximize the distance from exit 47 to exit 48, so I want to minimize the distance from exit 46 to exit 47 and the distance from exit 48 to exit 49.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "Again, I can assume that these distances are 6 km each, since that is the minimum possible.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "So, the distance from exit 47 to exit 48 is 64 - 6 - 6 = 52 km.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "This is the longest possible distance between these two exits.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 200, "metadata": { "path_label": "exits_on_a_highway_are_numbered_consecutively_from_1_to_50__2023-01-03T04-20-10.798504|chosen" } }, { "text": "To find $n$, I need to know how the number of divisors of a number depends on its prime factorization.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "I recall that if a number has a prime factorization of the form $p_1^{a_1}p_2^{a_2}\\cdots p_k^{a_k}$, where $p_1, p_2, \\ldots, p_k$ are distinct primes and $a_1, a_2, \\ldots, a_k$ are positive integers, then the number of divisors is given by $(a_1+1)(a_2+1)\\cdots (a_k+1)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "This is because for each prime factor $p_i$, we can choose any exponent from $0$ to $a_i$ to form a divisor, and there are $a_i+1$ possible choices.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "So, I want to find a prime factorization of $n$ that has this form and satisfies $(a_1+1)(a_2+1)\\cdots (a_k+1) = 75$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "I also know that $n$ must be a multiple of $75$, which means it has a factor of $3^15^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "So, I can write $n = 3^15^2p_3^{a_3}\\cdots p_k^{a_k}$, where $p_3, p_4, \\ldots, p_k$ are distinct primes and $a_3, a_4, \\ldots, a_k$ are positive integers.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "Then, the number of divisors of $n$ is given by $(1+1)(2+1)(a_3+1)\\cdots (a_k+1) = 2\\cdot 3\\cdot (a_3+1)\\cdots (a_k+1) = 75$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "I need to find the smallest possible values for $a_3, a_4, \\ldots, a_k$ that satisfy this equation.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "I notice that $75 = 3\\cdot 5^2$, so I can try to match the factors of $75$ with the factors of the number of divisors.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 201, "metadata": { "path_label": "let_n_be_the_smallest_positive_integer_that_is_a_multiple__2023-01-02T23-36-39.857824|chosen" } }, { "text": "To solve this problem, I need to use some facts about modular arithmetic and base conversion.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "First, I notice that $235935623_{74}$ is a number in base $74$, which means each digit represents a power of $74$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "For example, the rightmost digit is the ones place, the next one is the $74$s place, and so on.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "To convert this number to base $10$, I need to multiply each digit by its corresponding power of $74$ and add them up.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "So, $235935623_{74}=2\\cdot 74^8+3\\cdot 74^7+5\\cdot 74^6+9\\cdot 74^5+3\\cdot 74^4+5\\cdot 74^3+6\\cdot 74^2+2\\cdot 74^1+3\\cdot 74^0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This is a very large number, but I don't need to calculate it exactly.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "I only need to know its remainder when divided by $15$, since that will determine what value of $a$ makes it a multiple of $15$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "To find the remainder, I can use the fact that if $a \\equiv b \\pmod{m}$ and $c \\equiv d \\pmod{m}$, then $a+c \\equiv b+d \\pmod{m}$ and $ac \\equiv bd \\pmod{m}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This means that I can add or multiply any two numbers and only focus on their remainders modulo $15$, since the final remainder will be the same.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "For example, $74 \\equiv 14 \\pmod{15}$, so $74^2 \\equiv 14^2 \\pmod{15}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "Similarly, $74^3 \\equiv 14^3 \\pmod{15}$, and so on.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "Therefore, I can rewrite each term in the base $74$ number as a remainder modulo $15$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "So, $235935623_{74}\\equiv 2\\cdot 14^8+3\\cdot 14^7+5\\cdot 14^6+9\\cdot 14^5+3\\cdot 14^4+5\\cdot 14^3+6\\cdot 14^2+2\\cdot 14^1+3\\cdot 14^0 \\pmod{15}$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "Now, I need to simplify each power of $14$ modulo $15$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "I can use a calculator or a table of powers to do this.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "I find that $14^0 \\equiv 1 \\pmod{15}$, $14^1 \\equiv 14 \\pmod{15}$, $14^2 \\equiv 1 \\pmod{15}$, $14^3 \\equiv 14 \\pmod{15}$, $14^4 \\equiv 1 \\pmod{15}$, and so on.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "So, every even power of $14$ is congruent to $1$ and every odd power is congruent to $14$ modulo $15$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This means that $14^8 \\equiv 1 \\pmod{15}$, $14^7 \\equiv 14 \\pmod{15}$, $14^6 \\equiv 1 \\pmod{15}$, $14^5 \\equiv 14 \\pmod{15}$, and so on.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "Therefore, I can substitute these values into the base $74$ number and get $235935623_{74}\\equiv 2\\cdot 1+3\\cdot 14+5\\cdot 1+9\\cdot 14+3\\cdot 1+5\\cdot 14+6\\cdot 1+2\\cdot 14+3\\cdot 1 \\pmod{15}$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This simplifies to $235935623_{74}\\equiv 2+42+5+126+3+70+6+28+3 \\pmod{15}$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "I can add these numbers and get $235935623_{74}\\equiv 285 \\pmod{15}$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "To find the final remainder, I can divide $285$ by $15$ and get a quotient of $19$ and a remainder of $0$.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "So, $235935623_{74}\\equiv 0 \\pmod{15}$.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This means that $235935623_{74}$ is already a multiple of $15$, and I don't need to subtract anything from it.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "Therefore, the value of $a$ that makes $235935623_{74}-a$ a multiple of $15$ is $a=0$.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 202, "metadata": { "path_label": "suppose_a_is_an_integer_such_that_0_le_a_le_14_and_2__2023-01-02T23-22-04.807345|chosen" } }, { "text": "This problem is asking me to find the number of pages whose page numbers have a certain property: their digits add up to exactly 4.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "I could try to list all such page numbers, but that would take too long and be prone to errors.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "Instead, I will try to think of a systematic way to count them.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "I notice that the page numbers range from 1 to 1,000, which means they have at most four digits.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "I will consider each possible number of digits and see how many page numbers with that number of digits have the desired property.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "If a page number has only one digit, then it can only be 4, since that is the only single-digit number whose digits add up to 4.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 203, "metadata": { "path_label": "a_textbook_has_1_000_pages_how_many_of_the_pages_have_page__2023-01-02T12-04-33.930298|chosen" } }, { "text": "I notice that $504$ is divisible by $2$, so I can write it as $504 = 2 \\times 252$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 204, "metadata": { "path_label": "what_is_the_sum_of_all_of_the_positive_even_factors_of_504__2023-01-02T07-32-31.094432|chosen" } }, { "text": "Then, I notice that $252$ is also divisible by $2$, so I can write it as $252 = 2 \\times 126$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 204, "metadata": { "path_label": "what_is_the_sum_of_all_of_the_positive_even_factors_of_504__2023-01-02T07-32-31.094432|chosen" } }, { "text": "I continue this process until I reach a factor that is not divisible by $2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 204, "metadata": { "path_label": "what_is_the_sum_of_all_of_the_positive_even_factors_of_504__2023-01-02T07-32-31.094432|chosen" } }, { "text": "I get $504 = 2 \\times 2 \\times 2 \\times 2 \\times 3 \\times 3 \\times 7$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 204, "metadata": { "path_label": "what_is_the_sum_of_all_of_the_positive_even_factors_of_504__2023-01-02T07-32-31.094432|chosen" } }, { "text": "I want to count the number of possible distributions of 6 pencils among 3 friends, where each friend gets at least one pencil.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "One way to approach this problem is to use the stars and bars method, which is a technique for dividing n identical objects into k groups.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "The idea is to represent the n objects as stars (*) and the k groups as bars (|).", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "For example, if I have 6 stars and 2 bars, I can arrange them in a row like this: * * * | * * * |.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "This means that the first group gets 3 objects, the second group gets 3 objects, and the third group gets 0 objects.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "In general, there are (n + k - 1) choose (k - 1) ways to arrange n stars and k bars in a row, where (a choose b) means the number of ways to choose b items out of a items without regard to order.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "In this problem, I have n = 6 objects (pencils) and k = 3 groups (friends), so there are (6 + 3 - 1) choose (3 - 1) ways to arrange 6 stars and 3 bars in a row.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "However, this includes cases where some friends get 0 pencils, which I want to exclude.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "To avoid this, I can give each friend one pencil first, and then distribute the remaining 3 pencils among them.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "This means that I have n = 3 objects (remaining pencils) and k = 3 groups (friends), so there are (3 + 3 - 1) choose (3 - 1) ways to arrange 3 stars and 3 bars in a row.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "This is the same as (5 choose 2), which is equal to 10.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "Therefore, there are 10 ways to distribute 6 pencils among 3 friends, where each friend gets at least one pencil.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 205, "metadata": { "path_label": "three_friends_have_a_total_of_6_identical_pencils_and_each__2023-01-02T07-28-35.723873|chosen" } }, { "text": "This problem is asking me to find a constant $c$ that makes a certain inequality true for all nonnegative $x$ and $y.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "To get a feel for the inequality, I will try plugging in some simple values of $x$ and $y.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "For example, if $x = y = 1,$ then the inequality becomes $\\sqrt{1 \\cdot 1} + c |1 - 1| \\ge \\frac{1 + 1}{2},$ which simplifies to $1 \\ge 1.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "This is true for any value of $c,$ so it does not help me much.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "What if $x = 0$ and $y = 1?$ Then the inequality becomes $\\sqrt{0 \\cdot 1} + c |0 - 1| \\ge \\frac{0 + 1}{2},$ which simplifies to $c \\ge \\frac{1}{2}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "This gives me a lower bound for $c.$ I know that $c$ cannot be less than $\\frac{1}{2},$ or else the inequality would be false for this choice of $x$ and $y.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "Can $c = \\frac{1}{2}$ work? To check this, I need to see if the inequality holds for all other values of $x$ and $y.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "One way to approach this is to rearrange the inequality and see what it implies about the relationship between $x$ and $y.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "Subtracting $\\sqrt{xy}$ from both sides, I get $c |x - y| \\ge \\frac{x + y}{2} - \\sqrt{xy}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "Squaring both sides, I get $c^2 (x - y)^2 \\ge \\left(\\frac{x + y}{2} - \\sqrt{xy}\\right)^2.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 206, "metadata": { "path_label": "find_the_smallest_positive_real_number_c_such_that_for_al__2022-12-31T04-31-02.407636|chosen" } }, { "text": "To solve this problem, I need to count how many perfect powers are in the range from 1 to 150, and then subtract that from the total number of numbers in the range, which is 150. That will give me the number of numbers that are not perfect powers, and then I can divide that by 150 to get the probability.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 207, "metadata": { "path_label": "the_numbers_from_1_to_150_inclusive_are_placed_in_a_bag_an__2023-01-08T05-48-34.483685|chosen" } }, { "text": "To count the perfect powers, I can start with the smallest ones and work my way up. I know that 1 is a perfect power, since it can be written as $1^{2}$. I also know that 2, 3, 5, 7, 11, and so on, are not perfect powers, since they are prime numbers and the only way to write them as $x^{y}$ is to have $y=1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 207, "metadata": { "path_label": "the_numbers_from_1_to_150_inclusive_are_placed_in_a_bag_an__2023-01-08T05-48-34.483685|chosen" } }, { "text": "So the first non-prime perfect power I encounter is 4, which is $2^{2}$. Then I have 8, which is $2^{3}$. Then 9, which is $3^{2}$. Then 16, which is $2^{4}$. Then 25, which is $5^{2}$. Then 27, which is $3^{3}$. Then 32, which is $2^{5}$. Then 36, which is $6^{2}$. Then 49, which is $7^{2}$. Then 64, which is $2^{6}$. Then 81, which is $3^{4}$. Then 100, which is $10^{2}$. Then 121, which is $11^{2}$. Then 125, which is $5^{3}$. Then 128, which is $2^{7}$. Then 144, which is $12^{2}$. And that's it, since the next perfect power would be $2^{8}=256$, which is outside the range.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 207, "metadata": { "path_label": "the_numbers_from_1_to_150_inclusive_are_placed_in_a_bag_an__2023-01-08T05-48-34.483685|chosen" } }, { "text": "This problem looks like it involves some algebra and factoring.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I notice that the left-hand side of the equation is a perfect cube: $1000x^3+27 = (10x)^3 + 3^3$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I recall a formula for factoring the sum of two cubes: $a^3+b^3 = (a+b)(a^2-ab+b^2)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I apply this formula to the left-hand side of the equation, using $a=10x$ and $b=3$: $(10x)^3 + 3^3 = (10x+3)((10x)^2-10x \\cdot 3 + 3^2)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I simplify the second factor: $(10x)^2-10x \\cdot 3 + 3^2 = 100x^2 - 30x + 9$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I compare this with the right-hand side of the equation, which is also a product of two quadratic factors.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 208, "metadata": { "path_label": "if_a_b_c_d_e_and_f_are_integers_for_which__2023-01-07T11-55-48.841708|chosen" } }, { "text": "I notice that the sum of the digits of $D\\,767\\,E89,$ excluding $D$ and $E,$ is $7+6+7+8+9=37.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 209, "metadata": { "path_label": "a_number_is_divisible_by_9_if_the_sum_of_its_digits_is_div__2023-01-07T11-42-16.659873|chosen" } }, { "text": "I also notice that $37$ is not divisible by $9,$ so I need to find $D$ and $E$ such that $D+E+37$ is divisible by $9.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 209, "metadata": { "path_label": "a_number_is_divisible_by_9_if_the_sum_of_its_digits_is_div__2023-01-07T11-42-16.659873|chosen" } }, { "text": "One way to do this is to look at the remainders when $D+E+37$ is divided by $9.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 209, "metadata": { "path_label": "a_number_is_divisible_by_9_if_the_sum_of_its_digits_is_div__2023-01-07T11-42-16.659873|chosen" } }, { "text": "I notice that the angles in the cosines are all multiples of $\\frac{\\pi}{7}$, so I wonder if there is some connection to the seventh roots of unity.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 210, "metadata": { "path_label": "compute_cos_frac_2_pi_7_cos_frac_4_pi_7_cos_fr__2023-01-07T06-37-41.784057|chosen" } }, { "text": "I recall that the seventh roots of unity are the complex numbers of the form $e^{2 k \\pi i / 7}$, where $k = 0, 1, 2, \\ldots, 6$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 210, "metadata": { "path_label": "compute_cos_frac_2_pi_7_cos_frac_4_pi_7_cos_fr__2023-01-07T06-37-41.784057|chosen" } }, { "text": "I also recall that the real part of $e^{2 k \\pi i / 7}$ is $\\cos \\frac{2 k \\pi}{7}$, so maybe I can write the product of the cosines in terms of the roots of unity.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 210, "metadata": { "path_label": "compute_cos_frac_2_pi_7_cos_frac_4_pi_7_cos_fr__2023-01-07T06-37-41.784057|chosen" } }, { "text": "I try to do this by multiplying both sides of the equation by $\\sin \\frac{2 \\pi}{7}$, which I know is not zero, to get \\[\\sin \\frac{2 \\pi}{7} \\cos \\frac{2 \\pi}{7} \\cos \\frac{4 \\pi}{7} \\cos \\frac{8 \\pi}{7} = \\frac{1}{2} \\sin \\frac{4 \\pi}{7} \\cos \\frac{4 \\pi}{7} \\cos \\frac{8 \\pi}{7}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 210, "metadata": { "path_label": "compute_cos_frac_2_pi_7_cos_frac_4_pi_7_cos_fr__2023-01-07T06-37-41.784057|chosen" } }, { "text": "This looks like a Diophantine equation, where we want to find integer solutions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "One way to approach this is to isolate one of the variables and see what values it can take.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "For example, let's solve for $n$ in terms of $m$:", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Adding $2n$ to both sides, we get $14mn + 2n = 55 - 7m$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Factoring out $n$ on the left, we get $n(14m + 2) = 55 - 7m$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Dividing both sides by $14m + 2$, we get $n = \\frac{55 - 7m}{14m + 2}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Now, we want to find positive integer values of $m$ that make $n$ a positive integer as well.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "One way to do this is to try some small values of $m$ and see if they work.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 1$, then $n = \\frac{55 - 7}{14 + 2} = \\frac{48}{16} = 3$, which is a positive integer.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "So $(m,n) = (1,3)$ is one solution.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 2$, then $n = \\frac{55 - 14}{28 + 2} = \\frac{41}{30}$, which is not an integer.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 3$, then $n = \\frac{55 - 21}{42 + 2} = \\frac{34}{44} = \\frac{17}{22}$, which is not an integer.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 4$, then $n = \\frac{55 - 28}{56 + 2} = \\frac{27}{58}$, which is not an integer.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 5$, then $n = \\frac{55 - 35}{70 + 2} = \\frac{20}{72} = \\frac{5}{18}$, which is not an integer.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 6$, then $n = \\frac{55 - 42}{84 + 2} = \\frac{13}{86}$, which is not an integer.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 7$, then $n = \\frac{55 - 49}{98 + 2} = \\frac{6}{100} = \\frac{3}{50}$, which is not an integer.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "If $m = 8$, then $n = \\frac{55 - 56}{112 + 2} = -\\frac{1}{114}$, which is not a positive integer.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "So far, $(m,n) = (1,3)$ is the only solution we have found.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "We can also notice that as $m$ increases, $n$ decreases, and if $m \\geq 8$, then $n$ is negative or zero.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Therefore, there are no other positive integer solutions for $m \\geq 8$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "Hence, the only ordered pair $(m,n)$ that satisfies the equation is $(1,3)$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 211, "metadata": { "path_label": "find_the_ordered_pair_m_n_where_m_n_are_positive_inte__2023-01-06T12-01-20.677664|chosen" } }, { "text": "I know that inverse functions are reflections of each other across the line $y = x$, so the point of intersection must also lie on that line.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "That means $a = b$ and $f(a) = f^{-1}(a) = a$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "I can substitute $a$ for $x$ and $y$ in the equation $y = f(x)$ and get \\[a = a^3 + 6a^2 + 16a + 28.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "This is a cubic equation that I want to solve for $a$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "I notice that the constant term is $28$ and the coefficient of $a^3$ is $1$, so I can use the rational root theorem to find possible rational roots.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "The rational root theorem says that any rational root of the form $p/q$, where $p$ is a factor of $28$ and $q$ is a factor of $1$, must be an integer factor of $28$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "The integer factors of $28$ are $\\pm 1, \\pm 2, \\pm 4, \\pm 7, \\pm 14, \\pm 28$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "I can use synthetic division or a calculator to test each of these factors and see if any of them are roots of the cubic equation.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 212, "metadata": { "path_label": "let_f_x_x_3_6x_2_16x_28_the_graphs_of_y_f_x__2023-01-06T06-56-09.455029|chosen" } }, { "text": "This problem asks me to find the maximum value of a function of two variables, $f(x, y) = x^2 y - xy^2,$ over a rectangular region in the $xy$-plane.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "One way to approach this problem is to use calculus and find the critical points and boundary values of the function, and compare them to find the maximum.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "Another way is to use algebra and try to rewrite the function in a simpler form that makes it easier to find the maximum.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "I'll try the second way first, since it might be faster and more elegant. [* so I want to simplify $f(x, y) = x^2 y - xy^2.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "I notice that both terms have a common factor of $xy,$ so I can factor it out: $f(x, y) = xy(x - y).$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "Now I see that the function is zero when either $x = 0,$ $y = 0,$ or $x = y.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 213, "metadata": { "path_label": "let_x_and_y_be_real_numbers_such_that_0_le_x_le_1_an__2023-01-06T06-41-58.599933|chosen" } }, { "text": "I need to find all the values of $a$ that satisfy the given equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "To do that, I need to find the inverse function of $f(x)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "One way to do that is to switch the roles of $x$ and $y$, and then solve for $y$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "So I start by writing $y=\\frac{a}{1-x}$, and then I swap $x$ and $y$ to get $x=\\frac{a}{1-y}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "Now I want to isolate $y$ in terms of $x$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "I can do that by cross-multiplying and then adding $y$ to both sides.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 214, "metadata": { "path_label": "define_the_function_f_x_frac_a_1_x_if_f_1_f_1_4__2023-01-06T06-26-33.829049|chosen" } }, { "text": "This problem asks me to identify the type of curve given by a polar equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 215, "metadata": { "path_label": "find_the_curve_defined_by_the_equation_r_frac_6_2_3__2023-01-06T06-22-24.865636|chosen" } }, { "text": "I recall that the polar equation of a conic section has the form \\[r = \\frac{ep}{1 \\pm e \\cos \\theta},\\] where $e$ is the eccentricity and $p$ is the distance from the focus to the directrix.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 215, "metadata": { "path_label": "find_the_curve_defined_by_the_equation_r_frac_6_2_3__2023-01-06T06-22-24.865636|chosen" } }, { "text": "I compare this form with the given equation and notice that they have similar fractions on the right-hand side.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 215, "metadata": { "path_label": "find_the_curve_defined_by_the_equation_r_frac_6_2_3__2023-01-06T06-22-24.865636|chosen" } }, { "text": "I try to write the given equation in the standard form by finding values of $e$ and $p$ that match the numerator and denominator.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 215, "metadata": { "path_label": "find_the_curve_defined_by_the_equation_r_frac_6_2_3__2023-01-06T06-22-24.865636|chosen" } }, { "text": "To find the greatest common divisor (GCD) of two numbers, I can use the Euclidean algorithm, which is based on repeated applications of the division algorithm.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "The division algorithm says that for any positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$ and $0 \\leq r < b$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "The Euclidean algorithm uses the fact that the GCD of $a$ and $b$ is the same as the GCD of $b$ and $r$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "So I can start by applying the division algorithm to $5616$ and $11609$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "I can use long division or a calculator to find that $11609 = 5616 \\times 2 + 377$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "So the remainder is $377$, and the GCD of $5616$ and $11609$ is the same as the GCD of $5616$ and $377$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "Now I can apply the division algorithm again to $5616$ and $377$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 216, "metadata": { "path_label": "find_the_greatest_common_divisor_of_5616_and_11609__2023-01-05T06-36-57.152242|chosen" } }, { "text": "I know that the area of a triangle is half the product of the base and the height.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "So if the altitudes are $4$ and $12$, then the bases must be inversely proportional to them, since the area is constant.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Let me call the bases $a$, $b$, and $c$, and the altitudes $h_a$, $h_b$, and $h_c$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Then I have $ah_a = bh_b = ch_c$, and I can substitute the given values of $h_a$ and $h_b$ to get $4a = 12b$, or $a = 3b$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "This means that the base $a$ is three times longer than the base $b$, and the base $c$ must be somewhere in between them.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "I also know that the sum of any two sides of a triangle must be greater than the third side, otherwise the triangle inequality is violated.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "So I have $a + b > c$, $b + c > a$, and $c + a > b$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Substituting $a = 3b$, I get $4b > c$, $b + c > 3b$, and $c + 3b > b$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Simplifying, I get $b > \\frac{c}{4}$, $c > 2b$, and $c > -b$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "The last inequality is trivial, since $b$ and $c$ are positive lengths.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "The first and second inequalities give me bounds on $c$ in terms of $b$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "I want to maximize $c$, so I should look at the lower bound, which is $c > 2b$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "This means that $c$ is more than twice as long as $b$, so the altitude $h_c$ must be less than half of $h_b$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Since $h_b = 12$, the altitude $h_c$ must be less than $6$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "But it also has to be an integer, so the largest possible value for $h_c$ is $5$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "Therefore, the answer is $5$, which is choice (B).", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 217, "metadata": { "path_label": "two_of_the_altitudes_of_the_scalene_triangle_abc_have_leng__2022-12-30T22-33-20.711234|chosen" } }, { "text": "I want to find the angle and direction of the rotation, so I can apply it to the second vector.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "I can use the dot product formula to find the angle between the original and rotated vectors: $\\cos \\theta = \\frac{\\vec{u} \\cdot \\vec{v}}{|\\vec{u}||\\vec{v}|},$ where $\\theta$ is the angle, $\\vec{u}$ is the original vector, and $\\vec{v}$ is the rotated vector.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "Plugging in the given values, I get: $\\cos \\theta = \\frac{(-4)(1) + (7)(8)}{\\sqrt{(-4)^2 + (7)^2} \\sqrt{(1)^2 + (8)^2}} = \\frac{52}{\\sqrt{65} \\sqrt{65}} = \\frac{52}{65}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "Solving for $\\theta,$ I get: $\\theta = \\cos^{-1} \\frac{52}{65} \\approx 0.9273$ radians.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "I also need to determine the direction of the rotation, clockwise or counterclockwise.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "I can use the cross product formula to find the sign of the rotation: $\\vec{u} \\times \\vec{v} = |\\vec{u}||\\vec{v}| \\sin \\theta \\hat{n},$ where $\\hat{n}$ is the unit vector perpendicular to the plane of $\\vec{u}$ and $\\vec{v}.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "The sign of the cross product is the same as the sign of $\\hat{n},$ which determines the direction of the rotation.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 218, "metadata": { "path_label": "a_rotation_centered_at_the_origin_takes_begin_pmatrix_4__2022-12-30T13-00-38.908803|chosen" } }, { "text": "To solve this problem, I need to find the inverse function of $f(c)$ and then equate it to the given expression involving $f^{-1}(c)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "To find $f^{-1}(c)$, I need to switch the roles of $c$ and $f(c)$ and then solve for $c$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "So, if $f(c)=\\frac{3}{2c-3}$, then $c=\\frac{3}{2f^{-1}(c)-3}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "To solve for $f^{-1}(c)$, I need to clear the fraction by multiplying both sides by $2f^{-1}(c)-3$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "This gives me $c(2f^{-1}(c)-3)=3$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "Next, I need to distribute the $c$ and move the constant term to the other side.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "This gives me $2cf^{-1}(c)-3c=3$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 219, "metadata": { "path_label": "if_f_c_frac_3_2c_3_find_frac_kn_2_lm_when_f_1__2022-12-30T06-47-56.411186|chosen" } }, { "text": "I need to find the largest number n such that n < 100 and gcd(n, 12) = 4.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "I know that the gcd of two numbers is the largest factor that they both share, so n must be a multiple of 4.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "The largest multiple of 4 that is less than 100 is 96, but gcd(96, 12) = 12, not 4.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "So I need to go down to the next smaller multiple of 4, which is 92.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "Is gcd(92, 12) = 4?", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "I can use the Euclidean algorithm to check: 92 = 12 * 7 + 8, 12 = 8 * 1 + 4, 8 = 4 * 2 + 0.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "The last nonzero remainder is 4, so gcd(92, 12) = 4.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "Therefore, 92 is the largest number n that satisfies the conditions.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 220, "metadata": { "path_label": "what_is_the_greatest_integer_less_than_100_for_which_the_gre__2022-12-30T06-43-24.946375|chosen" } }, { "text": "This problem involves a rational function and an inequality, so I want to be careful with the signs of the expressions and the possibility of dividing by zero.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "I also notice that the fractions have different denominators, so I might want to combine them using a common denominator.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "To do that, I need to multiply the first fraction by $\\frac{x + 5}{x + 5}$ and the second fraction by $\\frac{x + 1}{x + 1}$, which are both equivalent to 1.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "This gives me $\\frac{x + 5}{(x + 1)(x + 5)} + \\frac{6(x + 1)}{(x + 1)(x + 5)} \\ge 1$, which I can simplify by expanding the numerators and combining like terms.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "This simplifies to $\\frac{7x + 11}{(x + 1)(x + 5)} \\ge 1$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "Now I want to isolate the rational expression on one side of the inequality, so I subtract 1 from both sides.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "This gives me $\\frac{7x + 11}{(x + 1)(x + 5)} - 1 \\ge 0$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "To simplify this further, I need to find a common denominator for the 1, which is just $(x + 1)(x + 5)$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "So I multiply 1 by $\\frac{(x + 1)(x + 5)}{(x + 1)(x + 5)}$ and subtract it from the other fraction.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "This gives me $\\frac{7x + 11}{(x + 1)(x + 5)} - \\frac{(x + 1)(x + 5)}{(x + 1)(x + 5)} \\ge 0$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 221, "metadata": { "path_label": "find_all_real_values_of_x_which_satisfy_frac_1_x_1__2022-12-30T06-42-00.322755|chosen" } }, { "text": "To find the multiples of 11, I can start from the smallest three-digit number that is divisible by 11, which is 110.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "Then I can add 11 to get the next multiple, which is 121, and keep adding 11 until I reach the largest three-digit multiple of 11.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "The largest three-digit multiple of 11 is 990, because if I add 11 more I get 1001, which is not a three-digit number.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "So the multiples of 11 are 110, 121, 132, ..., 979, 990.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "I want to count how many numbers are in this list.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "One way to do this is to notice that the difference between any two consecutive multiples of 11 is 11.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "So if I divide the difference between the largest and smallest multiple by 11, I get the number of gaps between them.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "The difference between 990 and 110 is 880, and 880 divided by 11 is 80.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "That means there are 80 gaps between 990 and 110, or 80 + 1 = 81 multiples of 11.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "Another way to do this is to use the formula for the number of terms in an arithmetic sequence.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "An arithmetic sequence is a sequence of numbers where each term is obtained by adding a constant amount to the previous term.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "In this case, the constant amount is 11.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "The formula for the number of terms in an arithmetic sequence is n = (last term - first term)/common difference + 1.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "In this case, the last term is 990, the first term is 110, and the common difference is 11.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "Plugging these values into the formula, I get n = (990 - 110)/11 + 1 = 80 + 1 = 81.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "So either way, I get the same answer: there are 81 three-digit positive integers that are multiples of 11.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 222, "metadata": { "path_label": "how_many_three_digit_positive_integers_are_multiples_of_11__2022-12-30T06-32-51.453966|chosen" } }, { "text": "I need to find the probability that Celeste does not hear her favorite song from start to finish in the first 4 minutes, 30 seconds of music.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "I can use the complement rule to find this probability, since it is the opposite of the probability that she does hear her favorite song from start to finish in the first 4 minutes, 30 seconds of music.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "To hear her favorite song from start to finish in the first 4 minutes, 30 seconds of music, she must hear it as the first song or as the second song after a song that is 1 minute or shorter.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "The probability of hearing her favorite song as the first song is 1/10, since there are 10 songs and they are played in random order.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "The probability of hearing her favorite song as the second song after a song that is 1 minute or shorter is the product of the probabilities of hearing a song that is 1 minute or shorter as the first song and hearing her favorite song as the second song.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "The only songs that are 1 minute or shorter are the first two songs, which are 30 seconds and 1 minute long, respectively.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "The probability of hearing one of these two songs as the first song is 2/10, since there are 10 songs and they are played in random order.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "The probability of hearing her favorite song as the second song, given that the first song was 1 minute or shorter, is 1/9, since there are 9 songs left and they are played in random order.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "So, the probability of hearing her favorite song as the second song after a song that is 1 minute or shorter is 2/10 times 1/9, which is 2/90 or 1/45.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 223, "metadata": { "path_label": "an_o_pod_mp3_player_stores_and_plays_entire_songs_celeste_h__2022-12-30T01-41-08.628652|chosen" } }, { "text": "I know that the remainder theorem tells me that the remainder when a polynomial $p(x)$ is divided by another polynomial $q(x)$ is equal to $p(r)$, where $r$ is a root of $q(x)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "So, if I can find a root of $g(x)$, I can plug it into $g(x^{12})$ and get the remainder.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "To find a root of $g(x)$, I can use the rational root theorem, which says that any rational root of $g(x)$ must be of the form $\\pm \\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "The constant term of $g(x)$ is $1$, so $p$ can only be $\\pm 1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "The leading coefficient of $g(x)$ is also $1$, so $q$ can also only be $\\pm 1$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "Therefore, the only possible rational roots of $g(x)$ are $\\pm 1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "I can check if $\\pm 1$ are actually roots of $g(x)$ by plugging them in and seeing if they make $g(x)$ equal to zero.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "If I plug in $x = 1$, I get $g(1) = 1 + 1 + 1 + 1 + 1 + 1 = 6$, which is not zero.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "So, $x = 1$ is not a root of $g(x)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "If I plug in $x = -1$, I get $g(-1) = (-1)^5 + (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1 = -1 + 1 - 1 + 1 - 1 + 1 = 0$, which is zero.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "So, $x = -1$ is a root of $g(x)$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "This means that I can use $x = -1$ as the value of $r$ in the remainder theorem.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "To find the remainder when $g(x^{12})$ is divided by $g(x)$, I just need to evaluate $g((-1)^{12})$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "I can simplify $(-1)^{12}$ by using the exponent rule that says that $(-1)^{2n} = 1$ for any integer $n$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "So, $(-1)^{12} = 1$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "Therefore, the remainder is just $g(1)$, which I already computed as $6$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 224, "metadata": { "path_label": "let_g_x_x_5_x_4_x_3_x_2_x_1_what_is_the_rem__2022-12-29T23-14-38.045026|chosen" } }, { "text": "I notice that the inequality involves the absolute value of a quadratic expression, which means that it can change sign depending on the values of $ x$ and $ a$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 225, "metadata": { "path_label": "determine_all_real_numbers_a_such_that_the_inequality__2022-12-29T21-11-04.770320|chosen" } }, { "text": "To find out when the inequality holds, I need to consider the cases when the quadratic expression is positive, zero, or negative.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 225, "metadata": { "path_label": "determine_all_real_numbers_a_such_that_the_inequality__2022-12-29T21-11-04.770320|chosen" } }, { "text": "If the quadratic expression is zero, then the inequality becomes $ 0\\le2$, which is always true.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 225, "metadata": { "path_label": "determine_all_real_numbers_a_such_that_the_inequality__2022-12-29T21-11-04.770320|chosen" } }, { "text": "So, I want to find the values of $ a$ that make the quadratic expression have a single zero, since that would mean that the inequality has only one solution in $ x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 225, "metadata": { "path_label": "determine_all_real_numbers_a_such_that_the_inequality__2022-12-29T21-11-04.770320|chosen" } }, { "text": "To do that, I can use the discriminant of the quadratic expression, which is the part under the square root in the quadratic formula.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 225, "metadata": { "path_label": "determine_all_real_numbers_a_such_that_the_inequality__2022-12-29T21-11-04.770320|chosen" } }, { "text": "This problem involves the remainder theorem, which says that the remainder when $f(x)$ is divided by $x - a$ is equal to $f(a).$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "So, for example, the fact that the remainder when $f(x)$ is divided by $x - 3$ is 15 means that $f(3) = 15.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Similarly, the fact that the remainder when $f(x)$ is divided by $(x - 1)^2$ is $2x + 1$ means that $f(1) = 2 \\cdot 1 + 1 = 3$ and $f'(1) = 2,$ where $f'(x)$ is the derivative of $f(x).$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "To find the remainder when $f(x)$ is divided by $(x - 3)(x - 1)^2,$ I need to find a polynomial $r(x)$ of degree less than 3 such that $f(x) = q(x) \\cdot (x - 3)(x - 1)^2 + r(x),$ where $q(x)$ is some quotient polynomial.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "To do that, I can use the method of undetermined coefficients, where I assume that $r(x) = ax^2 + bx + c$ and then plug in some values of $x$ to find $a, b,$ and $c.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Since $r(x)$ is the remainder when $f(x)$ is divided by $(x - 3)(x - 1)^2,$ it must satisfy $r(3) = f(3) = 15$ and $r(1) = f(1) = 3.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Plugging in $x = 3,$ I get $r(3) = a \\cdot 3^2 + b \\cdot 3 + c = 15,$ which simplifies to $9a + 3b + c = 15.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Plugging in $x = 1,$ I get $r(1) = a \\cdot 1^2 + b \\cdot 1 + c = 3,$ which simplifies to $a + b + c = 3.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "These are two linear equations with three unknowns, so I need one more equation to solve for $a, b,$ and $c.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "I can get another equation by using the fact that $r'(x)$ is the remainder when $f'(x)$ is divided by $(x - 3)(x - 1),$ and that $r'(1) = f'(1) = 2.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Differentiating $r(x) = ax^2 + bx + c,$ I get $r'(x) = 2ax + b.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Plugging in $x = 1,$ I get $r'(1) = 2a + b = 2.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Now I have three linear equations with three unknowns: $9a + 3b + c = 15, a + b + c = 3,$ and $2a + b = 2.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "I can use elimination or substitution to solve for $a, b,$ and $c.$", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "For example, subtracting the second equation from the first, I get $8a + 2b = 12.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Dividing both sides by 2, I get $4a + b = 6.$", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Subtracting the third equation from this one, I get $2a = 4.$", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Dividing both sides by 2, I get $a = 2.$", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Plugging this into the third equation, I get $2 \\cdot 2 + b = 2.$", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Subtracting 4 from both sides, I get $b = -2.$", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Plugging these into the second equation, I get $2 + (-2) + c = 3.$", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "Adding 2 to both sides, I get $c = 3.$", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "So, $r(x) = 2x^2 - 2x + 3.$", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 226, "metadata": { "path_label": "when_the_polynomial_f_x_is_divided_by_x_3_the_remain__2022-12-29T06-52-53.789161|chosen" } }, { "text": "To find the general term of a geometric sequence, I need to know the common ratio $r$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "I can find $r$ by dividing any term by the previous term, so $r = \\frac{15}{\\frac{3}{4}} = 20$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "Now I can write the general term as $a_n = \\frac{3}{4} \\cdot 20^{n-1}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "I want to find the smallest $n$ for which $a_n$ is divisible by one million.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "This means that $a_n$ must be a multiple of $10^6 = 2^6 \\cdot 5^6$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "Therefore, both $2^6$ and $5^6$ must be factors of $a_n$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 227, "metadata": { "path_label": "let_the_first_term_of_a_geometric_sequence_be_frac_3_4__2022-12-29T06-39-00.457320|chosen" } }, { "text": "I need to find an integer $n$ such that $450n\\equiv 1\\pmod{3599}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "This is equivalent to finding a solution to the linear congruence $450x+3599y=1$, where $x$ and $y$ are integers.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "I can use the extended Euclidean algorithm to find such a solution.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "I start by applying the division algorithm to the coefficients of $x$ and $y$, and write the remainder as a linear combination of the dividends.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "So, $3599=450\\cdot 7+449$, which means $449=3599-450\\cdot 7$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "Then, I divide $450$ by $449$, and write the remainder as a linear combination of the dividends.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "So, $450=449\\cdot 1+1$, which means $1=450-449$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "Now, I can work backwards and substitute the expressions for the remainders in terms of the dividends, until I get an expression for $1$ in terms of $450$ and $3599$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "So, $1=450-449=450-(3599-450\\cdot 7)=450\\cdot 8-3599$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "This means that $x=8$ and $y=-1$ is a solution to the linear congruence.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "Therefore, the multiplicative inverse to $450$ modulo $3599$ is $8$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "This is already an integer between $0$ and $3599$, so I don't need to adjust it further.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 228, "metadata": { "path_label": "the_numbers_60_221_and_229_are_the_legs_and_hypotenuse_of__2022-12-29T06-14-11.994913|chosen" } }, { "text": "To compare the number of digits in different bases, I need to convert $987_{10}$ to both base-3 and base-8.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 229, "metadata": { "path_label": "how_many_more_digits_does_the_base_3_representation_of_987__2022-12-29T06-07-50.129989|chosen" } }, { "text": "I can use repeated division and write the remainders as the digits from right to left.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 229, "metadata": { "path_label": "how_many_more_digits_does_the_base_3_representation_of_987__2022-12-29T06-07-50.129989|chosen" } }, { "text": "For base-3, I start by dividing 987 by 3 and get a quotient of 329 and a remainder of 0.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 229, "metadata": { "path_label": "how_many_more_digits_does_the_base_3_representation_of_987__2022-12-29T06-07-50.129989|chosen" } }, { "text": "Then I divide 329 by 3 and get a quotient of 109 and a remainder of 2.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 229, "metadata": { "path_label": "how_many_more_digits_does_the_base_3_representation_of_987__2022-12-29T06-07-50.129989|chosen" } }, { "text": "I continue this process until I get a quotient of 0 and a final remainder of 1.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 229, "metadata": { "path_label": "how_many_more_digits_does_the_base_3_representation_of_987__2022-12-29T06-07-50.129989|chosen" } }, { "text": "To form a two-digit prime number, I need to choose a tens digit and a units digit from the set $\\{2, 7, 8, 9\\}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "I know that a prime number is a positive integer that has exactly two factors: 1 and itself.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "So, I want to avoid any numbers that are divisible by 2, 3, 5, or 10, since those would have more than two factors.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "This means that the units digit cannot be 2, 8, or 0, since those are all divisible by 2 or 10.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "The units digit also cannot be 5, since that is divisible by 5.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "So, the only possible units digits are 7 and 9.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "Now, I need to consider the tens digit.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 230, "metadata": { "path_label": "how_many_two_digit_prime_numbers_can_be_formed_by_choosing_t__2022-12-29T05-57-39.115000|chosen" } }, { "text": "This is a problem about counting the number of possible distributions of stickers among sheets of paper, where the order of the stickers does not matter.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 231, "metadata": { "path_label": "henry_s_little_brother_has_8_identical_stickers_and_4_sh__2022-12-29T05-05-53.930754|chosen" } }, { "text": "One way to approach this problem is to use the stars and bars method, which is a technique for counting the number of ways to distribute n identical objects into k bins.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 231, "metadata": { "path_label": "henry_s_little_brother_has_8_identical_stickers_and_4_sh__2022-12-29T05-05-53.930754|chosen" } }, { "text": "The idea is to represent the objects as stars (*) and the partitions between bins as bars (|). For example, if we have 8 stars and 3 bars, we can arrange them in a row to represent different distributions of 8 objects into 4 bins. For instance, **|*|****|* means that the first bin has 2 objects, the second has 1, the third has 4, and the fourth has 1.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 231, "metadata": { "path_label": "henry_s_little_brother_has_8_identical_stickers_and_4_sh__2022-12-29T05-05-53.930754|chosen" } }, { "text": "The problem then becomes how many ways can we arrange 8 stars and 3 bars in a row, which is equivalent to choosing 3 positions for the bars out of 11 possible positions (8 stars + 3 bars).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 231, "metadata": { "path_label": "henry_s_little_brother_has_8_identical_stickers_and_4_sh__2022-12-29T05-05-53.930754|chosen" } }, { "text": "This is a combination problem, where the order of the bars does not matter, so we can use the formula for n choose k, which is n!/(k!(n-k)!). In this case, n is 11 and k is 3, so the answer is 11!/(3!(11-3)!) = 165.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 231, "metadata": { "path_label": "henry_s_little_brother_has_8_identical_stickers_and_4_sh__2022-12-29T05-05-53.930754|chosen" } }, { "text": "I need to find a number that is divisible by 2, but not by 3 or 4.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "That means it must be even, but not a multiple of 4 or 6.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "The smallest even number is 2, but that has a remainder of 2 when divided by 3, not 1.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "The next even number is 4, but that has a remainder of 0 when divided by 4, not 2.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "The next even number is 6, but that has a remainder of 0 when divided by 3, not 1.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "The next even number is 8, but that has a remainder of 0 when divided by 4, not 2.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "The next even number is 10, and that works! It has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "So the least positive integer that satisfies the conditions is 10.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 232, "metadata": { "path_label": "what_is_the_least_positive_integer_that_has_a_remainder_of_0__2022-12-29T01-03-48.185212|chosen" } }, { "text": "I notice that this is a sequence where each term is the reciprocal of the previous term, multiplied by $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "That means that the product of two consecutive terms, $x_{n-1}x_n$, is always $n$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "So, the product of the first eight terms is the product of the first four pairs of consecutive terms: $x_1x_2 \\cdots x_8 = (x_1x_2)(x_3x_4)(x_5x_6)(x_7x_8)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "Now, I can use the fact that $x_{n-1}x_n = n$ to simplify each pair: $x_1x_2 = 2, x_3x_4 = 4, x_5x_6 = 6, x_7x_8 = 8$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "So, the product of the first eight terms is just $2 \\cdot 4 \\cdot 6 \\cdot 8$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "I can use the commutative property of multiplication to rearrange the order of the factors: $2 \\cdot 4 \\cdot 6 \\cdot 8 = 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 3 \\cdot 4$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "I can also use the associative property of multiplication to group the factors by powers of $2$: $2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 3 \\cdot 4 = (2^5) \\cdot (3 \\cdot 4)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "I can evaluate the powers and products: $(2^5) \\cdot (3 \\cdot 4) = 32 \\cdot 12$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "Finally, I can multiply $32$ and $12$ to get the final answer: $32 \\cdot 12 = 384$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 233, "metadata": { "path_label": "let_x_1_97_and_for_n_1_define_x_n_frac_n_x__n_1__2023-01-02T05-15-17.624969|chosen" } }, { "text": "I notice that the point $D$ is equidistant from $B, C,$ and $A$, so it must be the circumcenter of $\\triangle ABC$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 234, "metadata": { "path_label": "let_us_have_a_triangle_triangle_abc_and_a_point_d_on_b__2023-01-02T03-51-54.779519|chosen" } }, { "text": "That means that $AD, BD,$ and $CD$ are all radii of the circumcircle, and that $\\triangle ABC$ is a right triangle with $\\angle A = 90^\\circ.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 234, "metadata": { "path_label": "let_us_have_a_triangle_triangle_abc_and_a_point_d_on_b__2023-01-02T03-51-54.779519|chosen" } }, { "text": "I can use the fact that the sum of the angles in a triangle is $180^\\circ$ to find the measure of $\\angle ACB$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 234, "metadata": { "path_label": "let_us_have_a_triangle_triangle_abc_and_a_point_d_on_b__2023-01-02T03-51-54.779519|chosen" } }, { "text": "I have $\\angle A + \\angle B + \\angle C = 180^\\circ,$ so $\\angle C = 180^\\circ - 90^\\circ - 50^\\circ = 40^\\p.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 234, "metadata": { "path_label": "let_us_have_a_triangle_triangle_abc_and_a_point_d_on_b__2023-01-02T03-51-54.779519|chosen" } }, { "text": "This problem looks like it involves some optimization techniques, since we want to minimize an expression.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 235, "metadata": { "path_label": "let_a_b_c_be_positive_real_numbers_find_the_smalle__2023-01-02T03-49-09.402424|chosen" } }, { "text": "One possible strategy is to use calculus and find the critical points of the function, but that might be messy and tedious.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 235, "metadata": { "path_label": "let_a_b_c_be_positive_real_numbers_find_the_smalle__2023-01-02T03-49-09.402424|chosen" } }, { "text": "Another possible strategy is to use some inequalities, such as the AM-GM inequality, which says that for any non-negative numbers $x$ and $y$, the arithmetic mean $\\frac{x+y}{2}$ is greater than or equal to the geometric mean $\\sqrt{xy}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 235, "metadata": { "path_label": "let_a_b_c_be_positive_real_numbers_find_the_smalle__2023-01-02T03-49-09.402424|chosen" } }, { "text": "This inequality can be generalized to more than two variables, so for example, for non-negative $x,$ $y,$ $z,$ we have $\\frac{x+y+z}{3} \\geq \\sqrt[3]{xyz}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 235, "metadata": { "path_label": "let_a_b_c_be_positive_real_numbers_find_the_smalle__2023-01-02T03-49-09.402424|chosen" } }, { "text": "If we apply this inequality to the terms $6a^3,$ $9b^3,$ $32c^3,$ and $\\frac{1}{4abc}$, we get that $\\frac{6a^3 + 9b^3 + 32c^3 + \\frac{1}{4abc}}{4} \\geq \\sqrt[4]{6a^3 \\cdot 9b^3 \\cdot 32c^3 \\cdot \\frac{1}{4abc}}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 235, "metadata": { "path_label": "let_a_b_c_be_positive_real_numbers_find_the_smalle__2023-01-02T03-49-09.402424|chosen" } }, { "text": "I see that this problem involves the sum of squares and cubes of four variables, and I wonder if there is a way to relate them using some inequality.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "One inequality that comes to mind is the Cauchy-Schwarz inequality, which says that for any real numbers $x_1, x_2, \\dots, x_n$ and $y_1, y_2, \\dots, y_n,$ we have \\[(x_1^2 + x_2^2 + \\cdots + x_n^2)(y_1^2 + y_2^2 + \\cdots + y_n^2) \\geq (x_1y_1 + x_2y_2 + \\cdots + x_ny_n)^2.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "I think this might be useful, because I can choose $x_1 = a, x_2 = b, x_3 = c, x_4 = d$ and $y_1 = a^2, y_2 = b^2, y_3 = c^2, y_4 = d^2,$ and then the inequality becomes \\[(a^2 + b^2 + c^2 + d^2)(a^4 + b^4 + c^4 + d^4) \\geq (a^3 + b^3 + c^3 + d^3)^2.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "Since I know that $a^2 + b^2 + c^2 + d^2 = 4,$ I can simplify this to \\[4(a^4 + b^4 + c^4 + d^4) \\geq (a^3 + b^3 + c^3 + d^3)^2.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "This means that $a^3 + b^3 + c^3 + d^3$ is bounded above by $2\\sqrt{a^4 + b^4 + c^4 + d^4}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "So, to find the maximum value of $a^3 + b^3 + c^3 + d^3,$ I need to find the maximum value of $a^4 + b^4 + c^4 + d^4.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "I notice that $a^4 + b^4 + c^4 + d^4$ is also the sum of squares, so I can apply the Cauchy-Schwarz inequality again, this time with $x_1 = a^2, x_2 = b^2, x_3 = c^2, x_4 = d^2$ and $y_1 = y_2 = y_3 = y_4 = 1.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "This gives me \\[(a^4 + b^4 + c^4 + d^4)(1 + 1 + 1 + 1) \\geq (a^2 + b^2 + c^2 + d^2)^2.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "Simplifying, I get \\[a^4 + b^4 + c^4 + d^4 \\geq 4.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "Therefore, the maximum value of $a^4 + b^4 + c^4 + d^4$ is $4,$ and it occurs when $a^2 = b^2 = c^2 = d^2 = 1.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "Plugging this back into the expression for $a^3 + b^3 + c^3 + d^3,$ I get \\[a^3 + b^3 + c^3 + d^3 \\leq 2\\sqrt{4} = 4.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "So, the maximum value of $a^3 + b^3 + c^3 + d^3$ is $4,$ and it occurs when $a = b = c = d = \\pm 1.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 236, "metadata": { "path_label": "let_a_b_c_d_be_real_numbers_such_that_a_2_b_2__2022-12-28T22-34-19.577760|chosen" } }, { "text": "I want to find the complex number that represents $S,$ and then find its modulus, which is the distance from the origin.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "Since $PQSR$ is a parallelogram, I know that the opposite sides are parallel and equal in length.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "That means that the vector from $P$ to $Q$ is the same as the vector from $R$ to $S.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "In complex numbers, adding a vector is the same as adding the corresponding complex number.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "So, to get from $R$ to $S,$ I need to add $(1 + i) z - z = iz$ to $R.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "Therefore, $S = 2 \\overline{z} + iz.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "Now, I want to find $|S|.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "I can use the fact that $|z| = 1,$ which means that $z \\overline{z} = 1.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "So, $|S|^2 = S \\overline{S} = (2 \\overline{z} + iz)(2z - i \\overline{z}).$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "I can expand this and simplify it using $z \\overline{z} = 1.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 237, "metadata": { "path_label": "the_points_p_q_and_r_are_represented_by_the_complex__2022-12-28T06-23-48.203949|chosen" } }, { "text": "This is a problem about the behavior of a function that flips and stretches the interval $[0,1]$ repeatedly.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "I want to visualize what the function $f$ does to the interval $[0,1]$, so I draw a graph of $f(x)$ on the coordinate plane.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "I see that $f$ maps the interval $[0, \\frac{1}{2}]$ linearly onto $[0,1]$, and then maps the interval $[\\frac{1}{2}, 1]$ linearly onto $[0,1]$ in reverse.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "So $f$ divides the interval $[0,1]$ into two halves, flips them, and then stretches them to cover the whole interval again.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "This means that $f^{[2]}$ will divide the interval $[0,1]$ into four equal parts, flip them, and then stretch them to cover the whole interval again.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "Similarly, $f^{[n]}$ will divide the interval $[0,1]$ into $2^n$ equal parts, flip them, and then stretch them to cover the whole interval again.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "So I can think of $f^{[n]}$ as a function that divides the interval $[0,1]$ into $2^n$ equal subintervals, and then assigns a value to each subinterval by flipping and stretching the previous values.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "Now I want to find the values of $x$ in $[0,1]$ for which $f^{[2005]}(x) = \\frac {1}{2}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "This means that $x$ must belong to one of the subintervals of $[0,1]$ that are mapped to $\\frac {1}{2}$ by $f^{[2005]}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "But how many such subintervals are there?", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 238, "metadata": { "path_label": "for_each_x_in_0_1_define_begin_cases_f_x_2x__2022-12-28T06-04-36.085122|chosen" } }, { "text": "I know that if a polynomial with integer coefficients has an integer root, then that root must divide the constant term.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 239, "metadata": { "path_label": "find_all_integer_values_of_a_so_that_the_polynomial_x_3__2022-12-28T05-58-50.218964|chosen" } }, { "text": "In this case, the constant term is 7, so the possible integer roots are $\\pm 1, \\pm 7.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 239, "metadata": { "path_label": "find_all_integer_values_of_a_so_that_the_polynomial_x_3__2022-12-28T05-58-50.218964|chosen" } }, { "text": "I can use synthetic division to test each of these roots and see if they give a remainder of zero.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 239, "metadata": { "path_label": "find_all_integer_values_of_a_so_that_the_polynomial_x_3__2022-12-28T05-58-50.218964|chosen" } }, { "text": "If I try $x = 1,$ I get", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 239, "metadata": { "path_label": "find_all_integer_values_of_a_so_that_the_polynomial_x_3__2022-12-28T05-58-50.218964|chosen" } }, { "text": "I notice that this function involves fractions and reciprocals, so I wonder if I can use some inequality to simplify it or bound it.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "One inequality that comes to mind is the arithmetic mean-geometric mean (AM-GM) inequality, which says that for any positive numbers $a$ and $b$, the average of $a$ and $b$ is greater than or equal to the square root of their product: \\[\\frac{a + b}{2} \\geq \\sqrt{ab}.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "This inequality is useful because it relates the sum and the product of two numbers, which are often involved in fractions and reciprocals.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "I also recall that the AM-GM inequality becomes an equality if and only if $a = b$, which might help me find the minimum value of the function.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "To apply the AM-GM inequality to the function $f(x)$, I need to identify two positive numbers whose average and product appear in the function.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "One pair that stands out is $x$ and $\\frac{1}{x}$, since their average is $\\frac{x + \\frac{1}{x}}{2}$ and their product is $1$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "By the AM-GM inequality, I have \\[\\frac{x + \\frac{1}{x}}{2} \\geq \\sqrt{x \\cdot \\frac{1}{x}} = \\sqrt{1} = 1.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "Multiplying both sides by $2$, I get \\[x + \\frac{1}{x} \\geq 2.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "This means that the first term of $f(x)$ is always greater than or equal to $2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "To minimize $f(x)$, I want to minimize this term as much as possible, so I want to make it equal to $2$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "This happens when $x = \\frac{1}{x}$, or equivalently, when $x = 1$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "So I suspect that the minimum value of $f(x)$ occurs when $x = 1$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "To check this, I need to plug in $x = 1$ into $f(x)$ and see if I can find a smaller value for any other $x > 0$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "When $x = 1$, I get \\[f(1) = 1 + \\frac{1}{1} + \\frac{1}{1 + \\frac{1}{1}} = 1 + 1 + \\frac{1}{2} = \\frac{5}{2}.\\]", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "Is there any other $x > 0$ that gives a smaller value for $f(x)$?", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "Well, I can see that as $x$ increases from $1$, the first term of $f(x)$ increases as well, so that's not good.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "The second term of $f(x)$ decreases as $x$ increases, but not enough to offset the increase in the first term.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "The third term of $f(x)$ is a bit tricky, but I can see that it is always positive and it approaches $0$ as $x$ increases.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "So overall, $f(x)$ increases as $x$ increases from $1$, and it never goes below $\\frac{5}{2}$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "What about when $x$ decreases from $1$?", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "Well, then the first term of $f(x)$ decreases, which is good, but the second term of $f(x)$ increases, which is not good.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 240, "metadata": { "path_label": "find_the_minimum_value_of_f_x_x_frac_1_x_frac_1__2022-12-28T05-48-44.217975|chosen" } }, { "text": "To be divisible by 15, a number must be divisible by both 3 and 5.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "For a number to be divisible by 5, its last digit must be either 0 or 5.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "So, the first condition is that the last digit of the six-digit number must be one of the two 5s.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "The probability of that happening is 2/6, or 1/3, since there are two 5s out of six digits to choose from.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "Now, for a number to be divisible. by 3, the sum of its digits must be divisible by 3.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "The sum of the six given digits is 1 + 2 + 3 + 5 + 5 + 8 = 24, which is divisible by 3.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "That means that any arrangement of the six digits, as long as the last digit is 5, will also have a sum divisible by 3.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "Therefore, the second condition is always satisfied, and does not affect the probability.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "So, the final answer is just the probability of the last digit being 5, which is 1/3.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 241, "metadata": { "path_label": "if_the_six_digits_1_2_3_5_5_and_8_are_randomly_arranged__2022-12-28T05-34-04.266194|chosen" } }, { "text": "I notice that 2004 is divisible by 2, so I can try to write it as a product of two even numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "One way to do that is to divide 2004 by 2, which gives 1002.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "Then I can divide 1002 by 2 again, which gives 501.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "So 2004 can be written as 2 * 2 * 501. [* 501 is not divisible by 2, but it is divisible by 3.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "So I can divide 501 by 3, which gives 167.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "Therefore, 2004 can be written as 2 * 2 * 3 * 167.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "These are all prime factors, so I cannot break them down any further.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "Now I have four factors, but I only want three.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "So I need to combine two of them into one.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "The best way to do that is to multiply the smallest two factors, 2 and 2, which gives 4.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "That way, I minimize the impact on the sum.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "So I can write 2004 as 4 * 3 * 167.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "The sum of these factors is 4 + 3 + 167, which is 174.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "This is the minimum possible value of the sum, since any other combination of factors would increase the sum.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "For example, if I combine 2 and 3 instead of 2 and 2, I get 6 * 2 * 167, which has a sum of 175, which is larger.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 242, "metadata": { "path_label": "the_product_of_positive_integers_x_y_and_z_equals_200__2022-12-28T05-30-55.974843|chosen" } }, { "text": "I know that $\\cos 4x = -\\frac{1}{2}$ when $4x$ is either $2\\pi/3$ or $4\\pi/3$ radians, since those are the angles in the second and third quadrants that have a cosine of $-\\frac{1}{2}.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 243, "metadata": { "path_label": "all_solutions_of_the_equation_cos_4x_frac_1_2_can_b__2022-12-28T04-53-37.107208|chosen" } }, { "text": "So I can write $4x = 2\\pi/3 + 2\\pi m$ or $4x = 4\\pi/3 + 2\\pi n,$ where $m$ and $n$ are integers.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 243, "metadata": { "path_label": "all_solutions_of_the_equation_cos_4x_frac_1_2_can_b__2022-12-28T04-53-37.107208|chosen" } }, { "text": "Solving for $x,$ I get $x = \\pi/6 + \\pi m/2$ or $x = \\pi/3 + \\pi n/2.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 243, "metadata": { "path_label": "all_solutions_of_the_equation_cos_4x_frac_1_2_can_b__2022-12-28T04-53-37.107208|chosen" } }, { "text": "These are the general solutions, but I need to express them in the form $\\frac{(kn \\pm 1) \\pi}{6},$ where $n$ is an integer and $k$ is a positive constant.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 243, "metadata": { "path_label": "all_solutions_of_the_equation_cos_4x_frac_1_2_can_b__2022-12-28T04-53-37.107208|chosen" } }, { "text": "I notice that both solutions have a factor of $\\pi/6$ in them, so I can factor that out and write $x = \\frac{\\pi}{6}(1 + 3m)$ or $x = \\frac{\\pi}{6}(2 + 3n).$ [* .] Now I can compare these expressions with the given form and see that $k = 3.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 243, "metadata": { "path_label": "all_solutions_of_the_equation_cos_4x_frac_1_2_can_b__2022-12-28T04-53-37.107208|chosen" } }, { "text": "I know that a regular heptagon has 7 congruent sides and angles, and that each angle is 180(7-2)/7 = 900/7 degrees.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 244, "metadata": { "path_label": "2_diagonals_of_a_regular_heptagon_a_7_sided_polygon_are_ch__2022-12-27T23-26-26.019483|chosen" } }, { "text": "I also know that any diagonal of a regular heptagon will connect two vertices that are not adjacent, so there are 7C2 - 7 = 21 - 7 = 14 possible diagonals.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 244, "metadata": { "path_label": "2_diagonals_of_a_regular_heptagon_a_7_sided_polygon_are_ch__2022-12-27T23-26-26.019483|chosen" } }, { "text": "To find the probability that two diagonals intersect inside the heptagon, I need to count how many pairs of diagonals have this property, and divide by the total number of pairs of diagonals.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 244, "metadata": { "path_label": "2_diagonals_of_a_regular_heptagon_a_7_sided_polygon_are_ch__2022-12-27T23-26-26.019483|chosen" } }, { "text": "To count the pairs of intersecting diagonals, I will use a systematic approach. I will start with the diagonal that connects vertex 1 and vertex 4, and see which other diagonals intersect it inside the heptagon.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 244, "metadata": { "path_label": "2_diagonals_of_a_regular_heptagon_a_7_sided_polygon_are_ch__2022-12-27T23-26-26.019483|chosen" } }, { "text": "I need to find the number of ways to choose two vowels and three consonants from the word MATHCOUNTS.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 245, "metadata": { "path_label": "on_the_refrigerator_mathcounts_is_spelled_out_with_10_magne__2022-12-27T22-57-58.384228|chosen" } }, { "text": "I notice that there are three vowels (A, O, and U) and seven consonants (M, T, H, C, N, S, and another T).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 245, "metadata": { "path_label": "on_the_refrigerator_mathcounts_is_spelled_out_with_10_magne__2022-12-27T22-57-58.384228|chosen" } }, { "text": "However, the two Ts are indistinguishable, so I have to be careful not to overcount them.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 245, "metadata": { "path_label": "on_the_refrigerator_mathcounts_is_spelled_out_with_10_magne__2022-12-27T22-57-58.384228|chosen" } }, { "text": "One way to avoid overcounting is to separate the cases where I choose zero, one, or two Ts as part of the consonants.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 245, "metadata": { "path_label": "on_the_refrigerator_mathcounts_is_spelled_out_with_10_magne__2022-12-27T22-57-58.384228|chosen" } }, { "text": "I know that the possible outcomes for each die are 1, 2, 3, 4, 5, and 6, and that each outcome has a 1/6 probability of occurring.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I also know that the sum of the face up integers will range from 3 (when all dice show 1) to 18 (when all dice show 6).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I want to find the number of ways to get a sum of 16, and divide that by the total number of possible outcomes, which is 6^3 = 216.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "One way to approach this is to list all the possible outcomes that add up to 16, and count them.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "For example, one outcome is 6, 6, 4. Another is 6, 5, 5. And another is 6, 4, 6.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I notice that the largest possible value for any die is 6, so I can start by fixing one die at 6, and see how many ways the other two dice can add up to 10.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "If the second die is 6, then the third die must be 4. If the second die is 5, then the third die must be 5. If the second die is 4, then the third die must be 6.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "So far, I have three outcomes: 6, 6, 4; 6, 5, 5; and 6, 4, 6.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I can repeat this process by fixing one die at 5, and see how many ways the other two dice can add up to 11.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "If the second die is 6, then the third die must be 5. If the second die is 5, then the third die must be 6.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "So far, I have two more outcomes: 5, 6, 5 and 5, 5, 6.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I can continue this process by fixing one die at 4, and see how many ways the other two dice can add up to 12.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "If the second die is 6, then the third die must be 6. If the second die is 5, then the third die must be 7, which is impossible. If the second die is 4, then the third die must be 8, which is also impossible.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "So far, I have one more outcome: 4, 6, 6.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I can skip fixing one die at 3, because then the other two dice would have to add up to 13, which is impossible with dice that only go up to 6.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "I can also skip fixing one die at 2 or 1, for the same reason.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "Therefore, I have found all the possible outcomes that add up to 16: 6, 6, 4; 6, 5, 5; 6, 4, 6; 5, 6, 5; 5, 5, 6; and 4, 6, 6.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "There are 6 of them, out of 216 possible outcomes.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "So the probability of getting a sum of 16 is 6/216, which can be simplified by dividing both numerator and denominator by 6.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "The final answer is 1/36.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 246, "metadata": { "path_label": "if_three_standard_6_faced_dice_are_rolled_what_is_the_pro__2022-12-26T07-00-05.453538|chosen" } }, { "text": "To start, I notice that the graph of this equation involves the absolute value function, which is usually piecewise linear and symmetric about the origin.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 247, "metadata": { "path_label": "find_the_area_in_the_plane_contained_by_the_graph_of_x__2022-12-26T06-58-53.368959|chosen" } }, { "text": "So I expect the graph to be a polygon with some symmetry, and I want to find its vertices and then use a formula for the area of a polygon.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 247, "metadata": { "path_label": "find_the_area_in_the_plane_contained_by_the_graph_of_x__2022-12-26T06-58-53.368959|chosen" } }, { "text": "One way to find the vertices is to consider different cases for the signs of x and y, and then solve for the values that make the equation an equality.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 247, "metadata": { "path_label": "find_the_area_in_the_plane_contained_by_the_graph_of_x__2022-12-26T06-58-53.368959|chosen" } }, { "text": "I know that the difference of two consecutive perfect squares is always odd, because (n+1)^2 - n^2 = 2n + 1.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "So I need to find how many odd perfect squares are less. than 10,000.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "The largest odd perfect square less than 10,000 is 99^2, which is 9,801.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "The smallest odd perfect square is 1^2, which is 1.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "I can list the odd perfect squares between 1 and 9,801: 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "There are 50 of them.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "So there are 50 perfect squares less than 10,000 that can be represented as the difference of two consecutive perfect squares.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 248, "metadata": { "path_label": "recall_that_a_perfect_square_is_the_square_of_some_integer__2022-12-26T06-29-27.422538|chosen" } }, { "text": "A composite number is a positive integer that is not prime, which means it has more than two factors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "To have no prime factors less than 10, a composite number must be divisible only by primes that are greater than or equal to 10. [*, I need to find the smallest such number.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "The smallest prime that is greater than or equal to 10 is 11.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "If I multiply 11 by itself, I get 121, which is a composite number that has no prime factors less than 10.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "But is it the smallest one?", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "Well, if I multiply 11 by any other prime that is greater than or equal to 10, I get a larger number.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "For example, 11 times 13 is 143, which is larger than 121.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "So, 121 is the smallest composite number that has no prime factors less than 10.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 249, "metadata": { "path_label": "find_the_smallest_composite_number_that_has_no_prime_factors__2022-12-26T06-00-24.464510|chosen" } }, { "text": "I notice that the numerator and denominator of the fraction are symmetric in $x$, $y$, and $z$, so I wonder if there is a way to simplify or factor them using symmetry.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 250, "metadata": { "path_label": "let_x_y_and_z_be_distinct_real_numbers_that_sum_to_0__2022-12-26T05-45-45.369638|chosen" } }, { "text": "I recall that $(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+zx)$, so if I plug in $x+y+z=0$, I get $x^2+y^2+z^2 = -2(xy+yz+zx)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 250, "metadata": { "path_label": "let_x_y_and_z_be_distinct_real_numbers_that_sum_to_0__2022-12-26T05-45-45.369638|chosen" } }, { "text": "This means I can rewrite the fraction as $$\\frac{xy+yz+zx}{x^2+y^2+z^2} = -\\frac{1}{2}.$$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 250, "metadata": { "path_label": "let_x_y_and_z_be_distinct_real_numbers_that_sum_to_0__2022-12-26T05-45-45.369638|chosen" } }, { "text": "So the fraction does not depend on the values of $x$, $y$, and $z$, as long as they are distinct and sum to 0.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 250, "metadata": { "path_label": "let_x_y_and_z_be_distinct_real_numbers_that_sum_to_0__2022-12-26T05-45-45.369638|chosen" } }, { "text": "Therefore, the maximum possible value of the fraction is just -1/2.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 250, "metadata": { "path_label": "let_x_y_and_z_be_distinct_real_numbers_that_sum_to_0__2022-12-26T05-45-45.369638|chosen" } }, { "text": "To find the probability, I need to compare the area of the region below the $x$-axis to the total area of the parallelogram.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 251, "metadata": { "path_label": "parallelogram_abcd_has_vertices_a_3_3_b_3_3_c__2022-12-26T05-18-01.277256|chosen" } }, { "text": "I can use the formula for the area of a parallelogram, which is $A = bh$, where $b$ is the base and $h$ is the height.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 251, "metadata": { "path_label": "parallelogram_abcd_has_vertices_a_3_3_b_3_3_c__2022-12-26T05-18-01.277256|chosen" } }, { "text": "To find the base, I can use the distance formula between $A$ and $B$, which is $b = \\sqrt{(3 - (-3))^2 + (3 - (-3))^2} = \\sqrt{36 + 36} = 6\\sqrt{2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 251, "metadata": { "path_label": "parallelogram_abcd_has_vertices_a_3_3_b_3_3_c__2022-12-26T05-18-01.277256|chosen" } }, { "text": "To find $e,$ I need to use the fact that it is the product of the roots and the leading coefficient $a,$ according to Vieta's formulas.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 252, "metadata": { "path_label": "the_polynomial_ax_4_bx_3_cx_2_dx_e_0_has_coeffi__2022-12-26T05-11-37.752892|chosen" } }, { "text": "So, $e = (-2)(5)(9)(-1/3)a.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 252, "metadata": { "path_label": "the_polynomial_ax_4_bx_3_cx_2_dx_e_0_has_coeffi__2022-12-26T05-11-37.752892|chosen" } }, { "text": "To make $e$ as small as possible, I want to make $a$ as small as possible, subject to the condition that it is an integer.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 252, "metadata": { "path_label": "the_polynomial_ax_4_bx_3_cx_2_dx_e_0_has_coeffi__2022-12-26T05-11-37.752892|chosen" } }, { "text": "To compare these quantities, I want to express them with a common denominator, so I can compare the numerators.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 253, "metadata": { "path_label": "which_of_the_following_quantities_is_the_largest_write_a__2022-12-26T04-03-44.116162|chosen" } }, { "text": "For $A$, the common denominator is $2005 \\times 2007$, and the numerator is $2006 \\times 2007 + 2006 \\times 2005$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 253, "metadata": { "path_label": "which_of_the_following_quantities_is_the_largest_write_a__2022-12-26T04-03-44.116162|chosen" } }, { "text": "For $B$, the common denominator is $2007^2$, and the numerator is $2006 \\times 2007 + 2008 \\times 2007$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 253, "metadata": { "path_label": "which_of_the_following_quantities_is_the_largest_write_a__2022-12-26T04-03-44.116162|chosen" } }, { "text": "For $C$, the common denominator is $2006 \\times 2008$, and the numerator is $2007 \\times 2008 + 2007 \\times 2006$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 253, "metadata": { "path_label": "which_of_the_following_quantities_is_the_largest_write_a__2022-12-26T04-03-44.116162|chosen" } }, { "text": "I notice that the common denominators are very close to each other, and they are all around $4 \\times 10^6$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 253, "metadata": { "path_label": "which_of_the_following_quantities_is_the_largest_write_a__2022-12-26T04-03-44.116162|chosen" } }, { "text": "To find the smallest positive multiple of $23$ that is $4$ more than a multiple of $89$, I need to find the smallest positive integer $n$ such that $23n = 89k + 4$ for some integer $k$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "This is equivalent to finding the smallest positive integer $n$ that satisfies the congruence $23n \\equiv 4 \\pmod{89}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "To solve this congruence, I can try to find the inverse of $23$ modulo $89$, which is a number $x$ such that $23x \\equiv 1 \\pmod{89}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "One way to find the inverse is to use the extended Euclidean algorithm, which finds integers $a$ and $b$ such that $23a + 89b = 1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Then $a$ will be the inverse of $23$ modulo $89$, and I can multiply both sides of the congruence by $a$ to get $n \\equiv 4a \\pmod{89}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "To apply the extended Euclidean algorithm, I need to write $1$ as a linear combination of $23$ and $89$ by repeatedly applying the division algorithm and back-substituting the remainders.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "I start with $89 = 3 \\times 23 + 20$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Then $23 = 1 \\times 20 + 3$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Then $20 = 6 \\times 3 + 2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Then $3 = 1 \\times 2 + 1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Then $2 = 2 \\times 1 + 0$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Now I can work backwards from $1$ and substitute the remainders to get a linear combination of $23$ and $89$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "I have $1 = 3 - 1 \\times 2$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "But $2 = 20 - 6 \\times 3$, so I can substitute that and get $1 = 3 - 1 \\times (20 - 6 \\times 3) = 7 \\times 3 - 1 \\times 20$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "But $3 = 23 - 1 \\times 20$, so I can substitute that and get $1 = 7 \\times (23 - 1 \\times 20) - 1 \\times 20 = 7 \\times 23 - 8 \\times 20$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "But $20 = 89 - 3 \\times 23$, so I can substitute that and get $1 = 7 \\times 23 - 8 \\times (89 - 3 \\times 23) = 31 \\times 23 - 8 \\times 89$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Therefore, $a = 31$ and $b = -8$ are the coefficients that satisfy $23a + 89b = 1$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "This means that $31$ is the inverse of $23$ modulo $89$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "To find $n$, I multiply both sides of the congruence by $31$ and get $23n \\equiv 4 \\pmod{89} \\implies 31 \\times 23n \\equiv 31 \\times 4 \\pmod{89} \\implies n \\equiv 124 \\pmod{89}$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "This means that $n$ is the smallest positive integer that is equivalent to $124$ modulo $89$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "To find this, I divide $124$ by $89$ and get a quotient of $1$ and a remainder of $35$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "Therefore, $n = 35$ is the smallest positive integer that satisfies the congruence and the original problem.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 254, "metadata": { "path_label": "what_is_the_smallest_positive_multiple_of_23_that_is_4_m__2022-12-26T03-02-31.669437|chosen" } }, { "text": "I know that $6^{-1}\\pmod m$ is the inverse of $6\\pmod m$, which means that $6\\cdot 6^{-1}\\equiv 1\\pmod m$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "So, if $6^{-1}\\equiv 6^2\\pmod m$, then I can multiply both sides by $6$ and get $6\\cdot 6^{-1}\\equiv 6\\cdot 6^2\\pmod m$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "But $6\\cdot 6^{-1}\\equiv 1\\pmod m$, so this simplifies to $1\\equiv 6^3\\pmod m$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "This means that $6^3-1$ is divisible by $m$, or $m$ is a factor of $6^3-1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I can factor $6^3-1$ as $(6-1)(6^2+6+1)$, which is $(5)(43)$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "So, $m$ is either $5$ or $43$, or a multiple of both.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "However, since $m$ is a two-digit positive integer, the only possible value for $m$ is $43$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I can check that $6^{-1}\\equiv 6^2\\pmod{43}$ by finding the inverse of $6\\pmod{43}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I can use the extended Euclidean algorithm to find $a$ and $b$ such that $6a+43b=1$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "Then $a$ is the inverse of $6\\pmod{43}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I start by writing $43=6\\cdot 7+1$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "Then I write $1=43-6\\cdot 7$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "This means that $1\\equiv (-7)\\cdot 6\\pmod{43}$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "So, $a=-7$ and $b=1$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "To get a positive value for $a$, I can add $43$ to it.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "So, $a=36$ and $b=1$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "This means that $6^{-1}\\equiv 36\\pmod{43}$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I can check that $36\\equiv 6^2\\pmod{43}$ by squaring $6$ and reducing it modulo $43$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I get $6^2=36\\equiv 36\\pmod{43}$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "So, I have confirmed that $6^{-1}\\equiv 6^2\\pmod{43}$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "Therefore, $m=43$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 255, "metadata": { "path_label": "suppose_m_is_a_two_digit_positive_integer_such_that_6_1__2022-12-26T01-18-45.679143|chosen" } }, { "text": "I know that the expansion of this expression will have 13 terms, each of the form \\[\\binom{12}{k}(\\sqrt{2})^{12 - k}(\\sqrt[3]{3})^k.\\]", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 256, "metadata": { "path_label": "find_the_sum_of_all_the_rational_terms_in_the_expansion_of__2022-12-26T00-32-03.425452|chosen" } }, { "text": "To find the rational terms, I need to find the values of k that make both exponents multiples of 2 and 3, respectively.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 256, "metadata": { "path_label": "find_the_sum_of_all_the_rational_terms_in_the_expansion_of__2022-12-26T00-32-03.425452|chosen" } }, { "text": "This means that k must be a multiple of 6, since 6 is the least common multiple of 2 and 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 256, "metadata": { "path_label": "find_the_sum_of_all_the_rational_terms_in_the_expansion_of__2022-12-26T00-32-03.425452|chosen" } }, { "text": "The only multiples of 6 between 0 and 12 are 0, 6, and 12, so there are only three rational terms in the expansion.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 256, "metadata": { "path_label": "find_the_sum_of_all_the_rational_terms_in_the_expansion_of__2022-12-26T00-32-03.425452|chosen" } }, { "text": "I want to find the probability of the event that the product of their numbers is more than 10, so I need to compare the number of favorable outcomes to the number of possible outcomes.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "The possible outcomes are all the pairs of integers between 1 and 10, inclusive, that they could pick. There are 10 choices for Krishanu's number, and for each of those choices, there are 10 choices for Shaunak's number, so the total number of possible outcomes is 10 times 10, which is 100.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "The favorable outcomes are the pairs of integers between 1 and 10, inclusive, that have a product greater than 10. I could try to list them all, but that might take too long and be prone to error. Instead, I'll use some logic and counting techniques to find them more efficiently.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "I'll start with the smallest possible number that Krishanu could pick, which is 1. If Krishanu picks 1, then no matter what Shaunak picks, the product will be 1 times that number, which will never be more than 10. So there are no favorable outcomes when Krishanu picks 1.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "Next, I'll consider the case when Krishanu picks 2. If Krishanu picks 2, then Shaunak needs to pick a number greater than 5 for the product to be more than 10. The numbers greater than 5 are 6, 7, 8, 9, and 10, so there are 5 favorable outcomes when Krishanu picks 2.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "Similarly, I'll look at the case when Krishanu picks 3. If Krishanu picks 3, then Shaunak needs to pick a number greater than 3 for the product to be more than 10. The numbers greater than 3 are 4, 5, 6, 7, 8, 9, and 10, so there are 7 favorable outcomes when Krishanu picks 3.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 257, "metadata": { "path_label": "krishanu_and_shaunak_each_pick_an_integer_at_random_between__2022-12-26T00-16-42.216538|chosen" } }, { "text": "I know that an $n^{\\text{th}}$ root of unity is a complex number $z$ that satisfies $z^n = 1$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I also know that the $n^{\\text{th}}$ roots of unity are equally spaced on the unit circle in the complex plane, with angles of $2\\pi k/n$ radians for $k = 0, 1, 2, \\dots, n-1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "So I want to find the smallest $n$ such that the solutions of $z^4 - z^2 + 1 = 0$ are also solutions of $z^n = 1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I can try to factor the equation $z^4 - z^2 + 1 = 0$ to see if I can find its roots.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I notice that this equation is a quadratic equation in terms of $z^2$, so I can use the quadratic formula to solve for $z^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I get $z^2 = \\frac{1 \\pm i\\sqrt{3}}{2}$, where $i$ is the imaginary unit.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "Then I can take the square roots of both sides to get $z = \\pm \\sqrt{\\frac{1 \\pm i\\sqrt{3}}{2}}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "This gives me four possible values of $z$, which I can write in exponential form as $z = e^{\\pm \\frac{\\pi}{6}i}$ or $z = e^{\\pm \\frac{5\\pi}{6}i}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "These are the roots of the equation $z^4 - z^2 + 1 = 0$, and they are also the vertices of a square inscribed in the unit circle.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "Now I want to find the smallest $n$ such that these values of $z$ are also $n^{\\text{th}}$ roots of unity.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "That means I want $z^n = 1$ for all four values of $z$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I can use the fact that $z^n = e^{n\\theta i}$, where $\\theta$ is the angle of $z$ in radians.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "So I want $n\\theta$ to be a multiple of $2\\pi$ for all four values of $\\theta$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "The values of $\\theta$ are $\\pm \\frac{\\pi}{6}$ and $\\pm \\frac{5\\pi}{6}$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I can see that the smallest common multiple of these angles is $2\\pi$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "That means I need $n$ to be a multiple of $12$, since $12\\theta = 2\\pi$ for all four values of $\\theta$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "The smallest positive multiple of $12$ is $12$ itself, so $n = 12$ is the smallest positive integer that satisfies the condition.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 258, "metadata": { "path_label": "what_is_the_smallest_positive_integer_n_such_that_all_the__2022-12-25T23-00-21.765097|chosen" } }, { "text": "I know that Pascal's Triangle is formed by adding the two numbers above each entry, starting with 1 on the edges.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "I also know that the number of entries in each row is one more than the row number, so the top 10 rows have 1, 2, 3, ..., 10 entries respectively.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "I can use this to count the even integers in each row, starting from the top.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "In the first row, there is only one entry, 1, which is odd.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "In the second row, there are two entries, 1 and 1, which are both odd.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "In the third row, there are three entries, 1, 2, and 1. The middle one is even, so that's one even integer so far.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "In the fourth row, there are four entries, 1, 3, 3, and 1. They are all odd.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 259, "metadata": { "path_label": "there_are_four_even_integers_in_the_top_five_rows_of_pascal__2022-12-25T06-03-55.280229|chosen" } }, { "text": "I notice that the clock chimes on a regular pattern every hour, so I can focus on counting the chimes in one hour and then multiplying by 24 to get the total for a day.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 260, "metadata": { "path_label": "my_clock_chimes_two_times_15_minutes_after_the_hour_four_ti__2022-12-25T06-00-41.069595|chosen" } }, { "text": "In one hour, the clock chimes two times at 15 minutes after the hour, four times at 30 minutes after the hour, six times at 45 minutes after the hour, and eight times at the hour.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 260, "metadata": { "path_label": "my_clock_chimes_two_times_15_minutes_after_the_hour_four_ti__2022-12-25T06-00-41.069595|chosen" } }, { "text": "That's a total of $2 + 4 + 6 + 8 = 20$ chimes every hour, not counting the chimes that match the hour.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 260, "metadata": { "path_label": "my_clock_chimes_two_times_15_minutes_after_the_hour_four_ti__2022-12-25T06-00-41.069595|chosen" } }, { "text": "To find the remainder of a division, I can use the modulo operator, which is denoted by % in many programming languages.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 261, "metadata": { "path_label": "calculate_the_whole_number_remainder_when_987_670_is_divided__2022-12-25T05-58-09.608410|chosen" } }, { "text": "So, 987,670 % 128 means the same as \"the remainder when 987,670 is divided by 128.\"", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 261, "metadata": { "path_label": "calculate_the_whole_number_remainder_when_987_670_is_divided__2022-12-25T05-58-09.608410|chosen" } }, { "text": "I can use a calculator or a long division algorithm to find this remainder.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 261, "metadata": { "path_label": "calculate_the_whole_number_remainder_when_987_670_is_divided__2022-12-25T05-58-09.608410|chosen" } }, { "text": "To find the area of the quadrilateral, I need to use a formula that relates the coordinates of the vertices and the area.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "I recall that the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ is given by the determinant of the matrix $\\begin{bmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\end{bmatrix}$ divided by $2$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "I can use this formula to divide the quadrilateral into two triangles, one with vertices $(1, 3)$, $(1, 1)$ and $(2006, 2007)$, and the other with vertices $(1, 1)$, $(2, 1)$ and $(2006, 2007)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "For the first triangle, the area is $\\frac{1}{2} \\det \\begin{bmatrix} 1 & 3 & 1 \\\\ 1 & 1 & 1 \\\\ 2006 & 2007 & 1 \\end{bmatrix}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "To find the determinant, I can use the cofactor expansion along the first row. This gives me $1 \\cdot \\det \\begin{bmatrix} 1 & 1 \\\\ 2007 & 1 \\end{bmatrix} - 3 \\cdot \\det \\begin{bmatrix} 1 & 1 \\\\ 2006 & 1 \\end{bmatrix} + 1 \\cdot \\det \\begin{bmatrix} 1 & 1 \\\\ 2006 & 2007 \\end{bmatrix}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "Simplifying, this becomes $1 \\cdot (1 - 2007) - 3 \\cdot (1 - 2006) + 1 \\cdot (2007 - 2006) = -2006 + 6015 + 1 = 4010$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "Therefore, the area of the first triangle is $\\frac{1}{2} \\cdot 4010 = 2005$ square units.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "For the second triangle, the area is $\\frac{1}{2} \\det \\begin{bmatrix} 1 & 1 & 1 \\\\ 2 & 1 & 1 \\\\ 2006 & 2007 & 1 \\end{bmatrix}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "Again, I can use the cofactor expansion along the first row. This gives me $1 \\cdot \\det \\begin{bmatrix} 1 & 1 \\\\ 2007 & 1 \\end{bmatrix} - 1 \\cdot \\det \\begin{bmatrix} 2 & 1 \\\\ 2006 & 1 \\end{bmatrix} + 1 \\cdot \\det \\begin{bmatrix} 2 & 1 \\\\ 2006 & 2007 \\end{bmatrix}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 262, "metadata": { "path_label": "a_quadrilateral_in_the_plane_has_vertices_1_3_1_1__2022-12-24T06-45-04.841048|chosen" } }, { "text": "The first thing we want to do is simplify the left side of the equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "The first term is $(\\sin \\alpha + \\csc \\alpha)^2$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "That's equal to $\\sin^2 \\alpha + 2 \\sin \\alpha \\csc \\alpha + \\csc^2 \\alpha$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "And we know that $\\sin \\alpha \\csc \\alpha = 1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "Right. So that means that $(\\sin \\alpha + \\csc \\alpha)^2 = \\sin^2 \\alpha + 2 + \\csc^2 \\alpha$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "The second term is $(\\cos \\alpha + \\sec \\alpha)^2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "That's equal to $\\cos^2 \\alpha + 2 \\cos \\alpha \\sec \\alpha + \\sec^2 \\alpha$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "And we know that $\\cos \\alpha \\sec \\alpha = 1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "Right. So that means that $(\\cos \\alpha + \\sec \\alpha)^2 = \\cos^2 \\alpha + 2 + \\sec^2 \\alpha$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "So the left side of the equation is equal to $\\sin^2 \\alpha + 2 + \\csc^2 \\alpha + \\cos^2 \\alpha + 2 + \\sec^2 \\alpha$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "And we know that $\\sin^2 \\alpha + \\cos^2 \\alpha = 1$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "So the left side of the equation is equal to $1 + 2 + 2 + \\csc^2 \\alpha + \\sec^2 \\alpha$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "And that's equal to $5 + \\csc^2 \\alpha + \\sec^2 \\alpha$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 263, "metadata": { "path_label": "find_k_if_sin_alpha_csc_alpha_2_cos_alpha__2022-12-24T05-19-47.510962|chosen" } }, { "text": "So the surface area of the original cube is 6 times the area of each face.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "And each face is a square, so its area is the length of each edge squared.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "Let's call the length of each edge x. So the surface area of the original cube is 6*$x^2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "And if we increase the length of each edge by 50%, we get 1.5*x.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "So the area of each face is $(1.5*x)^2$ which is $2.25*x^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "So the surface area of the new cube is 6*$2.25*x^2$ which is $13.5*x^2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "And if we divide that by the original surface area, we get $13.5*x^2$/$6*x^2$ which is 2.25.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 264, "metadata": { "path_label": "when_each_edge_of_a_cube_is_increased_by_50_by_what_per__2022-12-24T05-10-30.862639|chosen" } }, { "text": "We know that the perimeter of the first triangle is $2+2+2 = 6$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "And that the perimeter of the second triangle is $3+3+3 = 9$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "Right. So the percent increase in the perimeter from the first triangle to the second triangle is $\\frac{9-6}{6}*100 = 50$ percent.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "Let's find the perimeter of the third triangle.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "The sides of the third triangle are $150\\%$ of the sides of the second triangle, so they are $1.5*3 = 4.5$ units long.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "That means the perimeter of the third triangle is $4.5+4.5+4.5 = 13.5$ units.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "So the percent increase in the perimeter from the second triangle to the third triangle is $\\frac{13.5-9}{9}*100 = 50$ percent.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "Let's find the perimeter of the fourth triangle.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "The sides of the fourth triangle are $150\\%$ of the sides of the third triangle, so they are $1.5*4.5 = 6.75$ units long.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "That means the perimeter of the fourth triangle is $6.75+6.75+6.75 = 20.25$ units.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "So the percent increase in the perimeter from the third triangle to the fourth triangle is $\\frac{20.25-13.5}{13.5}*100 = 50$ percent.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 265, "metadata": { "path_label": "an_equilateral_triangle_has_sides_of_length_2_units_a_secon__2022-12-24T05-07-49.135114|chosen" } }, { "text": "I notice that all the terms on the left side of the equation are powers of 2, so I wonder if I can rewrite the equation in terms of a single base.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "I recall that $4=2^2$ and $8=2^3$, so I can substitute those expressions into the equation and use the exponent rules to get: $$2^a+2^{2b}+2^{3c}=328.$$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "Now I have a single base on the left side, but the right side is not a power of 2.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "However, I know that every positive integer can be written as a sum of distinct powers of 2, so I try to find such a representation for 328.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "I start with the largest power of 2 that is less than 328, which is 256.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "Then I subtract 256 from 328 and get 72.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "I repeat the process with 72 and find the next largest power of 2, which is 64.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "Subtracting 64 from 72 gives me 8, which is also a power of 2.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "So I have found that $$328=256+64+8=2^8+2^6+2^3.$$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "I can now equate the two sides of the equation and use the fact that if $2^x=2^y$, then $x=y$ to get a system of three equations: $$\\begin{aligned} 2^a&=2^8 \\\\ 2^{2b}&=2^6 \\\\ 2^{3c}&=2^3 \\end{aligned}$$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "Solving for $a$, $b$, and $c$ is easy now, I just need to divide the exponents by the coefficients of the variables.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "I get: $$\\begin{aligned} a&=8 \\\\ b&=3 \\\\ c&=1 \\end{aligned}$$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 266, "metadata": { "path_label": "solve_over_the_integers_2_a_4_b_8_c_328__2022-12-24T05-01-11.533682|chosen" } }, { "text": "So Juan's diameter could be anywhere from $20 - 20\\% * 20$ to $20 + 20\\% * 20$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 267, "metadata": { "path_label": "in_order_to_compute_the_area_of_a_particular_circle_juan_fi__2022-12-24T04-54-51.032389|chosen" } }, { "text": "That's $16$ to $24$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 267, "metadata": { "path_label": "in_order_to_compute_the_area_of_a_particular_circle_juan_fi__2022-12-24T04-54-51.032389|chosen" } }, { "text": "So the radius could be anywhere from $8$ to $12$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 267, "metadata": { "path_label": "in_order_to_compute_the_area_of_a_particular_circle_juan_fi__2022-12-24T04-54-51.032389|chosen" } }, { "text": "And the area could be anywhere from $64\\pi$ to $144\\pi$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 267, "metadata": { "path_label": "in_order_to_compute_the_area_of_a_particular_circle_juan_fi__2022-12-24T04-54-51.032389|chosen" } }, { "text": "Let's set up an equation. You worked $t-6$ hours and earned $2t-5$ dollars per hour, so you earned $(t-6)(2t-5)$ dollars.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "That's correct. I earned $2t^2-11t+30$ dollars.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "And Andrew worked $2t-8$ hours and earned $t-5$ dollars per hour, so Andrew earned $(2t-8)(t-5)$ dollars.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "That's correct. Andrew earned $2t^2-13t+40$ dollars.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "But we earned the same amount, so $2t^2-11t+30=2t^2-13t+40$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "We can subtract $2t^2$ from both sides to get $-11t+30=-13t+40$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "Right. Then we can add $13t$ to both sides to get $2t+30=40$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "Finally, we can subtract 30 from both sides to get $2t=10$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "So $t=5$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "That's correct.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 268, "metadata": { "path_label": "on_monday_i_worked_t_6_hours_and_earned_2t_5_dollars_pe__2022-12-24T04-52-17.036482|chosen" } }, { "text": "Let's start by finding the probability that the first three digits are correct.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 269, "metadata": { "path_label": "megan_has_lost_fatima_s_phone_number_megan_knows_that_the_f__2022-12-24T03-39-41.519430|chosen" } }, { "text": "The first three digits can be either 299 or 296, so there are 2 possible choices for these digits.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 269, "metadata": { "path_label": "megan_has_lost_fatima_s_phone_number_megan_knows_that_the_f__2022-12-24T03-39-41.519430|chosen" } }, { "text": "That's right. Now, let's find the probability that the last four digits are correct.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 269, "metadata": { "path_label": "megan_has_lost_fatima_s_phone_number_megan_knows_that_the_f__2022-12-24T03-39-41.519430|chosen" } }, { "text": "There are 4!=24 possible orderings of the digits 0, 1, 6, and 7.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 269, "metadata": { "path_label": "megan_has_lost_fatima_s_phone_number_megan_knows_that_the_f__2022-12-24T03-39-41.519430|chosen" } }, { "text": "That's correct. Since there are 24 possible orderings, the probability of guessing the correct ordering is 1/24.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 269, "metadata": { "path_label": "megan_has_lost_fatima_s_phone_number_megan_knows_that_the_f__2022-12-24T03-39-41.519430|chosen" } }, { "text": "So we're trying to find the value of $\\displaystyle\\frac{235^2-221^2}{14}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "Right, let's start by simplifying the numerator.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "Ok, so $235^2$ is $55225$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "And $221^2$ is $48441$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "So $55225-48441$ is $6784$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "Now let's divide that by 14.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 270, "metadata": { "path_label": "what_is_the_value_of_displaystyle_frac_235_2_221_2_14__2022-12-24T03-37-31.782137|chosen" } }, { "text": "I know that the area of a rhombus is half the product of its diagonals, so I need to find $AC$ and $BD$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 271, "metadata": { "path_label": "find_the_area_of_rhombus_abcd_given_that_the_radii_of_the__2022-12-27T02-32-10.253230|chosen" } }, { "text": "I also know that the diagonals of a rhombus are perpendicular bisectors of each other, so they form four right triangles inside the rhombus.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 271, "metadata": { "path_label": "find_the_area_of_rhombus_abcd_given_that_the_radii_of_the__2022-12-27T02-32-10.253230|chosen" } }, { "text": "Let me label the half-diagonals as $x$ and $y$, so $AC = 2x$ and $BD = 2y$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 271, "metadata": { "path_label": "find_the_area_of_rhombus_abcd_given_that_the_radii_of_the__2022-12-27T02-32-10.253230|chosen" } }, { "text": "I can use the formula for the circumradius of a right triangle, which is half the hypotenuse, to write two equations for $x$ and $y$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 271, "metadata": { "path_label": "find_the_area_of_rhombus_abcd_given_that_the_radii_of_the__2022-12-27T02-32-10.253230|chosen" } }, { "text": "To multiply two numbers in base 9, I can use the standard algorithm of multiplying each digit by each digit and adding the results, but I have to be careful with the arithmetic and the carry-overs, since they are different in base 9.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "First, I'll multiply 218 by 5, starting from the rightmost digit.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "8 times 5 is 40 in base 10, but in base 9, 40 is equal to 44, since 9 times 4 is 36 and 4 more is 40.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "So I write 4 in the ones place and carry over 4 to the next column.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Next, I multiply 1 by 5, which is 5, and add the carry-over of 4, which gives 9. [*,] But in base 9, 9 is equal to 10, since 9 times 1 is 9 and 1 more is 10.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "So I write 0 in the nines place and carry over 1 to the next column.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Finally, I multiply 2 by 5, which is 10, and add the carry-over of 1, which gives 11.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "But in base 9, 11 is equal to 12, since 9 times 1 is 9 and 2 more is 11.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "So I write 2 in the 81s place and carry over 1 to the next column.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Since there are no more digits to multiply, I write the final carry-over of 1 in the 729s place.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "So far, I have 1204 as the partial product of 218 and 5 in base 9.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Next, I need to multiply 218 by 0, which is 0, and write it under the partial product, shifting it one place to the right.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "So I write 0 in the nines place and 0 in the 81s place.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "This is the partial product of 218 and 0 in base 9.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Now, I need to add the two partial products to get the final product.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "To do this, I align the digits by their place values and add them from right to left, carrying over as needed.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Starting from the ones place, I add 4 and 0, which is 4, and write 4 in the ones place.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Moving to the nines place, I add 0 and 0, which is 0, and write 0 in the nines place.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Moving to the 81s place, I add 2 and 0, which is 2, and write 2 in the 81s place.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Moving to the 729s place, I add 1 and 0, which is 1, and write 1 in the 729s place.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "Since there are no more digits to add, I stop here.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "The final product is 1204 in base 9.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 272, "metadata": { "path_label": "find_the_product_of_218_9_cdot_5_9_express_your_answer__2022-12-27T01-03-05.607807|chosen" } }, { "text": "I want to find the values of x that make f(x) undefined or invalid.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 273, "metadata": { "path_label": "compute_the_domain_of_the_function_f_x_frac_1_lfloor_x__2022-12-26T23-44-15.448820|chosen" } }, { "text": "One thing that could make f(x) undefined is if the denominator is zero, since dividing by zero is not allowed.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 273, "metadata": { "path_label": "compute_the_domain_of_the_function_f_x_frac_1_lfloor_x__2022-12-26T23-44-15.448820|chosen" } }, { "text": "So I need to avoid any x that satisfies $\\lfloor x^2+3x+3\\rfloor=0$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 273, "metadata": { "path_label": "compute_the_domain_of_the_function_f_x_frac_1_lfloor_x__2022-12-26T23-44-15.448820|chosen" } }, { "text": "The floor function rounds down any real number to the nearest integer, so $\\lfloor x^2+3x+3\\rfloor=0$ means that $x^2+3x+3$ must be between 0 and 1, not inclusive.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 273, "metadata": { "path_label": "compute_the_domain_of_the_function_f_x_frac_1_lfloor_x__2022-12-26T23-44-15.448820|chosen" } }, { "text": "I can write this as $0 2*80+y=200.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 315, "metadata": { "path_label": "alice_buys_three_burgers_and_two_sodas_for_3_20_and_bil__2022-12-24T08-06-32.693133|chosen" } }, { "text": "That simplifies to 160+y=200.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 315, "metadata": { "path_label": "alice_buys_three_burgers_and_two_sodas_for_3_20_and_bil__2022-12-24T08-06-32.693133|chosen" } }, { "text": "So y=40.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 315, "metadata": { "path_label": "alice_buys_three_burgers_and_two_sodas_for_3_20_and_bil__2022-12-24T08-06-32.693133|chosen" } }, { "text": "So a soda costs 40 cents.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 315, "metadata": { "path_label": "alice_buys_three_burgers_and_two_sodas_for_3_20_and_bil__2022-12-24T08-06-32.693133|chosen" } }, { "text": "I think we should use complementary counting.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 316, "metadata": { "path_label": "at_the_end_of_the_year_the_math_club_decided_to_hold_an_ele__2022-12-24T07-50-13.105685|chosen" } }, { "text": "That means we should find the number of elections that have no past officers and subtract that from the total number of possible elections.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 316, "metadata": { "path_label": "at_the_end_of_the_year_the_math_club_decided_to_hold_an_ele__2022-12-24T07-50-13.105685|chosen" } }, { "text": "That's correct. First, let's find the total number of possible elections.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 316, "metadata": { "path_label": "at_the_end_of_the_year_the_math_club_decided_to_hold_an_ele__2022-12-24T07-50-13.105685|chosen" } }, { "text": "Each candidate can either be elected or not elected, so there are 2 options for each candidate.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 316, "metadata": { "path_label": "at_the_end_of_the_year_the_math_club_decided_to_hold_an_ele__2022-12-24T07-50-13.105685|chosen" } }, { "text": "I notice that the number $AB,CBA$ is palindromic, meaning it reads the same forwards and backwards.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "This suggests that there might be a shortcut to checking its divisibility by eleven, since I recall that a number is divisible by eleven if the alternating sum of its digits is divisible by eleven.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "For example, $121$ is divisible by eleven because $1 - 2 + 1 = 0$, which is divisible by eleven.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "So, to check if $AB,CBA$ is divisible by eleven, I can compute $A - B + C - B + A$ and see if it is divisible by eleven.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "This simplifies to $2A - 2B + C$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "Now, I want to maximize $AB,CBA$, so I want to make $A$, $B$ and $C$ as large as possible, subject to the constraints that they are distinct and that $2A - 2B + C$ is divisible by eleven.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "I can start by trying $A = 9$, which is the largest possible digit.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "Then, I need to find $B$ and $C$ such that $18 - 2B + C$ is divisible by eleven.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "I can try $B = 8$, which is the next largest possible digit.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "Then, I need $C$ such that $18 - 16 + C$ is divisible by eleven.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "This means $C$ has to be $9$, which is the same as $A$, so this doesn't work.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "I can try $B = 7$, which is the next largest possible digit.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "Then, I need $C$ such that $18 - 14 + C$ is divisible by eleven.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "This means $C$ has to be $7$, which is the same as $B$, so this doesn't work either.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "I can try $B = 6$, which is the next largest possible digit.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "Then, I need $C$ such that $18 - 12 + C$ is divisible by eleven.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "This means $C$ has to be $5$, which is not the same as $A$ or $B$, so this works.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "So, the largest possible value of $AB,CBA$ that is divisible by eleven is $96,569$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 317, "metadata": { "path_label": "a_positive_five_digit_integer_is_in_the_form_ab_cba_where__2022-12-24T07-44-57.755785|chosen" } }, { "text": "I need to find the fraction of license plates that have either a three-letter palindrome, a three-digit palindrome, or both.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "I can use the principle of inclusion-exclusion to avoid counting the same plates twice.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "To apply the principle of inclusion-exclusion, I need to know how many plates have a three-letter palindrome, how many have a three-digit palindrome, and how many have both.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "I will use the notation $|A|$ to mean the number of elements in a set $A$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "Let $A$ be the set of plates with a three-letter palindrome, and $B$ be the set of plates with a three-digit palindrome.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "I want to find $|A \\cup B|$, which is the number of plates with either a three-letter palindrome, a three-digit palindrome, or both.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "By the principle of inclusion-exclusion, $|A \\cup B| = |A| + |B| - |A \\cap B|$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 318, "metadata": { "path_label": "many_states_use_a_sequence_of_three_letters_followed_by_a_se__2022-12-24T07-42-37.423225|chosen" } }, { "text": "I notice that the equation has a lot of cubic and quadratic terms, so I wonder if I can factor it somehow.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 319, "metadata": { "path_label": "how_many_ordered_pairs_of_integers_m_n_satisfy_the_equat__2022-12-24T07-41-42.972750|chosen" } }, { "text": "On the left side, I see that $m^3+6m^2+5m$ has a common factor of $m$, so I pull that out and get $m(m^2+6m+5)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 319, "metadata": { "path_label": "how_many_ordered_pairs_of_integers_m_n_satisfy_the_equat__2022-12-24T07-41-42.972750|chosen" } }, { "text": "The remaining quadratic $m^2+6m+5$ can be factored further into $(m+1)(m+5)$, so the left side becomes $m(m+1)(m+5)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 319, "metadata": { "path_label": "how_many_ordered_pairs_of_integers_m_n_satisfy_the_equat__2022-12-24T07-41-42.972750|chosen" } }, { "text": "We know that $|a+bi|=\\sqrt{a^2+b^2}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 320, "metadata": { "path_label": "evaluate_left_4_frac_7_6_i_right__2022-12-24T07-33-34.077033|chosen" } }, { "text": "That's right. So we just need to plug in the values of $a$ and $b$ into this formula.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 320, "metadata": { "path_label": "evaluate_left_4_frac_7_6_i_right__2022-12-24T07-33-34.077033|chosen" } }, { "text": "That's correct. In this case, $a=-4$ and $b=\\frac{7}{6}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 320, "metadata": { "path_label": "evaluate_left_4_frac_7_6_i_right__2022-12-24T07-33-34.077033|chosen" } }, { "text": "So $|{-4+\\frac{7}{6}i}|=\\sqrt{(-4)^2+\\left(\\frac{7}{6}\\right)^2}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 320, "metadata": { "path_label": "evaluate_left_4_frac_7_6_i_right__2022-12-24T07-33-34.077033|chosen" } }, { "text": "To find the equation of the circle, I need to know its center and radius.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "The center of the circle lies on the given line, so I can write its coordinates as $$(x, 4x-3)$$ for some value of $$x$$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "The radius of the circle is the distance from the center to any of the given points, say $$(2, 3)$$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Using the distance formula, the radius squared is equal to $$(x-2)^2+(4x-3-3)^2 = (x-2)^2+(4x-6)^2$$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "I can simplify this expression by expanding the squares and collecting like terms: $$r^2 = x^2-4x+4+16x^2-48x+36 = 17x^2-52x+40$$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Now I have an equation relating $$x$$ and $$r$$, but I want to eliminate $$r$$ and find $$x$$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "To do that, I can use the fact that the circle also passes through $$(4, 5)$$, which means that the radius squared is also equal to $$(x-4)^2+(4x-3-5)^2 = (x-4)^2+(4x-8)^2$$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Expanding and simplifying this expression gives me another equation: $$r^2 = x^2-8x+16+16x^2-64x+64 = 17x^2-72x+80$$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Since both expressions are equal to $$r^2$$, I can set them equal to each other and solve for $$x$$: $$17x^2-52x+40=17x^2-72x+80$$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Subtracting $$17x^2$$ from both sides and rearranging, I get $$20x = 40$$, which implies $$x=2$$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Now that I have $$x$$, I can plug it into the coordinates of the center to get $$(2, 4(2)-3) = (2, 5)$$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "I can also plug $$x=2$$ into the expression for the radius squared to get $$r^2 = 17(2)^2-52(2)+40 = 16$$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Therefore, the radius is $$r=4$$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Finally, I can write the equation of the circle using the standard form: $$(x-h)^2+(y-k)^2=r^2$$, where $$(h, k)$$ is the center and $$r$$ is the radius.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "Substituting the values I found, I get $$(x-2)^2+(y-5)^2=16$$ as the equation of the circle.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 321, "metadata": { "path_label": "find_the_equation_of_the_circle_which_passes_through_2_3__2022-12-24T07-30-28.220715|chosen" } }, { "text": "This integral looks tricky, but I notice that the denominator is a linear combination of sine and cosine, which makes me think of a trigonometric identity.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 322, "metadata": { "path_label": "evaluate_the_given_integral_displaystyle_int_cfrac__2022-12-24T07-23-40.535850|chosen" } }, { "text": "If I recall correctly, there is an identity that says $$\\cos { \\left( x+\\frac { \\pi }{ 4 } \\right) } =\\frac { 1 }{ \\sqrt { 2 } } \\left( \\cos { x } +\\sin { x } \\right) $$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 322, "metadata": { "path_label": "evaluate_the_given_integral_displaystyle_int_cfrac__2022-12-24T07-23-40.535850|chosen" } }, { "text": "So maybe I can use that to simplify the denominator by rewriting it as a multiple of $\\cos { \\left( x+\\frac { \\pi }{ 4 } \\right) }$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 322, "metadata": { "path_label": "evaluate_the_given_integral_displaystyle_int_cfrac__2022-12-24T07-23-40.535850|chosen" } }, { "text": "I need to find the numbers of cookies that are multiples of $11$ plus $4$, and also multiples of $7$ plus $1$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "One way to do this is to list the multiples of $11$ plus $4$ until I reach $100$, and then check which ones are also multiples of $7$ plus $1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "The multiples of $11$ plus $4$ are $15, 26, 37, 48, 59, 70, 81, 92$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "The multiples of $7$ plus $1$ are $8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "The numbers that appear in both lists are $15, 92$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "So the possible numbers of cookies that Eleanor could have made are $15$ and $92$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "The sum of these numbers is $15 + 92 = 107$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 323, "metadata": { "path_label": "eleanor_is_making_chocolate_chip_cookies_for_her_friends_if__2022-12-23T18-05-19.882271|chosen" } }, { "text": "I want to find the fraction of outcomes where the product of the two top numbers is greater than their sum.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "One way to do this is to list all the possible outcomes and count how many satisfy the condition.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "There are 8 x 8 = 64 possible outcomes, since each die can show any of the 8 faces.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "I can arrange the outcomes in a table, where the rows represent the first die and the columns represent the second die.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "For example, the outcome (1, 1) means the first die shows a 1 and the second die shows a 1, and the outcome (2, 8) means the first die shows a 2 and the second die shows an 8.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "I can fill in each cell of the table with the product of the two top numbers and the sum of the two top numbers, and compare them.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "For example, the cell (1, 1) will have 1 x 1 = 1 and 1 + 1 = 2, so the product is not greater than the sum.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "The cell (2, 8) will have 2 x 8 = 16 and 2 + 8 = 10, so the product is greater than the sum.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "I can use a symbol like > to indicate that the product is greater than the sum, and a symbol like <= to indicate that the product is not greater than the sum.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 324, "metadata": { "path_label": "two_eight_sided_dice_each_have_faces_numbered_1_through_8_w__2022-12-23T18-01-56.856893|chosen" } }, { "text": "I need to find a multiple of 25 that has a product of digits that is also a multiple of 25.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 325, "metadata": { "path_label": "what_is_the_least_positive_multiple_of_25_for_which_the_prod__2022-12-23T17-47-03.942207|chosen" } }, { "text": "That means the digits of the multiple of 25 must either be all 5s, or have a pair of 5s and a pair of 2s.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 325, "metadata": { "path_label": "what_is_the_least_positive_multiple_of_25_for_which_the_prod__2022-12-23T17-47-03.942207|chosen" } }, { "text": "I can start by looking at the smallest multiples of 25 and see if they have this property.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 325, "metadata": { "path_label": "what_is_the_least_positive_multiple_of_25_for_which_the_prod__2022-12-23T17-47-03.942207|chosen" } }, { "text": "25 = 25, but the product of its digits is 10, not a multiple of 25.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 325, "metadata": { "path_label": "what_is_the_least_positive_multiple_of_25_for_which_the_prod__2022-12-23T17-47-03.942207|chosen" } }, { "text": "50 = 25 * 2, but the product of its digits is 0, not a positive multiple of 25.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 325, "metadata": { "path_label": "what_is_the_least_positive_multiple_of_25_for_which_the_prod__2022-12-23T17-47-03.942207|chosen" } }, { "text": "This problem involves some algebra and symmetry.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "I notice that the sum of the powers of $ a$, $ b$, $ c$ is zero for both odd and even exponents.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "This suggests that there might be a factorization or identity that relates the sum of the powers of $ a$, $ b$, $ c$ to their sum and product.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "I recall that $ a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "This might be useful, since $ a+b+c=0$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "If I substitute $ a+b+c=0$ into the identity, I get $ a^3+b^3+c^3=3abc$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "This also means that $ a^5+b^5+c^5=3abc(a^2+b^2+c^2)$, by multiplying both sides by $ a^2+b^2+c^2$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "Now I have two equations involving $ a^3+b^3+c^3$ and $ a^5+b^5+c^5$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "I can equate them and simplify: $ a^3+b^3+c^3=a^5+b^5+c^5 \\implies 3abc=3abc(a^2+b^2+c^2) \\implies 1=a^2+b^2+c^2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "Therefore, the value of $ a^2+b^ the same as the value of $ a^3+b^3+c^3$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 326, "metadata": { "path_label": "let_a_b_c_be_nonzero_real_numbers_such_that_a_b__2022-12-23T17-28-06.418651|chosen" } }, { "text": "I recognize this as a geometric series, where the first term is $x^n$ and the common ratio is $x^{-1}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 327, "metadata": { "path_label": "find_the_maximum_number_of_real_roots_to_a_polynomial_of_the__2022-12-23T17-22-00.257954|chosen" } }, { "text": "I can use the formula for the sum of a geometric series to rewrite the polynomial as \\[\\frac{x^{n + 1} - 1}{x - 1} = 0.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 327, "metadata": { "path_label": "find_the_maximum_number_of_real_roots_to_a_polynomial_of_the__2022-12-23T17-22-00.257954|chosen" } }, { "text": "Now I can see that this equation has a root at $x = 1$, but this is not a valid root because it would make the denominator zero.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 327, "metadata": { "path_label": "find_the_maximum_number_of_real_roots_to_a_polynomial_of_the__2022-12-23T17-22-00.257954|chosen" } }, { "text": "So I need to find the roots of the numerator, which is $x^{n + 1} - 1 = 0$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 327, "metadata": { "path_label": "find_the_maximum_number_of_real_roots_to_a_polynomial_of_the__2022-12-23T17-22-00.257954|chosen" } }, { "text": "I know that base $6$ means that each digit represents a power of $6$, starting from the rightmost one.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "So, $555_6 = 5 \\times 6^2 + 5 \\times 6^1 + 5 \\times 6^0 = 215_{10}$, where $10$ means base $10$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "Similarly, $55_6 = 5 \\times 6^1 + 5 \\times 6^0 = 35_{10}$ and $5_6 = 5 \\times 6^0 = 5_{10}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "To find the sum, I can add these numbers in base $10$ and then convert the result back to base $6$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "So, $555_6 + 55_6 + 5_6 = 215_{10} + 35_{10} + 5_{10} = 255_{10}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "To convert $255_{10}$ to base $6$, I can divide it by $6$ and write the remainder as the rightmost digit, then repeat with the quotient until I get zero.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "So, $255_{10} \\div 6 = 42_{10}$ with remainder $3$, so the rightmost digit is $3_6$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "Then, $42_{10} \\div 6 = 7_{10}$ with remainder $0$, so the next digit is $0_6$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "Finally, $7_{10} \\div 6 = 1_{10}$ with remainder $1$, so the leftmost digit is $1_6$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 328, "metadata": { "path_label": "find_the_sum_of_555_6_55_6_and_5_6_in_base_6__2022-12-23T17-17-28.562767|chosen" } }, { "text": "I need to find a number that is divisible by $4$, leaves a remainder of $2$ when divided by $3$, and leaves a remainder of $5$ when divided by $11$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "This number must also be close to $200$, since that is the emperor's estimate.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "I can start by trying multiples of $4$, since that is the easiest condition to satisfy.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The first multiple of $4$ that is greater than $200$ is $204$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "I can check if this works by dividing it by $3$ and $11$ and seeing the remainders.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$204 \\div 3 = 68$ with no remainder, so this does not work. [* for i in range(1, 10)]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $204 + 4 = 208$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$208 \\div 3 = 69$ with a remainder of $1$, so this does not work either.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $208 + 4 = 212$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$212 \\div 3 = 70$ with a remainder of $2$, which matches one condition.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$212 \\div 11 = 19$ with a remainder of $3$, which does not match the other condition.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $212 + 4 = 216$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$216 \\div 3 = 72$ with no remainder, so this does not work.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $216 + 4 = 220$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$220 \\div 3 = 73$ with a remainder of $1$, so this does not work.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $220 + 4 = 224$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$224 \\div 3 = 74$ with a remainder of $2$, which matches one condition.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$224 \\div 11 = 20$ with a remainder of $4$, which does not match the other condition.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $224 + 4 = 228$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$228 \\div 3 = 76$ with no remainder, so this does not work.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $228 + 4 = 232$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$232 \\div 3 = 77$ with a remainder of $1$, so this does not work.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "The next multiple of $4$ is $232 + 4 = 236$.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$236 \\div 3 = 78$ with a remainder of $2$, which matches one condition.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$236 \\div 11 = 21$ with a remainder of $5$, which matches the other condition.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "This means that $236$ is the smallest number that satisfies all the conditions.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "I can check if there are any larger numbers that also work by adding $4 \\cdot 3 \\cdot 11 = 132$ to $236$, since that is the least common multiple of $4$, $3$, and $11$.", "role": "entry", "step_index": 26, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "$236 + 132 = 368$, which is too large to be the emperor's estimate.", "role": "entry", "step_index": 27, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "Therefore, the most likely number of soldiers in the regiment is $236$.", "role": "entry", "step_index": 28, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 329, "metadata": { "path_label": "a_chinese_emperor_orders_a_regiment_of_soldiers_in_his_palac__2022-12-23T17-15-57.483916|chosen" } }, { "text": "I need to find the integers between 200 and 300 that have the sum of their digits equal to 15.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 330, "metadata": { "path_label": "how_many_integers_between_200_and_300_have_the_sum_of_their__2022-12-23T17-11-59.439789|chosen" } }, { "text": "I notice that the first digit of these integers must be 2, so the sum of the other two digits must be 13.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 330, "metadata": { "path_label": "how_many_integers_between_200_and_300_have_the_sum_of_their__2022-12-23T17-11-59.439789|chosen" } }, { "text": "I can list the possible pairs of digits that add up to 13: (4, 9), (5, 8), (6, 7), (7, 6), (8, 5), and (9, 4).", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 330, "metadata": { "path_label": "how_many_integers_between_200_and_300_have_the_sum_of_their__2022-12-23T17-11-59.439789|chosen" } }, { "text": "The sum $7! + 8! + 9! + \\cdots + 2006!$ is very large, so I don't want to calculate it exactly.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "I only need to find the tens digit, so I can ignore any terms that are multiples of 100.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "That means I can ignore all the factorials from 10! onwards, since they are all divisible by 10 and 100.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "So the sum becomes $7! + 8! + 9!$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "Now I can use a calculator or recall that $7! = 5040$, $8! = 40320$, and $9! = 362880$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "Adding these numbers, I get $412240$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "The tens digit is the second to last digit, which is 4.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 331, "metadata": { "path_label": "what_is_the_tens_digit_in_the_sum_7_8_9_cdots_2__2022-12-23T17-11-07.725447|chosen" } }, { "text": "To find the integers between 300 and 500 with the sum of their digits equal to 16, I need to consider the possible values of the hundreds, tens, and units digits.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "Since the number is between 300 and 500, the hundreds digit must be either 3 or 4.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "If the hundreds digit is 3, then the tens and units digits must add up to 13.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "I can list the pairs of digits that add up to 13: (4, 9), (5, 8), (6, 7), (7, 6), (8, 5), and (9, 4).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "That gives me six possible numbers: 349, 358, 367, 376, 385, and 394.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "If the hundreds digit is 4, then the tens and units digits must add up to 12.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "I can list the pairs of digits that add up to 12: (3, 9), (4, 8), (5, 7), (6, 6), (7, 5), (8, 4), and (9, 3).", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "That gives me seven possible numbers: 439, 448, 457, 466, 475, 484, and 493.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "Therefore, the total number of integers between 300 and 500 with the sum of their digits equal to 16 is 6 + 7 = 13.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 332, "metadata": { "path_label": "how_many_integers_between_300_and_500_have_the_sum_of_their__2022-12-23T17-00-43.225024|chosen" } }, { "text": "I know that a number is divisible by 3 if the sum of its digits is divisible by 3.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "So, for a three-digit palindrome, the first and third digits must be the same, and the middle digit can be anything.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Let me call the first and third digits x, and the middle digit y.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Then, the sum of the digits is x + y + x, which simplifies to 2x + y.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "This sum must be divisible by 3, so I need to find the possible values of x and y that satisfy this condition.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "I know that x can be any digit from 1 to 9, since 0 would make the number not three digits.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "And y can be any digit from 0 to 9, since it does not affect the number of digits.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "So, there are 9 choices for x and 10 choices for y, which gives me 90 possible three-digit palindromes.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Now, I need to find out how many of them are divisible by 3.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "I can use the fact that 2x + y is divisible by 3 if and only if y is congruent to -2x modulo 3.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "This means that y and -2x have the same remainder when divided by 3.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "For example, if x is 1, then -2x is -2, and y can be 1, 4, or 7, since they all have the same remainder as -2 when divided by 3.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Similarly, if x is 2, then -2x is -4, and y can be 2, 5, or 8.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "And if x is 3, then -2x is -6, and y can be 0, 3, 6, or 9.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "I can see a pattern here: for each value of x, there are either 3 or 4 values of y that make 2x + y divisible by 3.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Specifically, if x is a multiple of 3, then there are 4 values of y that work, and if x is not a multiple of 3, then there are 3 values of y that work.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "So, out of the 9 choices for x, 3 of them are multiples of 3, and 6 of them are not.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "This means that the number of three-digit palindromes that are divisible by 3 is 3 * 4 + 6 * 3, which is 30.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "Therefore, the probability that a randomly chosen three-digit palindrome is divisible by 3 is 30/90, which simplifies to 1/3.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 333, "metadata": { "path_label": "a_palindrome_is_a_number_which_reads_the_same_forward_as_bac__2022-12-23T16-38-26.994192|chosen" } }, { "text": "To find the largest number that is certain to divide $P$, I need to think about the possible values of $P$ and their common factors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "The possible values of $P$ depend on which face is hidden, so there are six cases to consider.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 1, then $P$ is the product of 2, 3, 4, 5, and 6, which is 720.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 2, then $P$ is the product of 1, 3, 4, 5, and 6, which is 360.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 3, then $P$ is the product of 1, 2, 4, 5, and 6, which is 240.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 4, then $P$ is the product of 1, 2, 3, 5, and 6, which is 180.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 5, then $P$ is the product of 1, 2, 3, 4, and 6, which is 144.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "If the hidden face is 6, then $P$ is the product of 1, 2, 3, 4, and 5, which is 120.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "The largest number that is certain to divide $P$ is the greatest common factor of all these possible values.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "To find the greatest common factor, I can use prime factorization and look for the common prime factors and their lowest powers.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "720 = 2^4 * 3^2 * 5^1, 360 = 2^3 * 3^2 * 5^1, 240 = 2^4 * 3^1 * 5^1, 180 = 2^2 * 3^2 * 5^1, 144 = 2^4 * 3^2, 120 = 2^3 * 3^1 * 5^1.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "The common prime factors are 2, 3, and 5, and their lowest powers are 2^2, 3^1, and 5^0, respectively.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "The greatest common factor is 2^2 * 3^1 * 5^0 = 12.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "Therefore, the largest number that is certain to divide $P$ is 12.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 334, "metadata": { "path_label": "a_standard_six_sided_die_is_rolled_and_p_is_the_product__2022-12-23T16-36-00.814241|chosen" } }, { "text": "To solve this problem, I will use the fact that a median of a right triangle is half the length of the hypotenuse, and also the cosine rule.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Since $\\overline{BM}$ is a median of $\\triangle ABC$, I know that $BM = \\frac{AC}{2}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "To find $AC$, I can use the Pythagorean theorem, since $\\triangle ABC$ is a right triangle.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "I get $AC^2 = AB^2 + BC^2$, or $AC^2 = 10^2 + 24^2$, or $AC^2 = 676$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Taking the positive square root, I get $AC = 26$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Therefore, $BM = \\frac{26}{2} = 13$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Now, to find $\\cos \\angle ABM$, I can use the cosine rule, which says that for any triangle with sides $a, b, c$ and angle $C$ opposite to $c$, we have $c^2 = a^2 + b^2 - 2ab \\cos C$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "In this case, I can let $a = BM = 13$, $b = AB = 10$, and $c = AM$, and $C = \\angle ABM$. [* I need to find $AM$, which I can do by using the fact that $M$ is the midpoint of $AC$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "So, $AM = \\frac{AC}{2} = \\frac{26}{2} = 13$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Now, I can plug these values into the cosine rule and get $13^2 = 13^2 + 10^2 - 2 \\cdot 13 \\cdot 10 \\cos \\angle ABM$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Simplifying, I get $0 = 100 - 260 \\cos \\angle ABM$, or $260 \\cos \\angle ABM = 100$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "Dividing both sides by 260, I get $\\cos \\angle ABM = \\frac{100}{260} = \\frac{10}{26}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 335, "metadata": { "path_label": "in_right_triangle_abc_we_have_ab_10_bc_24_and__2022-12-23T16-30-53.621453|chosen" } }, { "text": "To inscribe a triangle in a circle with one side as a diameter, I need to use the fact that the angle subtended by a diameter at any point on the circle is a right angle.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "This means that the inscribed triangle is a right triangle, and its hypotenuse is a diameter of the circle.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "To maximize the area of the triangle, I want to make the other two sides as long as possible, subject to the constraint that they have to lie on the circle.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "I can use the Pythagorean theorem to relate the lengths of the sides of the right triangle: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "Since $c$ is a diameter of the circle, I know that $c = 2 \\times 6 = 12$ cm.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "To maximize $a$ and $b$, I want them to be as close to each other as possible, so that they form an isosceles right triangle.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "In that case, $a = b$, and the equation simplifies to $a^2 + a^2 = 12^2$, or $2a^2 = 144$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "Dividing both sides by 2, I get $a^2 = 72$, and taking the positive square root, I get $a = \\sqrt{72}$ cm.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "The area of the triangle is then $\\frac{1}{2}ab = \\frac{1}{2}a^2 = \\frac{1}{2} \\times 72 = 36$ square centimeters.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 336, "metadata": { "path_label": "circle_c_has_radius_6_cm_how_many_square_centimeters_are__2022-12-23T16-30-09.830860|chosen" } }, { "text": "To solve this problem, I need to add and subtract numbers in base 3, which means they only use the digits 0, 1, and 2.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "I can do this by aligning the numbers according to their place values and adding or subtracting them column by column, starting from the rightmost one.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "If I need to carry or borrow, I have to remember that 3 in base 10 is equivalent to 10 in base 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "So, let me write the numbers in a column and align them by their place values.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "The first two numbers are 121_3 and 2122_3, and I need to add them.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "I start from the rightmost column and add 1 and 2, which gives me 3.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "But 3 is not a valid digit in base 3, so I have to carry 1 and write 0 in the result column.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 337, "metadata": { "path_label": "compute_121_3_2122_3_1200_3_2111_3_express_your_answer_in__2022-12-23T21-02-45.222510|chosen" } }, { "text": "This is a product of six fractions, and one way to multiply fractions is to multiply the numerators and multiply the denominators separately.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "So I can write the product as $\\dfrac{4\\cdot 6\\cdot 8\\cdot 10\\cdot 12\\cdot 14}{3\\cdot 4\\cdot 5\\cdot 6\\cdot 7\\cdot 8}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "However, before I do that, I notice that some of the factors in the numerator and the denominator are the same, and I can cancel them out to simplify the product.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "For example, I see that there is a 4 in both the numerator and the denominator, so I can divide both by 4 and get 1.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "Similarly, I see that there is a 6 in both the numerator and the denominator, so I can divide both by 6 and get 1.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "And I see that there is an 8 in both the numerator and the denominator, so I can divide both by 8 and get 1.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 338, "metadata": { "path_label": "compute_the_product_dfrac_43_cdot_dfrac_64_cdot_dfrac_85__2022-12-23T21-01-39.105857|chosen" } }, { "text": "This problem asks me to eliminate the parameter $t$ and find a Cartesian equation for the curve in the form of a quadratic equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I notice that the curve involves both sine and cosine of $t$, so I wonder if I can use a trigonometric identity to simplify them.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "One identity that comes to mind is $\\cos^2 t + \\sin^2 t = 1$, which relates the sum of the squares of the two functions to a constant.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "If I can somehow express $x$ and $y$ in terms of $\\cos^2 t$ and $\\sin^2 t$, then I can use this identity to eliminate $t$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "To do that, I need to isolate $\\cos t$ and $\\sin t$ from the given equations.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I can divide both sides of the equation for $x$ by $2$ to get $\\cos t = \\frac{x + \\sin t}{2}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I can square both sides of this equation to get $\\cos^2 t = \\frac{x^2 + 2x \\sin t + \\sin^2 t}{4}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "Similarly, I can square both sides of the equation for $y$ to get $\\sin^2 t = \\frac{y^2}{16}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "Now I have two equations for $\\cos^2 t$ and $\\sin^2 t$ in terms of $x$ and $y$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I can substitute them into the identity $\\cos^2 t + \\sin^2 t = 1$ to get $\\frac{x^2 + 2x \\sin t + \\sin^2 t}{4} + \\frac{y^2}{16} = 1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I can multiply both sides by $16$ to clear the fractions and get $4x^2 + 8x \\sin t + 4 \\sin^2 t + y^2 = 16$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I want to get rid of the $\\sin t$ term, so I can use the equation for $y$ to replace it with $\\frac{y}{4}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I get $4x^2 + 2xy + y^2 = 16$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "This is almost in the form I want, except for the constant term.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I can divide both sides by $16$ to get $\\frac{x^2}{4} + \\frac{xy}{8} + \\frac{y^2}{16} = 1$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "This is the Cartesian equation for the curve in the form of a quadratic equation.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "Comparing it with the given form $ax^2 + bxy + cy^2 = 1$, I can see that $a = \\frac{1}{4}$, $b = \\frac{1}{8}$, and $c = \\frac{1}{16}$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "Therefore, the ordered triple $(a,b,c)$ is $\\left(\\frac{1}{4}, \\frac{1}{8}, \\frac{1}{16}\\right)$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 339, "metadata": { "path_label": "a_curve_is_described_parametrically_by_x_y_2_cos_t__2022-12-23T21-01-23.927965|chosen" } }, { "text": "I need to find the height of the center of the spherical ball, which I will call $h$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 340, "metadata": { "path_label": "a_torus_donut_having_inner_radius_2_and_outer_radius_4__2022-12-23T20-55-34.809696|chosen" } }, { "text": "Then the radius of the ball will be $h-1$, since it touches the table at $z=1$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 340, "metadata": { "path_label": "a_torus_donut_having_inner_radius_2_and_outer_radius_4__2022-12-23T20-55-34.809696|chosen" } }, { "text": "To find $h$, I need to consider the cross-section of the torus and the ball in the $xz$-plane.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 340, "metadata": { "path_label": "a_torus_donut_having_inner_radius_2_and_outer_radius_4__2022-12-23T20-55-34.809696|chosen" } }, { "text": "To find the area of the intersection, I need to identify the shape of the intersection and find its dimensions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "First, I will plot the given vertices of the rectangle and the equation of the circle on a coordinate plane.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I notice that the circle has its center at $(5, -2)$ and its radius is $3$ units.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "The rectangle has a length of $11$ units along the x-axis and a width of $13$ units along the y-axis.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I also notice that the fourth vertex of the rectangle must be $(5, -2)$, since it is the only point that makes the opposite sides parallel.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Therefore, the rectangle and the circle have a common vertex at $(5, -2)$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I will label this point as $A$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Now, I need to find the other points of intersection between the rectangle and the circle.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I see that the circle intersects the horizontal side of the rectangle at $(5, -2) \\pm (3, 0)$, which are $(2, -2)$ and $(8, -2)$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I will label these points as $B$ and $C$, respectively.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Similarly, I see that the circle intersects the vertical side of the rectangle at $(5, -2) \\pm (0, 3)$, which are $(5, 1)$ and $(5, -5)$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I will label these points as $D$ and $E$, respectively.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I have found all the points of intersection, and I can see that the shape of the intersection is a quarter-circle with radius $3$ units.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Therefore, the area of the intersection is one-fourth of the area of the circle, which is $\\frac{1}{4} \\pi r^2$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Plugging in the radius of $3$, I get $\\frac{1}{4} \\pi (3)^2$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "Simplifying, I get $\\frac{9}{4} \\pi$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "This is the final answer.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 341, "metadata": { "path_label": "three_of_the_four_vertices_of_a_rectangle_are_5_11_1__2022-12-23T20-54-46.573001|chosen" } }, { "text": "I notice that $\\tau(n)$ is odd if and only if $n$ is a perfect square, since any other factor of $n$ can be paired up with its complement.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "Therefore, $S(n)$ is odd if and only if there are an odd number for perfect squares less than or equal to $n.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "I can use this observation to find $a$ and $b$ by counting how many perfect squares are in the interval $[1, 2005].$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "The largest perfect square in this interval is $44^2 = 1936,$ and the smallest is $1^2 = 1.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "Therefore, there are $44$ perfect squares in this interval, which is even.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "This means that $S(n)$ is even for all $n \\leq 44^2 = 1936,$ and odd for all $n > 44^2 = 1936.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "So, $a$ is the number of positive integers $n \\leq 2005$ that are greater than $1936,$ which is $2005 - 1936 = 69.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "Similarly, $b$ is the number of positive integers $n \\leq 2005$ that are less than or equal to $1936,$ which is $1936.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "Therefore, $|a-b| = |69 - 1936| = 1867.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 342, "metadata": { "path_label": "for_positive_integers_n_let_tau_n_denote_the_number__2022-12-23T20-53-21.251541|chosen" } }, { "text": "I need to find the expected value of the number of times Bob rolls his die in a non-leap year.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 343, "metadata": { "path_label": "bob_rolls_a_fair_six_sided_die_each_morning_if_bob_rolls_a__2022-12-23T19-32-50.907295|chosen" } }, { "text": "Expected value is the average outcome of a random event over many trials.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 343, "metadata": { "path_label": "bob_rolls_a_fair_six_sided_die_each_morning_if_bob_rolls_a__2022-12-23T19-32-50.907295|chosen" } }, { "text": "To find the expected value, I need to multiply each possible outcome by its probability, and then add up those products.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 343, "metadata": { "path_label": "bob_rolls_a_fair_six_sided_die_each_morning_if_bob_rolls_a__2022-12-23T19-32-50.907295|chosen" } }, { "text": "I need to find out how many even numbers less than $10$ can divide a multiple of $6$ minus $2$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "I can list the even numbers less than $10$ as $2, 4, 6, 8$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "For each of them, I can see if there is a multiple of $6$ that gives a remainder of $2$ when divided by that number.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "For example, if $n = 2$, then any multiple of $6$ minus $2$ will be divisible by $2$, since $6$ is divisible by $2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "So $2$ is one possible value for $n$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "If $n = 4$, then I need to find a multiple of $6$ that is congruent to $2$ modulo $4$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "That means it has the same remainder as $2$ when divided by $4$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "I can try some small multiples of $6$ and see if they work.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "For example, $6$ divided by $4$ gives a remainder of $2$, so $6 - 2 = 4$ is divisible by $4$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "So $4$ is another possible value for $n$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "If $n = 6$, then I need to find a multiple of $6$ that is congruent to $2$ modulo $6$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "That means it has the same remainder as $2$ when divided by $6$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "But this is impossible, since $6$ is divisible by $6$ and $2$ is not.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "So $6$ is not a possible value for $n$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "If $n = 8$, then I need to find a multiple of $6$ that is congruent to $2$ modulo $8$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "That means it has the same remainder as $2$ when divided by $8$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "I can try some small multiples of $6$ and see if they work.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "For example, $12$ divided by $8$ gives a remainder of $4$, so $12 - 2 = 10$ is not divisible by $8$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "Similarly, $18$ divided by $8$ gives a remainder of $2$, so $18 - 2 = 16$ is divisible by $8$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "So $8$ is another possible value for $n$.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "Therefore, there are three possible values for $n$ that satisfy the given conditions: $2, 4, 8$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 344, "metadata": { "path_label": "the_number_of_toy_cars_that_ray_has_is_a_multiple_of_6_wh__2022-12-23T19-31-06.895967|chosen" } }, { "text": "This is an interesting equation that involves logarithms and exponentials.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "I wonder what happens if I take the logarithm of both sides.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "If I use the base 1995 logarithm, then the left-hand side becomes $\\log_{1995}(\\sqrt{1995} \\cdot x^{\\log_{1995}x})$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "By the properties of logarithms, this can be written as $\\frac{1}{2}\\log_{1995}(1995)+\\log_{1995}x\\cdot \\log_{1995}x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "Simplifying, this is $\\frac{1}{2}+\\log_{1995}^2x$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "The right-hand side becomes $\\log_{1995}(x^2)$, which is $2\\log_{1995}x$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "So the equation is equivalent to $\\frac{1}{2}+\\log_{1995}^2x=2\\log_{1995}x$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "This looks like a quadratic equation in $\\log_{1995}x$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "Let me rewrite it as $(\\log_{1995}x)^2-2\\log_{1995}x+\\frac{1}{2}=0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "To solve for $\\log_{1995}x$, I can use the quadratic formula.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "This gives me $\\log_{1995}x=\\frac{2\\pm \\sqrt{4-4\\cdot \\frac{1}{2}}}{2}=\\frac{2\\pm \\sqrt{2}}{2}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "Therefore, the possible values of x are $x=1995^{\\frac{2\\pm \\sqrt{2}}{2}}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "These are the positive roots of the original equation.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "To find their product, I can use the property that $a^b\\cdot a^c=a^{b+c}$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "This gives me $1995^{\\frac{2\\pm \\sqrt{2}}{2}}\\cdot 1995^{\\frac{2\\mp \\sqrt{2}}{2}}=1995^{\\frac{2\\pm \\sqrt{2}}{2}+\\frac{2\\mp \\sqrt{2}}{2}}=1995^2$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "Therefore, the product of the positive roots is $1995^2$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 345, "metadata": { "path_label": "find_the_product_of_the_positive_roots_of_sqrt_1995_cdot__2022-12-23T19-24-35.161114|chosen" } }, { "text": "To find the number of ordered pairs $(a,b)$, I need to find the conditions on $a$ and $b$ that make them both roots of the same quadratic equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "One way to approach this is to use the fact that the sum and product of the roots of a quadratic equation are equal to the coefficients of the linear and constant terms, respectively.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "That is, if $r_1$ and $r_2$ are the roots of $x^2 + ax + b = 0,$ then $r_1 + r_2 = -a$ and $r_1r_2 = b.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "In this problem, I have two cases: either $a = r_1$ and $b = r_2,$ or $a = r_2$ and $b = r_1.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "In either case, I have $a + b = -a$ and $ab = b.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "From the first equation, I get $2a = -b.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "From the second equation, I get $ab = b.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "If $b = 0,$ then $a = 0$ as well, and this is one ordered pair that works.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "If $b \\neq 0,$ then I can divide both sides by $b$ and get $a = 1.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "Then, from $2a = -b,$ I get $b = -2.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "This is another ordered pair that works.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "Therefore, there are only two ordered pairs of real numbers that satisfy the given conditions: $(0,0)$ and $(1,-2).$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "The number of ordered pairs is 2.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 346, "metadata": { "path_label": "find_the_number_of_ordered_pairs_a_b_of_real_numbers_suc__2022-12-23T19-22-08.052774|chosen" } }, { "text": "To find the multiplicative inverse of $201$ modulo $299$, I need to find an integer $x$ such that $201x \\equiv 1 \\pmod{299}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "This is equivalent to finding a solution to the linear equation $201x - 299y = 1$ in integers $x$ and $y$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "One way to do this is to use the extended Euclidean algorithm, which finds the greatest common divisor of $201$ and $299$ and expresses it as a linear combination of $201$ and $299$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "I can then divide both sides by the gcd to get a solution for $x$ and $y$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "To apply the extended Euclidean algorithm, I start by writing the equation $299 = 1 \\cdot 201 + 98$, which implies that $299 - 1 \\cdot 201 = 98$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "Next, I write the equation $201 = 2 \\cdot 98 + 5$, which implies that $201 - 2 \\cdot 98 = 5$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "Then, I write the equation $98 = 19 \\cdot 5 + 3$, which implies that $98 - 19 \\cdot 5 = 3$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "After that, I write the equation $5 = 1 \\cdot 3 + 2$, which implies that $5 - 1 \\cdot 3 = 2$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "Finally, I write the equation $3 = 1 \\cdot 2 + 1$, which implies that $3 - 1 \\cdot 2 = 1$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "Now, I work backwards from the last equation to express $1$ as a linear combination of $201$ and $299$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "I start by writing $1 = 3 - 1 \\cdot 2$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "Then, I substitute $2 = 5 - 1 \\cdot 3$ and get $1 = 3 - 1 \\cdot (5 - 1 \\cdot 3) = 2 \\cdot 3 - 1 \\cdot 5$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 347, "metadata": { "path_label": "compute_the_multiplicative_inverse_of_201_modulo_299_ex__2022-12-23T19-16-49.333997|chosen" } }, { "text": "This problem involves circles and tangents, so I think I'll start by drawing a picture of the situation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "I'll put the circles $C_1$ and $C_2$ on a coordinate plane, with $C_1$ centered at the origin and $C_2$ centered at $(2,0).$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "The radius of $C_1$ is $1$ and the radius of $C_2$ is $4,$ so I'll label those as well.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "Now I want to find the locus of the centers of circles that are externally tangent to $C_1$ and internally tangent to $C_2.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "Let me draw one such circle, say $C_3,$ and label its center $(a,b)$ and its radius $r.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "I notice that the distance from $(a,b)$ to $(0,0)$ is $r + 1,$ since $C_1$ and $C_3$ are externally tangent.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "Similarly, the distance from $(a,b)$ to $(2,0)$ is $4 - r,$ since $C_2$ and $C_3$ are internally tangent.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "These distances can be found using the distance formula, so I can write two equations: \\[(a - 0)^2 + (b - 0)^2 = (r + 1)^2\\] and \\[(a - 2)^2 + (b - 0)^2 = (4 - r)^2.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 348, "metadata": { "path_label": "the_circles_c_1_and_c_2_are_defined_by_the_equations_x__2022-12-23T19-16-24.195606|chosen" } }, { "text": "I want to find the greatest common divisor of two expressions that depend on $n$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "One way to do that is to use the Euclidean algorithm, which repeatedly applies the division algorithm to find the remainder of dividing one expression by the other, until I get a constant remainder.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "Let me start by dividing $11n+3$ by $6n+1$ and see what the remainder is.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "Using long division, I get $11n+3 = (6n+1)(1) + (5n+2)$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "So the remainder is $5n+2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "Now I repeat the process with $6n+1$ and $5n+2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "I get $6n+1 = (5n+2)(1) + (n-1)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "So the new remainder is $n-1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "One more time, with $5n+2$ and $n-1$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "I get $5n+2 = (n-1)(5) + (7)$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "So the final remainder is $7$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "This means that the greatest common divisor of $11n+3$ and $6n+1$ is the same as the greatest common divisor of $7$ and any multiple of $7$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "But the greatest common divisor of $7$ and any multiple of $7$ is just $7$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "So the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$ is $7$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "This happens when $n-1$ is a multiple of $7$, or equivalently, when $n$ is one more than a multiple of $7$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 349, "metadata": { "path_label": "as_n_ranges_over_the_positive_integers_what_is_the_maximu__2022-12-23T19-04-14.398279|chosen" } }, { "text": "I notice that the statements involve inequalities and fractions, so I think about how to relate them using algebra and the properties of inequalities.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 350, "metadata": { "path_label": "let_a_and_b_be_real_numbers_consider_the_following_fiv__2022-12-23T18-47-41.339761|chosen" } }, { "text": "I also notice that some of the statements are about the signs of $a$ and $b$, so I think about how to use that information.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 350, "metadata": { "path_label": "let_a_and_b_be_real_numbers_consider_the_following_fiv__2022-12-23T18-47-41.339761|chosen" } }, { "text": "I start by assuming that the first statement is true, that is, $\\frac{1}{a} < \\frac{1}{b}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 350, "metadata": { "path_label": "let_a_and_b_be_real_numbers_consider_the_following_fiv__2022-12-23T18-47-41.339761|chosen" } }, { "text": "I notice that both equations involve $x$ modulo $20$, so I can try to use one equation to eliminate $x$ from the other.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "For example, if I multiply the first equation by $3$, I get $12x\\equiv 24\\pmod{20}$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "Then, if I subtract the second equation from this one, I get $9x\\equiv 8\\pod{20}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "Now, I can simplify this equation by dividing both sides by $9$, since $9$ and $20$ are coprime.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "This gives me $x\\equiv \\frac{8}{9}\\pmod{20}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "To find the inverse of $9$ modulo $20$, I can use the extended Euclidean algorithm.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "I write $20=9\\times 2+2$ and $9=2\\times 4+1$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "Then, I work backwards and write $1=9-2\\times 4=9-(20-9\\times 2)\\times 4=9\\times 9-20\\times 4$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "This means that $9\\times 9\\equiv 1\\pmod{20}$, so $9^{-1}\\equiv 9\\pmod{20}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "Therefore, $x\\equiv \\frac{8}{9}\\equiv 8\\times 9\\equiv 12\\pmod{20}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "Now, I have found $x$ modulo $20$, and I can plug it into $x^2$ and find the remainder.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "I get $x^2\\equiv 12^2\\equiv 144\\equiv 4\\pmod{20}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "So, the final answer is $4$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 351, "metadata": { "path_label": "if_4x_equiv_8_pmod_20_and_3x_equiv_16_pmod_20_then_wh__2022-12-23T18-37-31.422074|chosen" } }, { "text": "I know that $555_{10}$ means 555 in base 10, which is the usual decimal system.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "To express it in another base, I need to find the largest power of that base that is less than or equal to 555, and use that as the leftmost digit.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract that power times the digit from 555, and repeat the process with the next lower power of the base, until I get to the rightmost digit.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "For example, in base 8, I would start with $8^3 = 512$, which is less than 555, so the leftmost digit is 1.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $1 \\times 512 = 512$ from 555, and get 43.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 8 is $8^2 = 64$, which is larger than 43, so the next digit is 0.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $0 \\times 64 = 0$ from 43, and get 43.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 8 is $8^1 = 8$, which is less than 43, so the next digit is 5.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $5 \\times 8 = 40$ from 43, and get 3.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 8 is $8^0 = 1$, which is less than 3, so the rightmost digit is 3.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "So $555_{10} = 1053_8$ in base 8.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "But this is not the base I'm looking for, because it does not have the form ABAB.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "I notice that the base I'm looking for must have two different digits, A and B, that repeat in an alternating pattern.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "This suggests that the base might be a multiple of 3, because then every third digit would be the same.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "For example, in base 6, the digits are 0, 1, 2, 3, 4, and 5, and every third digit is 0, 3, or 6.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Let me try base 6 and see if it works.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "In base 6, I would start with $6^3 = 216$, which is less than 555, so the leftmost digit is 2.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $2 \\times 216 = 432$ from 555, and get 123.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 6 is $6^2 = 36$, which is less than 123, so the next digit is 3.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $3 \\times 36 = 108$ from 123, and get 15.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 6 is $6^1 = 6$, which is less than 15, so the next digit is 2.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "Then I subtract $2 \\times 6 = 12$ from 15, and get 3.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The next lower power of 6 is $6^0 = 1$, which is less than 3, so the rightmost digit is 3.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "So $555_{10} = 2323_6$ in base 6.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "This is the base I'm looking for, because it has the form ABAB, where A = 2 and B = 3.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "The base is 6.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 352, "metadata": { "path_label": "when_555__10_is_expressed_in_this_base_it_has_4_digits__2022-12-23T18-37-25.931542|chosen" } }, { "text": "To multiply two numbers in base 2, I can use the same algorithm as for base 10, but with different rules for carrying and adding.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 353, "metadata": { "path_label": "find_10110_2_times10100_2_div10_2_express_your_answer_in__2022-12-23T18-24-56.752500|chosen" } }, { "text": "I'll start by aligning the least significant bits of the two factors and multiplying them: 0 times 0 is 0, so I write 0 below them.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 353, "metadata": { "path_label": "find_10110_2_times10100_2_div10_2_express_your_answer_in__2022-12-23T18-24-56.752500|chosen" } }, { "text": "Then I move to the next column and multiply 1 times 0, which is also 0, but I have to add a 0 from the previous carry, so I get 0 again.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 353, "metadata": { "path_label": "find_10110_2_times10100_2_div10_2_express_your_answer_in__2022-12-23T18-24-56.752500|chosen" } }, { "text": "I repeat this process for the next column, getting 0 times 1 plus 0, which is 0.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 353, "metadata": { "path_label": "find_10110_2_times10100_2_div10_2_express_your_answer_in__2022-12-23T18-24-56.752500|chosen" } }, { "text": "This problem involves an infinite series of the form \\[\\sum_{n = 0}^\\infty a_n \\cos n \\theta.\\]", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "One way to approach such series is to consider the corresponding complex series \\[\\sum_{n = 0}^\\infty a_n e^{i n \\theta},\\] which is easier to work with since it involves only exponentials.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "The series \\[\\sum_{n = 0}^\\infty \\frac{e^{i n \\theta}}{2^n}\\] is a geometric series with first term $\\frac{1}{2}$ and common ratio $\\frac{e^{i \\theta}}{2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "The sum of an infinite geometric series is given by \\[\\frac{a}{1 - r},\\] where $a$ is the first term and $r$ is the common ratio.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "In this case, the sum of the complex series is \\[\\frac{1}{2} \\cdot \\frac{1}{1 - \\frac{e^{i \\theta}}{2}} = \\frac{1}{2 - e^{i \\theta}}.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "To find the sum of the original series, I need to take the real part of the complex sum.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "To do that, I can multiply the numerator and denominator by the complex conjugate of the denominator, which is $2 - e^{-i \\theta}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "This gives \\[\\frac{1}{2 - e^{i \\theta}} \\cdot \\frac{2 - e^{-i \\theta}}{2 - e^{-i \\theta}} = \\frac{2 - e^{-i \\theta}}{4 - 2 e^{i \\theta} - 2 e^{-i \\theta} + 1}.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "Simplifying the denominator, I get \\[\\frac{2 - e^{-i \\theta}}{5 - 4 \\cos \\theta}.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "Now I can use the fact that $e^{-i \\theta} = \\cos \\theta - i \\sin \\theta$ and take the real part of the numerator.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "This gives \\[\\frac{2 - \\cos \\theta}{5 - 4 \\cos \\theta}.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "Finally, I can plug in the value of $\\cos \\theta = \\frac{1}{5}$ and simplify.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "This gives \\[\\frac{2 - \\frac{1}{5}}{5 - 4 \\cdot \\frac{1}{5}} = \\frac{\\frac{9}{5}}{\\frac{17}{5}} = \\frac{9}{17}.\\]", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 354, "metadata": { "path_label": "evaluate_sum__n_0_infty_frac_cos_n_theta_2_n_w__2022-12-23T18-24-44.235673|chosen" } }, { "text": "I know that the number of positive factors of a number depends on how many ways I can write it as a product of prime factors.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "For example, 12 = 2^2 * 3^1, so it has (2 + 1) * (1 + 1) = 6 factors, because I can choose either 0, 1, or 2 factors of 2 and either 0 or 1 factor of 3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "So, to maximize the number of factors of $b^n$, I want to make $b$ as composite as possible, and $n$ as large as possible.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "The most composite number less than or equal to 15 is 12, since it has the most prime factors (2, 2, and 3).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "The largest possible value of $n$ is 15, since that is the upper bound given.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "Therefore, the best choice for $b^n$ is 12^15.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "Now, I need to count how many factors 12^15 has.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "I can write 12^15 as (2^2 * 3)^15, which is the same as 2^30 * 3^15.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "So, the number of factors of 12^15 is (30 + 1) * (15 + 1), which is 31 * 16, which is 496.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 355, "metadata": { "path_label": "the_value_b_n_has_both_b_and_n_as_positive_integers_le__2022-12-23T16-14-55.395871|chosen" } }, { "text": "I know that the volume of a pyramid is one-third times the area of the base times the height.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 356, "metadata": { "path_label": "a_particular_right_pyramid_has_a_square_base_and_each_edge__2022-12-23T16-07-55.806553|chosen" } }, { "text": "I can find the area of the base by squaring the length of one side of the square, which is four inches.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 356, "metadata": { "path_label": "a_particular_right_pyramid_has_a_square_base_and_each_edge__2022-12-23T16-07-55.806553|chosen" } }, { "text": "So the area of the base is four squared, which is 16 square inches.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 356, "metadata": { "path_label": "a_particular_right_pyramid_has_a_square_base_and_each_edge__2022-12-23T16-07-55.806553|chosen" } }, { "text": "To find the determinant of a matrix, I need to use the formula that involves multiplying the diagonal entries and adding or subtracting them according to a pattern.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "However, this formula can be tedious to apply, especially for a 3 by 3 matrix.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "I recall that there is a shortcut for finding the determinant of a matrix whose columns are linear combinations of other vectors.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "If I write the matrix as $A = [\\mathbf{a} + \\mathbf{b} \\quad \\mathbf{b} + \\mathbf{c} \\quad \\mathbf{c} + \\mathbf{a}]$, I can factor out a $\\mathbf{a}$ from the first column, a $\\mathbf{b}$ from the second column, and a $\\mathbf{c}$ from the third column.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "This gives me $A = [\\mathbf{a} \\quad \\mathbf{b} \\quad \\mathbf{c}] \\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "Now, I can use the property that the determinant of a product of matrices is equal to the product of the determinants of the matrices.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "So, $\\det(A) = \\det([\\mathbf{a} \\quad \\mathbf{b} \\quad \\mathbf{c}]) \\det(\\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix})$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "But, the determinant of the matrix whose columns are $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ is given as $D$ in the problem.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "So, I just need to find the determinant of the matrix $\\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "I can use the cofactor expansion along the first row, since it has two zeros.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "This gives me $\\det(\\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}) = 1 \\det(\\begin{bmatrix} 1 & 0 \\\\ 1 & 1 \\end{bmatrix}) - 0 \\det(\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}) + 1 \\det(\\begin{bmatrix} 1 & 1 \\\\ 0 & 1 \\end{bmatrix})$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "The determinant of a 2 by 2 matrix is just the product of the diagonal entries minus the product of the off-diagonal entries.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "So, I get $\\det(\\begin{bmatrix} 1 & 0 \\\\ 1 & 1 \\end{bmatrix}) = 1 \\cdot 1 - 1 \\cdot 0 = 1$, $\\det(\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}) = 1 \\cdot 1 - 0 \\cdot 0 = 1$, and $\\det(\\begin{bmatrix} 1 & 1 \\\\ 0 & 1 \\end{bmatrix}) = 1 \\cdot 1 - 1 \\cdot 0 = 1$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "So, plugging these values into the cofactor expansion, I get $\\det(\\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix}) = 1 \\cdot 1 - 0 \\cdot 1 + 1 \\cdot 1 = 2$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "Therefore, the final answer is $\\det(A) = D \\cdot 2$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 357, "metadata": { "path_label": "let_d_be_the_determinant_of_the_matrix_whose_column_vector__2022-12-23T16-05-38.918288|chosen" } }, { "text": "I know that a number is divisible by nine if and only if the sum of its digits is divisible by nine.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "So, I want to find all the possible values of $A$ and $B$ that make $2 + 5 + A + B$ a multiple of nine.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "I can try different values of $A$ from $0$ to $9$ and see what $B$ has to be to make the sum a multiple of nine.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 0$, then $B$ has to be $2$ or $11$, but $11$ is not a valid digit, so only $B = 2$ works. That gives me one possible number, $2502$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 1$, then $B$ has to be $1$ or $10$, but $10$ is not a valid digit, so only $B = 1$ works. That gives me another possible number, $2511$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 2$, then $B$ has to be $0$ or $9$, both of which are valid digits. That gives me two possible numbers, $2520$ and $2529$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 3$, then $B$ has to be $-1$ or $8$, but $-1$ is not a valid digit, so only $B = 8$ works. That gives me another possible number, $2538$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 4$, then $B$ has to be $-2$ or $7$, but $-2$ is not a valid digit, so only $B = 7$ works. That gives me another possible number, $2547$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 5$, then $B$ has to be $-3$ or $6$, but $-3$ is not a valid digit, so only $B = 6$ works. That gives me another possible number, $2556$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 6$, then $B$ has to be $-4$ or $5$, but $-4$ is not a valid digit, so only $B = 5$ works. That gives me another possible number, $2565$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 7$, then $B$ has to be $-5$ or $4$, but $-5$ is not a valid digit, so only $B = 4$ works. That gives me another possible number, $2574$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 8$, then $B$ has to be $-6$ or $3$, but $-6$ is not a valid digit, so only $B = 3$ works. That gives me another possible number, $2583$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "If $A = 9$, then $B$ has to be $-7$ or $2$, but $-7$ is not a valid digit, so only $B = 2$ works. That gives me another possible number, $2592$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 358, "metadata": { "path_label": "the_four_digit_number_25ab_is_divisible_by_nine_with_a__2022-12-23T15-53-24.094199|chosen" } }, { "text": "This problem involves finding the greatest common divisor of two numbers that are generated by a formula.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "A common technique for finding the greatest common divisor of two numbers is to use the Euclidean algorithm, which repeatedly applies the division algorithm to reduce the pair of numbers until they are equal or one of them is zero.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "To apply the Euclidean algorithm to $a_n$ and $a_{n+1}$, I need to write one of them as a multiple of the other plus some remainder.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "I notice that $a_{n+1}$ is always larger than $a_n$, so I can write $a_{n+1} = q a_n + r$ for some quotient $q$ and remainder $r$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "To find $q$ and $r$, I need to divide $a_{n+1}$ by $a_n$ and write the result as a mixed fraction.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "I can simplify the division by noticing that $a_n = \\frac{10^n-1}{9}$ and $a_{n+1} = \\frac{10^{n+1}-1}{9}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This means that $a_{n+1} = 10 a_n + \\frac{9}{9}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "Therefore, $q = 10$ and $r = 1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This implies that $a_{n+1} = 10 a_n + 1$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "Now I can apply the Euclidean algorithm by repeating the process with $a_n$ and $r = 1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "I can write $a_n = q' r + r'$ for some quotient $q'$ and remainder $r'$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "To find $q'$ and $r'$, I need to divide $a_n$ by $r = 1$ and write the result as a mixed fraction.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This is trivial, since $a_n$ is always an integer and $r = 1$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "Therefore, $q' = a_n$ and $r' = 0$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This implies that $a_n = a_n \\cdot 1 + 0$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "Now I have reached the end of the Euclidean algorithm, since the remainder is zero.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This means that the greatest common divisor of $a_n$ and $a_{n+1}$ is equal to the last nonzero remainder, which is $r = 1$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "Therefore, $d_n = 1$ for any $n$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "This also means that the maximum possible value that $d_n$ can take on is $1$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 359, "metadata": { "path_label": "let_a_n_frac_10_n_1_9_define_d_n_to_be_the_greates__2022-12-23T15-51-15.192370|chosen" } }, { "text": "The first thing I notice is that the expression inside the square root is a quadratic function of n.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I wonder if I can factor it somehow, or at least find its zeros and vertex.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "To find the zeros, I use the quadratic formula: $n = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$, where $a = 1, b = -10, c = 29$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I get $n = \\frac{10 \\pm \\sqrt{100 - 116}}{2}$, which are complex numbers, so the quadratic function has no real zeros.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "That means it is either always positive or always negative, depending on the sign of the leading coefficient.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "Since the leading coefficient is 1, which is positive, the quadratic function is always positive.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "That means the square root is always defined and real. [*, I] Now I want to find the vertex of the quadratic function, which is the point where it reaches its minimum or maximum value.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I recall that the vertex occurs at $n = -\\frac{b}{2a}$, where $a = 1, b = -10$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "So the vertex is at $n = 5$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "To find the value of the function at the vertex, I plug in $n = 5$ into the quadratic function: $5^2 - 10 \\cdot 5 + 29 = 4$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "So the vertex is at $(5, 4)$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "That means the quadratic function is increasing on both sides of the vertex, and its value is always greater than or equal to 4.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "That means the square root is also increasing on both sides of the vertex, and its value is always greater than or equal to 2. [*, I] Now I want to find the value of the square root when $n = 19941994$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "Since $n = 19941994$ is very large, I expect the square root to be very large as well.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I want to find the largest integer that is less than or equal to the square root, which is what the floor function does.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I wonder if I can use some approximation or inequality to find the floor function without computing the square root exactly.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I notice that the quadratic function is very close to $n^2 - 10n + 25$, which is a perfect square: $(n - 5)^2$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "That means the square root is very close to $|n - 5|$, which is $n - 5$ when $n \\geq 5$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "So I can approximate the square root as $n - 5$, and then find the floor function of that.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "The floor function of $n - 5$ is just $n - 5$, since $n - 5$ is already an integer.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "So I can approximate the floor function of the square root as $n - 5$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "When $n = 19941994$, this gives me $19941994 - 5 = 19941989$. [*, I] I wonder how close this approximation is to the actual value.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "I can check by computing the square root exactly and then finding its floor function.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "The square root of $n^2 - 10n + 29$ when $n = 19941994$ is about $19941989.0000025$.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "The floor function of that is $19941989$.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "So my approximation was exactly correct in this case. [*, I] Therefore, the final answer is $19941989$.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 360, "metadata": { "path_label": "compute_lfloor_sqrt_n_2_10n_29_rfloor_when_n_19__2022-12-23T15-41-40.836399|chosen" } }, { "text": "To find the shortest possible distance between two lines, I need to find a common perpendicular to both lines.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 361, "metadata": { "path_label": "let_p_be_a_point_on_the_line_begin_pmatrix_3_1__2022-12-23T15-38-54.276289|chosen" } }, { "text": "That means I need to find a vector that is orthogonal to both the direction vectors of the lines.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 361, "metadata": { "path_label": "let_p_be_a_point_on_the_line_begin_pmatrix_3_1__2022-12-23T15-38-54.276289|chosen" } }, { "text": "I can use the cross product of the direction vectors to find such a vector.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 361, "metadata": { "path_label": "let_p_be_a_point_on_the_line_begin_pmatrix_3_1__2022-12-23T15-38-54.276289|chosen" } }, { "text": "I know that the distance from the center of the ellipse to each focus is called the focal length, and I can use the formula $c^2 = a^2 - b^2$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 362, "metadata": { "path_label": "an_ellipse_has_foci_at_9_20_and_49_55_in_the_xy__2022-12-23T15-24-45.296663|chosen" } }, { "text": "To find the center of the ellipse, I can take the midpoint of the foci, which is $\\left(\\frac{9 + 49}{2}, \\frac{20 + 55}{2}\\right) = (29, 37.5)$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 362, "metadata": { "path_label": "an_ellipse_has_foci_at_9_20_and_49_55_in_the_xy__2022-12-23T15-24-45.296663|chosen" } }, { "text": "To find the focal length, I can use the distance formula between the center and either focus, which is $c = \\sqrt{(49 - 29)^2 + (55 - 37.5)^2} = \\sqrt{400 + 306.25} = \\sqrt{706.25}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 362, "metadata": { "path_label": "an_ellipse_has_foci_at_9_20_and_49_55_in_the_xy__2022-12-23T15-24-45.296663|chosen" } }, { "text": "Since the ellipse is tangent to the $x$-axis, that means that the semi-minor axis is equal to the $y$-coordinate of the center, which is $b = 37.5$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 362, "metadata": { "path_label": "an_ellipse_has_foci_at_9_20_and_49_55_in_the_xy__2022-12-23T15-24-45.296663|chosen" } }, { "text": "I know that two lines are coplanar if and only if they are parallel or intersecting.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "To check if they are parallel, I can compare their direction vectors, which are the coefficients of the parameters $s$ and $t$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "The direction vector of the first line is $\\begin{pmatrix} 1 \\\\ -k \\\\ k \\end{pmatrix}$ and the direction vector of the second line is $\\begin{pmatrix} 1/2 \\\\ 1 \\\\ -1 \\end{pmatrix}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "Two vectors are parallel if they are scalar multiples of each other, that is, if there is some constant $c$ such that $\\begin{pmatrix} 1 \\\\ -k \\\\ k \\end{pmatrix} = c \\begin{pmatrix} 1/2 \\\\ 1 \\\\ -1 \\end{pmatrix}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "Comparing the first coordinates, I get $c = 2.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "Comparing the second coordinates, I get $-k = 2.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "Comparing the third coordinates, I get $k = -2.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "These equations are consistent, so the lines are parallel and $k = -2.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 363, "metadata": { "path_label": "one_line_is_parameterized_by_begin_pmatrix_1_s_3__2022-12-23T15-23-17.330680|chosen" } }, { "text": "I notice that the expression $m^2-n^2$ can be factored as $(m+n)(m-n)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "I also notice that since $m$ and $n$ are both odd, their sum and difference must be even.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "So, the largest integer that divides all possible numbers of the form $m^2-n^2$ must be a power of 2.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "The question is, what is the highest power of 2 that always divides $(m+n)(m-n)$?", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "To answer that, I need to consider the possible remainders of $m$ and $n$ when divided by 4.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "There are two possibilities: either they have the same remainder, or they have different remainders.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "If they have the same remainder, then both $m+n$ and $m-n$ are divisible by 4, so $m^2-n^2$ is divisible by 16.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "If they have different remainders, then one of them is divisible by 4 and the other is not, so $m^2-n^2$ is divisible by 8 but not by 16.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "Therefore, the highest power of 2 that always divides $m^2-n^2$ is 8.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "That means the largest integer that divides all possible numbers of the form $m^2-n^2$ is 8.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 364, "metadata": { "path_label": "let_m_and_n_be_any_two_odd_numbers_with_n_less_than__2022-12-23T15-17-39.373738|chosen" } }, { "text": "I need to count the number of ways to choose three cards out of four, but exclude the cases where one of the cards has a zero on it.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 365, "metadata": { "path_label": "each_digit_in_the_number_2001_is_placed_on_a_different_card__2022-12-23T15-16-48.333411|chosen" } }, { "text": "If I choose a zero card, then the product will always be zero, regardless of the other two cards.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 365, "metadata": { "path_label": "each_digit_in_the_number_2001_is_placed_on_a_different_card__2022-12-23T15-16-48.333411|chosen" } }, { "text": "So I want to avoid choosing a zero card at all.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 365, "metadata": { "path_label": "each_digit_in_the_number_2001_is_placed_on_a_different_card__2022-12-23T15-16-48.333411|chosen" } }, { "text": "I know that the asymptotes of a hyperbola are the lines that pass through the center of the hyperbola and have slopes that are related to the coefficients of the standard form equation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Specifically, if the equation of the hyperbola is \\[\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1,\\]then the slopes of the asymptotes are $\\pm \\frac{a}{b}.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "So, in this problem, I can use the given equations of the asymptotes to find $\\frac{a}{b}$ and the center of the hyperbola.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "The slopes of the asymptotes are $2$ and $-2,$ so $\\frac{a}{b} = \\pm 2.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "The center of the hyperbola is the point where the asymptotes intersect, so I can find it by solving the system of equations \\[y = 2x+5\\] \\[y = -2x+1\\] by substitution or elimination.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "I'll use elimination and subtract the second equation from the first to get \\[0 = 4x+4\\] and then divide by $4$ to get \\[x = -1.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "To find $y,$ I can plug in $x = -1$ into either equation of the asymptotes. I'll use the first one and get \\[y = 2(-1)+5 = 3.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "So, the center of the hyperbola is $(-1, 3),$ which means that $h = -1$ and $k = 3.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Now, I need to find $a$ and $b.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "I know that the hyperbola passes through the point $(0, 7),$ so I can plug in $x = 0$ and $y = 7$ into the standard form equation and get \\[\\frac{(7-3)^2}{a^2} - \\frac{(0+1)^2}{b^2} = 1\\] or \\[\\frac{16}{a^2} - \\frac{1}{b^2} = 1.\\]", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "I also know that $\\frac{a}{b} = \\pm 2,$ so I can square both sides and get \\[\\frac{a^2}{b^2} = 4.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "I can substitute this into the equation I got from the point $(0, 7)$ and get \\[\\frac{16}{a^2} - \\frac{4}{a^2} = 1\\] or \\[\\frac{12}{a^2} = 1.\\]", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "I can solve for $a$ by multiplying both sides by $a^2$ and taking the square root and get \\[a = \\pm \\sqrt{12}.\\]", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Since $a > 0,$ I'll choose the positive value and get \\[a = \\sqrt{12}.\\]", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "To find $b,$ I can use the relation $\\frac{a^2}{b^2} = 4$ and get \\[b^2 = \\frac{a^2}{4} = \\frac{12}{4} = 3.\\]", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Taking the square root, I get \\[b = \\pm \\sqrt{3}.\\]", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Since $b > 0,$ I'll choose the positive value and get \\[b = \\sqrt{3}.\\]", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "Finally, I have all the values I need to write the standard form equation of the hyperbola: \\[\\frac{(y-3)^2}{12} - \\frac{(x+1)^2}{3} = 1.\\]", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "The question asks for $a + h,$ so I just add $\\sqrt{12}$ and $-1$ and get \\[a + h = \\sqrt{12} - 1.\\]", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 366, "metadata": { "path_label": "the_equations_of_the_asymptotes_of_a_hyperbola_are_y_2x_5__2022-12-23T15-10-43.870123|chosen" } }, { "text": "This problem involves fractions, decimals, and digits, so I want to think about how to convert between them.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "I know that $0.abc$ is the same as $\\frac{abc}{1000}$, where $abc$ is a three-digit number.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "So, if I want $0.abc$ to be equal to $\\frac{1}{y}$, then I need to find a value of $y$ that makes $abc$ a factor of $1000$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "The factors of $1000$ are $1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500$, and $1000$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "But, the problem says that $y$ has to be between $0$ and $9$, so I can eliminate most of these factors and only focus on $1, 2, 4, 5$, and $8$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "Now, I need to find the largest possible value of $abc$ that is a multiple of one of these factors.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "If I try $y=8$, then $abc$ has to be a multiple of $125$, but the largest multiple of $125$ that is less than $1000$ is $875$, which has a digit sum of $20$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "If I try $y=5$, then $abc$ has to be a multiple of $200$, but the largest multiple of $200$ that is less than $1000$ is $800$, which has a digit sum of $8$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "If I try $y=4$, then $abc$ has to be a multiple of $250$, but the largest multiple of $250$ that is less than $1000$ is $750$, which has a digit sum of $12$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "If I try $y=2$, then $abc$ has to be a multiple of $500$, but the largest multiple of $500$ that is less than $1000$ is $500$, which has a digit sum of $5$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "If I try $y=1$, then $abc$ has to be a multiple of $1000$, but the only multiple of $1000$ that is less than $1000$ is $0$, which has a digit sum of $0$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 367, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-57-45.684163|chosen" } }, { "text": "I need to find the number of ways to distribute six rolls among three types, without any type getting zero.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "One way to approach this is to use stars and bars, a combinatorial technique that represents the items as stars and the partitions as bars.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "For example, if I have six stars and two bars, I can arrange them in any order to get a different distribution.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "The number of ways to arrange six stars and two bars is equal to the number of ways to choose two positions out of eight for the bars, which is 8 choose 2.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "However, this includes some cases where one or more types get zero rolls, which I want to exclude.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "To avoid that, I can first assign one roll to each type, which leaves me with three rolls to distribute among three types.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "Now I have three stars and two bars to arrange, and the number of ways to do that is 5 choose 2.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "Therefore, the final answer is 5 choose 2, which is equal to 10.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 368, "metadata": { "path_label": "a_bakery_sells_three_kinds_of_rolls_how_many_different_comb__2022-12-23T12-59-06.452352|chosen" } }, { "text": "I know that the smallest prime number is 2, but if I start with 2 as the first of the three consecutive primes, then the second one would be 3 and the third one would be 5, and 2 + 3 + 5 = 10, which is not a multiple of 7.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "So I need to start with a larger prime number as the first one.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "The next prime number after 2 is 3, so let me try 3 as the first prime. Then the second one would be 5 and the third one would be 7, and 3 + 5 + 7 = 15, which is also not a multiple of 7.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "I need to keep trying larger primes until I find a sum that is a multiple of 7.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "The next prime number after 3 is 5, so let me try 5 as the first prime. Then the second one would be 7 and the third one would be 11, and 5 + 7 + 11 = 23, which is still not a multiple of 7.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "The next prime number after 5 is 7, so let me try 7 as the first prime. Then the second one would be 11 and the third one would be 13, and 7 + 11 + 13 = 31, which is not a multiple of 7 either.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "The next prime number after 7 is 11, so let me try 11 as the first prime. Then the second one would be 13 and the third one would be 17, and 11 + 13 + 17 = 41, which is not a multiple of 7.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "The next prime number after 11 is 13, so let me try 13 as the first prime. Then the second one would be 17 and the third one would be 19, and 13 + 17 + 19 = 49, which is a multiple of 7.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "Finally, I found a sum that is a multiple of 7, and it is the smallest possible one, because I started with the smallest possible primes and worked my way up.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "So the least possible sum is 49.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 369, "metadata": { "path_label": "three_consecutive_positive_prime_numbers_have_a_sum_that_is__2022-12-23T20-03-40.304908|chosen" } }, { "text": "I notice that $72 = 8 \\cdot 9,$ so a number is divisible by $72$ if and only if it is divisible by both $8$ and $9.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 370, "metadata": { "path_label": "the_number_5_41g_507_2h6_is_divisible_by_72_if_g_a__2022-12-23T20-00-59.492256|chosen" } }, { "text": "To check divisibility by $8,$ I only need to look at the last three digits of the number. So I want $2H6$ to be divisible by $8.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 370, "metadata": { "path_label": "the_number_5_41g_507_2h6_is_divisible_by_72_if_g_a__2022-12-23T20-00-59.492256|chosen" } }, { "text": "The multiples of $8$ that end with $6$ are $16, 96, 176, 256, 336, 416, 496, 576, 656, 736, 816, 896.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 370, "metadata": { "path_label": "the_number_5_41g_507_2h6_is_divisible_by_72_if_g_a__2022-12-23T20-00-59.492256|chosen" } }, { "text": "I notice that the equation involves fractions with quadratic expressions in the denominators, so I want to clear the fractions by multiplying both sides by the product of the denominators.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "That would give me \\[(x-2)(x^2-10x) = (x-5)(x^2+4x+1).\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I can expand the brackets using the distributive property and then collect like terms to get a simpler equation.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "Expanding the brackets gives me \\[x^3-10x^2-2x^2+20x = x^3+4x^2+x-5x^2-20x-5.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "Collecting like terms gives me \\[x^3-12x^2+20x = x^3-x^2-19x-5.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I can subtract \\[x^3-x^2-19x-5\\] from both sides to get a quadratic equation in standard form.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "That gives me \\[-11x^2+39x+5 = 0.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "To find the solutions of this equation, I can use the quadratic formula.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "The quadratic formula says that if \\[ax^2+bx+c=0,\\] then \\[x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "In this case, \\[a = -11, b = 39, c = 5.\\]", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "So, plugging these values into the quadratic formula, I get \\[x = \\frac{-39 \\pm \\sqrt{39^2-4(-11)(5)}}{2(-11)}.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "Simplifying the expression, I get \\[x = \\frac{-39 \\pm \\sqrt{1881}}{-22}.\\]", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I notice that 1881 is not a perfect square, so I cannot simplify the radical further.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "Therefore, the solutions are \\[x = \\frac{-39 + \\sqrt{1881}}{-22}\\] and \\[x = \\frac{-39 - \\sqrt{1881}}{-22}.\\]", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "To find the sum of these solutions, I can add them together.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "The sum is \\[\\frac{-39 + \\sqrt{1881}}{-22} + \\frac{-39 - \\sqrt{1881}}{-22}.\\]", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I can simplify this by combining the fractions over a common denominator.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "The sum is \\[\\frac{-39 + \\sqrt{1881} - 39 - \\sqrt{1881}}{-22}.\\]", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I see that the terms with \\[\\sqrt{1881}\\] cancel out, so I am left with \\[\\frac{-78}{-22}.\\]", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I can reduce this fraction by dividing both the numerator and the denominator by 2.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "The sum is \\[\\frac{-39}{-11}.\\]", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "I can simplify this by changing the signs.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "The sum is \\[\\frac{39}{11}.\\]", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 371, "metadata": { "path_label": "find_the_sum_of_all_real_solutions_to_the_equation_frac_x__2022-12-23T19-51-46.748551|chosen" } }, { "text": "To find the vertical asymptotes of a rational function, I need to find the values of x that make the denominator zero and are not also zeros of the numerator.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "I can see that the denominator has four factors: (x + 3), (x + 2), (x + 1), and x.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "The zeros of these factors are x = -3, -2, -1, and 0, respectively.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "However, I also notice that some of these factors appear in the numerator as well, so they might cancel out and not be asymptotes.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "To check this, I need to compare the exponents of each factor in the numerator and the denominator.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "For (x + 3), the exponent in the numerator is 5 and the exponent in the denominator is 11.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "This means that (x + 3) does not cancel out completely, and there is still a zero in the denominator when x = -3.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "Therefore, x = -3 is a vertical asymptote.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "For (x + 2), the exponent in the numerator is 8 and the exponent in the denominator is 6.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "This means that (x + 2) does cancel out completely, and there is no zero in the denominator when x = -2.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "Therefore, x = -2 is not a vertical asymptote.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "For (x + 1), the exponent in the numerator is 4 and the exponent in the denominator is also 4.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "This means that (x + 1) cancels out completely, and there is no zero in the denominator when x = -1.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "Therefore, x = -1 is not a vertical asymptote.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "For x, the exponent in the numerator is 0 and the exponent in the denominator is 2.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "This means that x does not cancel out at all, and there is a zero in the denominator when x = 0.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "Therefore, x = 0 is a vertical asymptote.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "To summarize, I have found two vertical asymptotes: x = -3 and x = 0.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 372, "metadata": { "path_label": "how_many_vertical_asymptotes_does_the_graph_of_y_frac__2022-12-23T19-40-45.531707|chosen" } }, { "text": "This is an interesting problem, because it involves a functional equation, which is a relation between the values of a function at different points.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 373, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_y_f_x_f_y_for_al__2022-12-23T19-37-34.403399|chosen" } }, { "text": "A common strategy for solving functional equations is to look for special values of the variables that simplify the equation or reveal some useful information about the function.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 373, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_y_f_x_f_y_for_al__2022-12-23T19-37-34.403399|chosen" } }, { "text": "For example, if I let $x = y,$ then the equation becomes $f(0) = f(x) f(x) = f(x)^2.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 373, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_y_f_x_f_y_for_al__2022-12-23T19-37-34.403399|chosen" } }, { "text": "This tells me that $f(0)$ is either 0 or 1, but since the problem says that $f(x) \\neq 0$ for all real numbers $x,$ I can conclude that $f(0) = 1.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 373, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_y_f_x_f_y_for_al__2022-12-23T19-37-34.403399|chosen" } }, { "text": "Another useful thing to try is to let one of the variables be 0, say $y = 0.$ Then the equation becomes $f(x) = f(x) f(0) = f(x),$ which is always true and doesn't tell me anything new.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 373, "metadata": { "path_label": "the_function_f_x_satisfies_f_x_y_f_x_f_y_for_al__2022-12-23T19-37-34.403399|chosen" } }, { "text": "I notice that the determinant of a matrix is the volume of the parallelepiped spanned by its columns, so I want to maximize that volume.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 374, "metadata": { "path_label": "find_the_maximum_value_of_begin_vmatrix_1_1_1_1__2022-12-23T19-36-56.713496|chosen" } }, { "text": "I also notice that the matrix has a lot of ones, so I can simplify it by subtracting the first row from the other two rows.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 374, "metadata": { "path_label": "find_the_maximum_value_of_begin_vmatrix_1_1_1_1__2022-12-23T19-36-56.713496|chosen" } }, { "text": "This gives me \\[\\begin{vmatrix} 1 & 1 & 1 \\\\ 1 & 1 + \\sin \\theta & 1 \\\\ 1 + \\cos \\theta & 1 & 1 \\end{vmatrix} = \\begin{vmatrix} 1 & 1 & 1 \\\\ 0 & \\sin \\theta & 0 \\\\ \\cos \\theta & 0 & 0 \\end{vmatrix}.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 374, "metadata": { "path_label": "find_the_maximum_value_of_begin_vmatrix_1_1_1_1__2022-12-23T19-36-56.713496|chosen" } }, { "text": "I need to find two numbers that multiply to 2008 and add to $b$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "I can try to factor 2008 into prime factors and see what combinations of them work.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "2008 is even, so it must be divisible by 2. In fact, it is divisible by 4, since the last two digits are 08.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "So I can write 2008 as $4\\times 502$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Now, 502 is also even, so it is divisible by 2. I can write 2008 as $4\\times 2\\times 251$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "251 is a prime number, so I cannot factor it any further.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Therefore, the prime factorization of 2008 is $2^3\\times 251$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Now, I can look for pairs of factors that multiply to 2008 and add to $b$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "One pair is $1\\times 2008$, but that would give $b=2009$, which is not the smallest possible value.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Another pair is $2\\times 1004$, but that would give $b=1006$, which is still too big.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Another pair is $4\\times 502$, but that would give $b=506$, which is still not the smallest.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Another pair is $8\\times 251$, but that would give $b=259$, which might be the smallest, but I'm not sure.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "I need to check if there are any smaller pairs of factors that work.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "I notice that 2008 is not divisible by 3, 5, 7, 11, or any other odd prime less than 251.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "That means that the only possible factors of 2008 that are less than 251 are powers of 2.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "The powers of 2 that are less than 251 are 1, 2, 4, 8, 16, 32, 64, and 128.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "I can check if any of these numbers, paired with their quotients when dividing 2008, add up to a smaller value than 259.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "For example, $16\\times 125.5$ is not a valid pair, since 125.5 is not an integer.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Similarly, $32\\times 62.75$ is not a valid pair.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "The only valid pair that is smaller than $8\\times 251$ is $64\\times 31.375$, but that still gives $b=95.375$, which is not an integer.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "Therefore, the smallest possible value of $b$ is 259.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 375, "metadata": { "path_label": "find_the_smallest_positive_integer_b_for_which_x_2_bx_200__2022-12-23T16-59-44.926838|chosen" } }, { "text": "To find the number of multiples of either $5$ or $7$, I can use the inclusion-exclusion principle.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "This says that the number of elements in the union of two sets is equal to the sum of the numbers of elements in each set minus the number of elements in their intersection.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "In this case, the two sets are the multiples of $5$ and the multiples of $7$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "The number of multiples of $5$ less than $101$ is $\\lfloor \\frac{100}{5} \\rfloor = 20$, where $\\lfloor x \\rfloor$ means the greatest integer less than or equal to $x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "Similarly, the number of multiples of $7$ less than $101$ is $\\lfloor \\frac{100}{7} \\rfloor = 14$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "The number of multiples of both $5$ and $7$ less than $101$ is the same as the number of multiples of $35$ less than $101$, which is $\\lfloor \\frac{100}{35} \\rfloor = 2$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "So, by the inclusion-exclusion principle, the number of multiples of either $5$ or $7$ less than $101$ is $20 + 14 - 2 = 32$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "However, this includes the multiples of both $5$ and $7$, which I want to exclude.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "So, I need to subtract the number of multiples of both $5$ and $7$ again, which is $2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "Therefore, the final answer is $32 - 2 = 30$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 376, "metadata": { "path_label": "how_many_positive_integers_less_than_101_are_multiples_of__2022-12-23T16-58-31.110730|chosen" } }, { "text": "To solve this problem, I need to recall some basic facts about the greatest common divisor (gcd) and the least common multiple (lcm) of two numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "One fact is that the product of two numbers is equal to the product of their gcd and their lcm.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "This means that if I call the other number $y$, then I have $50y = (x+5)x(x+5)$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "Another fact is that the gcd of two numbers divides both numbers evenly.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "This means that $x+5$ must be a factor of both 50 and $y$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "I can list the factors of 50 as 1, 2, 5, 10, 25, and 50.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "Since $x$ is a positive integer, $x+5$ must be at least 6.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "The only factors of 50 that are at least 6 are 10, 25, and 50.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "This means that $x+5$ must be one of these values.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 377, "metadata": { "path_label": "the_greatest_common_divisor_of_two_positive_integers_is_x__2022-12-23T16-57-38.585936|chosen" } }, { "text": "I notice that the curve is a rational function, and I wonder how it looks like on a graph.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "I recall that rational functions have vertical asymptotes where the denominator is zero, and horizontal asymptotes where the degree of the numerator and denominator are equal.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "So, the curve has a vertical asymptote at $x = -\\frac{s}{r}$ and a horizontal asymptote at $y = \\frac{p}{r}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "Since the line $y = x$ is an axis of symmetry, the curve must be symmetric about this line.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "That means that if I reflect any point on the curve across the line $y = x$, I should get another point on the curve.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "Let me try to do that with the asymptotes.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "The vertical asymptote $x = -\\frac{s}{r}$ reflects to the horizontal asymptote $y = -\\frac{s}{r}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "The horizontal asymptote $y = \\frac{p}{r}$ reflects to the vertical asymptote $x = \\frac{p}{r}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "So, the curve has two pairs of asymptotes that are reflections of each other.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "This implies that the slopes of the asymptotes are equal and opposite, since they are perpendicular to each other.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "So, $\\frac{p}{r} = -\\frac{s}{r}$, which simplifies to $p + s = 0$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "This means that statement (C) must hold.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 378, "metadata": { "path_label": "the_line_with_equation_y_x_is_an_axis_of_symmetry_of_the__2022-12-23T16-52-42.646337|chosen" } }, { "text": "The lcm of two numbers is the smallest positive multiple of both numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "To simplify the expression, I can use the property that $\\mathop{\\text{lcm}}[a,b]=\\frac{ab}{\\gcd(a,b)}$, where $\\gcd(a,b)$ is the greatest common divisor of $a$ and $b$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "So I can rewrite the equation as $\\left(\\frac{12t}{\\gcd(12,t)}\\right)^3=(12t)^2$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "To eliminate the fractions, I can multiply both sides by $\\gcd(12,t)^3$, which gives $(12t)^3=\\gcd(12,t)^3(12t)^2$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "I can divide both sides by $(12t)^2$, which gives $12t=\\gcd(12,t)^3$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "Now I need to find a value of $t$ that makes this equation true.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "I know that $\\gcd(12,t)$ must be a factor of both $12$ and $t$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "The possible factors of $12$ are $1,2,3,4,6,12$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "I can try each of these factors and see if they satisfy the equation.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=1$, then $12t=1^3$, which is impossible, since $t$ has to be positive.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=2$, then $12t=2^3=8$, which implies $t=\\frac{2}{3}$, which is not an integer.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=3$, then $12t=3^3=27$, which implies $t=\\frac{9}{4}$, which is not an integer.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=4$, then $12t=4^3=64$, which implies $t=\\frac{16}{3}$, which is not an integer.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=6$, then $12t=6^3=216$, which implies $t=18$, which is an integer.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "If $\\gcd(12,t)=12$, then $12t=12^3=1728$, which implies $t=144$, which is also an integer.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "However, I want the smallest possible value of $t$, so I choose $t=18$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 379, "metadata": { "path_label": "suppose_t_is_a_positive_integer_such_that_mathop_text_l__2022-12-23T16-45-19.538205|chosen" } }, { "text": "I need to count the number of ways to choose 3 men out of 30 and 4 women out of 40.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "This is a combination problem, since the order of the people on the committee does not matter.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "To choose 3 men out of 30, I can use the formula for combinations: 30 choose 3 = 30! / (3! * 27!) = (30 * 29 * 28) / (3 * 2 * 1) = 4060.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "Similarly, to choose 4 women out of 40, I can use the same formula: 40 choose 4 = 40! / (4! * 36!) = (40 * 39 * 38 * 37) / (4 * 3 * 2 * 1) = 91390.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "To form the committee, I need to combine one of the 4060 groups of 3 men with one of the 91390 groups of 4 women.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "This means I have 4060 * 91390 ways to form the committee, since for each group of men, I have 91390 choices for the group of women.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 380, "metadata": { "path_label": "there_are_30_men_and_40_women_in_the_town_library_club_the__2022-12-23T12-49-27.906805|chosen" } }, { "text": "To express a repeating decimal as a common fraction, I need to find a way to eliminate the decimal part by multiplying both sides of an equation by a power of 10.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Let me start by assigning a variable to the repeating decimal: let x = 0.5\\overline{10}.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "To get rid of the 0.5 part, I can multiply both sides by 10, which gives me 10x = 5.\\overline{10}.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Now I still have a repeating decimal, but the part that repeats is 10, which is longer than before.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "To eliminate that, I need to multiply both sides by another power of 10, but this time it has to be big enough to cover the whole repeating part.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Since the repeating part is two digits long, I need to multiply by 10^2 = 100, which gives me 1000x = 510.\\overline{10}.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Now I have two equations with x, but they have different coefficients: 10x and 1000x.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "If I subtract the first equation from the second, I can eliminate x and get a constant on one side: 1000x - 10x = 510.\\overline{10} - 5.\\overline{10}.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Simplifying, I get 990x = 505.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "To isolate x, I need to divide both sides by 990, which gives me x = 505/990.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "This is a fraction, but it is not in lowest terms, since both the numerator and the denominator are divisible by 5.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "Dividing both by 5, I get x = 101/198.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "This is the common fraction that represents the repeating decimal 0.5\\overline{10}.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 381, "metadata": { "path_label": "express_0_5_overline_10_as_a_common_fraction__2022-12-23T12-29-47.079166|chosen" } }, { "text": "I need to find the number of ways Carlos can choose five CDs that include at least one from each category, and divide that by the total number of ways he can choose any five CDs from the 12.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 382, "metadata": { "path_label": "carlos_has_chosen_twelve_different_compact_discs_cds_he_wo__2022-12-23T12-10-23.693323|chosen" } }, { "text": "To choose any five CDs from the 12, he has 12 choose 5 ways, which is 12!/(5!7!) = 792.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 382, "metadata": { "path_label": "carlos_has_chosen_twelve_different_compact_discs_cds_he_wo__2022-12-23T12-10-23.693323|chosen" } }, { "text": "To choose five CDs that include at least one from each category, I can consider the different cases based on how many CDs he chooses from each category.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 382, "metadata": { "path_label": "carlos_has_chosen_twelve_different_compact_discs_cds_he_wo__2022-12-23T12-10-23.693323|chosen" } }, { "text": "For example, one case is that he chooses one rap CD, one country CD, and three metal CDs. The number of ways to do this is 4 choose 1 times 5 choose 1 times 3 choose 3, which is 4 times 5 times 1 = 20.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 382, "metadata": { "path_label": "carlos_has_chosen_twelve_different_compact_discs_cds_he_wo__2022-12-23T12-10-23.693323|chosen" } }, { "text": "Another case is that he chooses one rap CD, two country CDs, and two metal CDs. The number of ways to do this is 4 choose 1 times 5 choose 2 times 3 choose 2, which is 4 times 10 times 3 = 120.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 382, "metadata": { "path_label": "carlos_has_chosen_twelve_different_compact_discs_cds_he_wo__2022-12-23T12-10-23.693323|chosen" } }, { "text": "I know that the roots of $z^3 = 1$ are $1, \\omega,$ and $\\omega^2,$ where $\\omega = e^{2\\pi i/3} = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2} i.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "So $\\omega$ is a complex number on the unit circle with angle $\\frac{2\\pi}{3}$ radians.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "I also know that $(\\omega + 1)$ is a complex number with magnitude $|(\\omega + 1)| = \\sqrt{(\\frac{-1}{2} + 1)^2 + (\\frac{\\sqrt{3}}{2})^2} = \\sqrt{1 + \\frac{1}{4} + \\frac{3}{4}} = \\sqrt{2}.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "And the angle of $(\\omega + 1)$ is $\\arg(\\omega + 1) = \\tan^{-1}(\\frac{\\sqrt{3}}{2} / (\\frac{-1}{2} + 1)) = \\tan^{-1}(\\frac{\\sqrt{3}}{2} / \\frac{1}{2}) = \\tan^{-1}(\\sqrt{3}) = \\frac{\\pi}{3}.$ [* the way I remember this is by using the half-angle formula for tangent: $\\tan(\\frac{\\theta}{2}) = \\frac{\\sin(\\theta)}{1 + \\cos(\\theta)}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "So $(\\omega + 1)$ is a complex number with magnitude $\\sqrt{2}$ and angle $\\frac{\\pi}{3}$ radians.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "Now I want to find the possible values of $(\\omega + 1)^n,$ where $n$ is a positive integer.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "I know that when I raise a complex number to a power, I multiply its magnitude by itself $n$ times, and I add its angle to itself $n$ times, modulo $2\\pi.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "So the magnitude of $(\\omega + 1)^n$ is $(\\sqrt{2})^n = 2^{\\frac{n}{2}}.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "And the angle of $(\\omega + 1)^n$ is $n \\cdot \\frac{\\pi}{3} \\pmod{2\\pi}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "This means that the angle of $(\\omega + 1)^n$ can only take six possible values: $\\frac{\\pi}{3}, \\frac{2\\pi}{3}, \\pi, \\frac{4\\pi}{3}, \\frac{5\\pi}{3},$ and $2\\pi.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "These are the same as the angles of the sixth roots of unity.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "So the possible values of $(\\omega + 1)^n$ are the same as the possible values of $2^{\\frac{n}{2}} \\cdot e^{i\\frac{\\pi}{3}},$ where $e^{i\\frac{\\pi}{3}}$ is any sixth root of unity.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "There are six distinct sixth roots of unity, so there are six different possible values of $(\\omega + 1)^n.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 383, "metadata": { "path_label": "let_omega_be_a_nonreal_root_of_z_3_1_find_the_numbe__2022-12-23T11-55-50.699791|chosen" } }, { "text": "I know that if a function is its own inverse, then it must be symmetric about the line $y=x$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "That means that for any point $(a,b)$ on the graph of $f$, the point $(b,a)$ must also be on the graph of $f$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "I also know that the graph of $f$ will be a parabola for $x\\leq3$, and some other function for $x>3$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "I can use the parabola part to find some points on the graph of $f$, and then use the symmetry property to find the corresponding points on the graph of $k$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "For example, if I plug in $x=3$, I get $f(3)=3^2-6\\cdot3+12=3$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "So the point $(3,3)$ is on the graph of $f$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "By symmetry, the point $(3,3)$ is also on the graph of $k$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "That means that $k(3)=3$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "Similarly, if I plug in $x=2$, I get $f(2)=2^2-6\\cdot2+12=4$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "So the point $(2,4)$ is on the graph of $f$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "By symmetry, the point $(4,2)$ is also on the graph of $k$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "That means that $k(4)=2$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 384, "metadata": { "path_label": "let_f_x_begin_cases_k_x_text_if_x_3_x_2_6x_1__2022-12-23T11-48-19.189452|chosen" } }, { "text": "I know that the circumcenter is the point where the perpendicular bisectors of the sides of the triangle meet, and the orthocenter is the point where the altitudes of the triangle meet.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "I also know that the circumradius $R$ is the distance from the circumcenter to any vertex of the triangle, and the orthocenter is the reflection of the circumcenter across the midpoint of the opposite side.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "So, if $AO = AH,$ then I can use the fact that $AH = 2R \\cos A$ to get $AO = 2R \\cos A.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "But $AO = R,$ so this means $R = 2R \\cos A,$ or $\\cos A = \\frac{1}{2}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "This implies that $A$ is either $60$ degrees or $300$ degrees, since those are the angles whose cosine is $\\frac{1}{2}.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "However, $A$ cannot be $300$ degrees, since that would make the triangle obtuse and the orthocenter would be outside the triangle, which contradicts the given condition.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "Therefore, the only possible value of $A$ is $60$ degrees.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 385, "metadata": { "path_label": "let_o_and_h_denote_the_circumcenter_and_orthocenter_of_t__2022-12-23T11-44-28.503198|chosen" } }, { "text": "I notice that the sine function has a period of $360^\\circ,$ which means that adding or subtracting multiples of $360^\\circ$ to an angle does not change its sine value.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 386, "metadata": { "path_label": "find_the_integer_n_90_le_n_le_90_such_that_sin_n__2022-12-23T11-40-58.029355|chosen" } }, { "text": "So, I can write $\\sin 604^\\circ = \\sin (604 - 360)^\\circ = \\sin 244^\\circ.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 386, "metadata": { "path_label": "find_the_integer_n_90_le_n_le_90_such_that_sin_n__2022-12-23T11-40-58.029355|chosen" } }, { "text": "Now, I want to find an angle between $-90^\\circ$ and $90^\\circ$ that has the same sine as $244^\\circ.$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 386, "metadata": { "path_label": "find_the_integer_n_90_le_n_le_90_such_that_sin_n__2022-12-23T11-40-58.029355|chosen" } }, { "text": "I notice that this equation has both $x^2$ and $y^2$ terms, so it is not a line or a point.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "I also notice that the coefficients of $x^2$ and $y^2$ have different signs, so it is not a circle or an ellipse.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "Therefore, it must be either a hyperbola, two lines, or empty. [* 2 points] To decide which one, I will try to complete the square for both $x$ and $y$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "I can rewrite the equation as $(x^2 - 10x) - 50(y^2 - 0) = -25$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "To complete the square for $x$, I need to add $(10/2)^2 = 25$ to both sides.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "To complete the square for $y$, I need to subtract $50(0/2)^2 = 0$ from both sides.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "So I get $(x - 5)^2 - 50y^2 = 0$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "This looks like a hyperbola, but I need to make sure it is not degenerate.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "If I divide both sides by $-50$, I get $y^2 - (1/50)(x - 5)^2 = 0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "This means that $y = \\pm \\sqrt{(1/50)(x - 5)^2}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "This is a pair of lines that intersect at $(5, 0)$ and have slopes of $\\pm \\sqrt{1/50}$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "So the graph is two lines.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 387, "metadata": { "path_label": "determine_if_the_graph_of_the_equation_below_is_a_parabola__2022-12-23T10-53-53.307812|chosen" } }, { "text": "I notice that both congruences are linear, so their graphs will be straight lines that wrap around the modulo $16$ grid.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 388, "metadata": { "path_label": "modulo_m_graph_paper_consists_of_a_grid_of_m_2_points__2022-12-23T10-09-54.974116|chosen" } }, { "text": "To find the points in common, I need to solve the system of congruences: $$\\begin{cases}y\\equiv 5x+2\\pmod{16}\\\\y\\equiv 11x+12\\pmod{16}\\end{cases}$$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 388, "metadata": { "path_label": "modulo_m_graph_paper_consists_of_a_grid_of_m_2_points__2022-12-23T10-09-54.974116|chosen" } }, { "text": "I can eliminate $y$ by subtracting the second congruence from the first, which gives me: $$0\\equiv -6x-10\\pmod{16}$$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 388, "metadata": { "path_label": "modulo_m_graph_paper_consists_of_a_grid_of_m_2_points__2022-12-23T10-09-54.974116|chosen" } }, { "text": "I need to find a number of eggs that is one more than a multiple of both $12$ and $13$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "This means that the number of eggs is one more than a common multiple of $12$ and $13$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "I can use the least common multiple of $12$ and $13$ to find the smallest possible number of eggs that satisfies this condition.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "To find the least common multiple of $12$ and $13$, I can use the prime factorization method.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "The prime factorization of $12$ is $2^2 \\times 3$, and the prime factorization of $13$ is $13$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "The least common multiple of $12$ and $13$ is the product of the highest power of each prime factor that appears in either factorization.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "So the least common multiple of $12$ and $13$ is $2^2 \\times 3 \\times 13$, which is $156$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 389, "metadata": { "path_label": "your_friend_has_an_egg_collection_comprising_at_least_200__2022-12-23T20-49-54.087411|chosen" } }, { "text": "This problem involves absolute value equations, which can be tricky to solve.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 390, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-48-14.422535|chosen" } }, { "text": "One way to approach it is to split the number line into different intervals based on the critical points of the equation, which are $x = 1, 2, 3$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 390, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-48-14.422535|chosen" } }, { "text": "In each interval, the absolute value expressions will have a different sign, so we can remove the absolute value bars and solve for $x$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 390, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-48-14.422535|chosen" } }, { "text": "For example, if $x < 1$, then $x-1$ is negative, so $|x-1| = -(x-1) = 1-x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 390, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-48-14.422535|chosen" } }, { "text": "Similarly, $x-2$ and $x-3$ are negative, so $|x-2| = 2-x$ and $|x-3| = 3-x$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 390, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-48-14.422535|chosen" } }, { "text": "This problem involves absolute value equations, which can be tricky to solve.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 391, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-46-56.148979|chosen" } }, { "text": "One way to approach it is to split the number line into different intervals based on the critical points of the equation, which are $x = 1, 2, 3$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 391, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-46-56.148979|chosen" } }, { "text": "In each interval, the absolute value expressions will have a different sign, so we can remove the absolute value bars and solve for $x$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 391, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-46-56.148979|chosen" } }, { "text": "For example, if $x < 1$, then $x-1$ is negative, so $|x-1| = -(x-1) = 1-x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 391, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-46-56.148979|chosen" } }, { "text": "Similarly, $x-2$ and $x-3$ are negative, so $|x-2| = 2-x$ and $|x-3| = 3-x$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 391, "metadata": { "path_label": "how_many_real_numbers_x_are_solutions_to_the_following_equ__2022-12-23T20-46-56.148979|chosen" } }, { "text": "I know that congruent to $15 \\pmod{22}$ means that the difference between the number and $15$ is a multiple of $22$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "So I want to find the largest $4$ digit number that has this property.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "I could start by adding $22$ to $15$ until I get a $4$ digit number, but that might take too long.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "I could also try to guess and check, but that might also take too long.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "A better way is to use the division algorithm and write the number as $22q + 15$, where $q$ is the quotient and $15$ is the remainder.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "Then I want to maximize $q$ such that $22q + 15$ is still a $4$ digit number.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "The largest $4$ digit number is $9999$, so I can write $9999 = 22q + 15$ and solve for $q$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "Subtracting $15$ from both sides gives $9984 = 22q$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 392, "metadata": { "path_label": "what_is_the_largest_4_digit_integer_congruent_to_15_pmod__2022-12-23T20-43-03.082923|chosen" } }, { "text": "To form a geometric sequence, the ratio between consecutive terms must be constant.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "So, if the first digit is $a$, the second digit is $ar$, and the third digit is $ar^2$, where $a$ and $r$ are positive integers.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "The largest possible value of $a$ is $9$, since it is a digit.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "The largest possible value of $r$ is $3$, since otherwise $ar^2$ would exceed $9$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "So, the largest geometric number is $936$, where $a=9$ and $r=2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "The smallest possible value of $a$ is $1$, since it is a digit.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 393, "metadata": { "path_label": "call_a_3_digit_number_geometric_if_it_has_3_distinct_dig__2022-12-23T20-42-28.718662|chosen" } }, { "text": "I need to find the points where the two curves intersect, since those will be the corners of the region.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "I can set $|x|=-x^2+6$ and solve for x.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "If $x\\geq 0$, then $|x|=x$ and the equation becomes $x=-x^2+6$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "I can rearrange this to get $x^2+x-6=0$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "This factors as $(x+3)(x-2)=0$, so $x=-3$ or $x=2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "But $x=-3$ is not possible, since I assumed $x\\geq 0$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "So the only positive solution is $x=2$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "If $x<0$, then $|x|=-x$ and the equation becomes $-x=-x^2+6$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "I can rearrange this to get $x^2-x-6=0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "This factors as $(x-3)(x+2)=0$, so $x=3$ or $x=-2$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "But $x=3$ is not possible, since I assumed $x<0$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "So the only negative solution is $x=-2$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "Therefore, the two curves intersect at $(2,2)$ and $(-2,2)$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "Now I need to count the lattice points on the boundary and inside the region.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "I can start by counting the points on the x-axis, since they are easy to see.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 394, "metadata": { "path_label": "a_lattice_point_is_a_point_whose_coordinates_are_both_intege__2022-12-23T20-39-38.838554|chosen" } }, { "text": "This problem involves matrix multiplication and transposes.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "I know that if $\\mathbf{M} \\mathbf{M}^T = 9 \\mathbf{I},$ then the matrix $\\mathbf{M}$ must be orthogonal, which means that its columns are unit vectors that are perpendicular to each other.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "That means that the dot product of any two columns of $\\mathbf{M}$ must be zero, and the dot product of any column with itself must be 9.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "So I can use these properties to find $a$ and $b$ by taking dot products of the columns of $\\mathbf{M}.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "The first column of $\\mathbf{M}$ is $(1,2,a)^T$ and the second column is $(2,1,2)^T.$ So their dot product is $1 \\cdot 2 + 2 \\cdot 1 + a \\cdot 2 = 4 + 2a.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "Since this must be zero, I can solve for $a$ and get $a = -2.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "The first column of $\\mathbf{M}$ is $(1,2,-2)^T$ and the third column is $(2,-2,b)^T.$ So their dot product is $1 \\cdot 2 + 2 \\cdot (-2) + (-2) \\cdot b = 2 - 4 - 2b.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "Since this must be zero, I can solve for $b$ and get $b = -1.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "So the ordered pair $(a,b)$ is $(-2,-1).$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 395, "metadata": { "path_label": "let_mathbf_m_begin_pmatrix_1_2_2_2_1_2__2022-12-23T20-31-37.062307|chosen" } }, { "text": "I need to find the volume of the original box and the volume of the removed cubes, and then compare them as a ratio.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "The volume of the original box is the product of its dimensions: 15 x 10 x 8 = 1200 cm^3.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "The volume of each removed cube is the cube of its side length: 3 x 3 x 3 = 27 cm^3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "There are 8 corners on the box, so 8 cubes are removed. [*: The total volume removed is 8 times the volume of one cube: 8 x 27 = 216 cm^3.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "The ratio of the removed volume to the original volume is 216/1200, which simplifies to 9/50.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "To convert this ratio to a percent, I multiply by 100 and add a percent sign: 9/50 x 100 = 18%.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "So 18% of the original volume is removed.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 396, "metadata": { "path_label": "a_solid_box_is_15_cm_by_10_cm_by_8_cm_a_new_solid_is_formed__2022-12-23T20-27-36.691358|chosen" } }, { "text": "I notice that this expression has complex numbers, logarithms, and trigonometric functions, so I want to use some identities and properties to simplify it.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 397, "metadata": { "path_label": "simplify_2_cos_2_log_2009_i_i_sin_log_4036081_i_note__2022-12-23T20-15-03.851401|chosen" } }, { "text": "First, I recall that $$e^{ix}=\\cos(x)+i\\sin(x)$$ for any real number $x.$ This means that $$\\cos(x)=\\frac{e^{ix}+e^{-ix}}{2}$$ and $$\\sin(x)=\\frac{e^{ix}-e^{-ix}}{2i}$$ for any real number $x.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 397, "metadata": { "path_label": "simplify_2_cos_2_log_2009_i_i_sin_log_4036081_i_note__2022-12-23T20-15-03.851401|chosen" } }, { "text": "Second, I recall that $$\\log(a^b)=b\\log(a)$$ for any positive numbers $a$ and $b.$ This means that $$\\log(2009)i=i\\log(2009)$$ and $$\\log(4036081)i=i\\log(4036081)$$ since $i$ is a constant.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 397, "metadata": { "path_label": "simplify_2_cos_2_log_2009_i_i_sin_log_4036081_i_note__2022-12-23T20-15-03.851401|chosen" } }, { "text": "Now, I can substitute these identities and properties into the original expression and get $$2\\frac{e^{i\\log(2009)}+e^{-i\\log(2009)}}{2}+i\\frac{e^{i\\log(4036081)}-e^{-i\\log(4036081)}}{2i}.$$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 397, "metadata": { "path_label": "simplify_2_cos_2_log_2009_i_i_sin_log_4036081_i_note__2022-12-23T20-15-03.851401|chosen" } }, { "text": "To find a multiplicative inverse to 160 modulo 1399, I need to find an $n$ such that $160n\\equiv 1 \\pmod{1399}$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "This means that $160n$ and $1$ have the same remainder when divided by 1399.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "Another way to say this is that $160n-1$ is a multiple of 1399, or $160n-1=1399k$ for some integer $k$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "If I can find such a $k$, then I can find $n$ by dividing both sides by 160.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "But how do I find $k$?", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "One way to approach this is to use the extended Euclidean algorithm, which finds integers $x$ and $y$ such that $ax+by=\\gcd(a,b)$, where $a$ and $b$ are given.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 398, "metadata": { "path_label": "notice_that_35_cdot40_1400_find_some_integer_n_with__2022-12-23T20-14-26.878563|chosen" } }, { "text": "I need to find the area of a triangle inscribed in a circle, with one side as a diameter.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 399, "metadata": { "path_label": "circle_c_has_radius_10_cm_how_many_square_centimeters_are__2022-12-23T20-14-18.736093|chosen" } }, { "text": "I recall that such a triangle is always a right triangle, by Thales' theorem.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 399, "metadata": { "path_label": "circle_c_has_radius_10_cm_how_many_square_centimeters_are__2022-12-23T20-14-18.736093|chosen" } }, { "text": "I also remember that the area of a triangle is half the product of the base and the height.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 399, "metadata": { "path_label": "circle_c_has_radius_10_cm_how_many_square_centimeters_are__2022-12-23T20-14-18.736093|chosen" } }, { "text": "I notice that the problem is asking about remainders when dividing by $20$, so I think about using modular arithmetic to simplify the calculations.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "If $ab\\equiv 17\\pmod{20}$, then I can write $ab=20k+17$ for some integer $k$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "Now I want to find $(a+10)(b+10)$ in terms of $ab$ and $k$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "I use the distributive property to get $(a+10)(b+10)=ab+10a+10b+100$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "I substitute $ab=20k+17$ and simplify: $(a+10)(b+10)=20k+17+10a+10b+100=20(k+5)+10(a+b)+17$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "To find the remainder when dividing by $20$, I only need to look at the last term, which is $10(a+b)+17$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "I want to reduce this term modulo $20$, so I need to find the remainder of $a+b$ when dividing by $2$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "I use the fact that $ab\\equiv 17\\pmod{20}$ to infer that $a$ and $b$ are both odd, since $17$ is odd and $20$ is even.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "Therefore, $a+b$ is even, and the remainder when dividing by $2$ is $0$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "This means that $10(a+b)$ is divisible by $20$, and the remainder when dividing by $20$ is the same as the remainder of $17$ when dividing by $20$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "The remainder of $17$ when dividing by $20$ is just $17$, so the answer is $17$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 400, "metadata": { "path_label": "if_a_and_b_are_integers_such_that_ab_equiv_17_pmod_20__2022-12-23T20-11-06.201346|chosen" } }, { "text": "This is a product of six factors under a square root.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "I wonder if I can simplify this expression by finding perfect squares among the factors.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "A perfect square is a number that can be written as the square of an integer, like 4 = 2^2 or 9 = 3^2.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "If I can find some perfect squares, I can use the property that $\\sqrt{a\\cdot b} = \\sqrt{a}\\cdot\\sqrt{b}$ to separate them out and simplify the square roots.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Let me look at each factor and see if it is a perfect square or not.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "3 is not a perfect square, it is a prime number.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "6 is not a perfect square, it is 2 times 3.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "10 is not a perfect square, it is 2 times 5.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "15 is not a perfect square, it is 3 times 5.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "21 is not a perfect square, it is 3 times 7.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "28 is not a perfect square, it is 4 times 7.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Hmm, none of the factors are perfect squares by themselves.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Maybe I can group them into pairs and see if the product of each pair is a perfect square.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "3 times 6 is 18, which is not a perfect square.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "6 times 10 is 60, which is not a perfect square.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "10 times 15 is 150, which is not a perfect square.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "15 times 21 is 315, which is not a perfect square.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "21 times 28 is 588, which is not a perfect square.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "28 times 3 is 84, which is not a perfect square.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Hmm, none of the pairs of factors are perfect squares either.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Maybe I can group them into triples and see if the product of each triple is a perfect square.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "3 times 6 times 10 is 180, which is not a perfect square.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "6 times 10 times 15 is 900, which is a perfect square! It is 30^2.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "10 times 15 times 21 is 3150, which is not a perfect square.", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "15 times 21 times 28 is 8820, which is not a perfect square.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "21 times 28 times 3 is 1764, which is a perfect square! It is 42^2.", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "28 times 3 times 6 is 504, which is not a perfect square.", "role": "entry", "step_index": 26, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Aha, I found two triples of factors that are perfect squares: 6 times 10 times 15 and 21 times 28 times 3.", "role": "entry", "step_index": 27, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "I can use the property that $\\sqrt{a\\cdot b} = \\sqrt{a}\\cdot\\sqrt{b}$ to separate them out and simplify the square roots.", "role": "entry", "step_index": 28, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "I have $\\sqrt{3\\cdot6\\cdot10\\cdot15\\cdot21\\cdot28} = \\sqrt{(6\\cdot10\\cdot15)\\cdot(21\\cdot28\\cdot3)} = \\sqrt{30^2\\cdot42^2} = \\sqrt{30^2}\\cdot\\sqrt{42^2} = 30\\cdot42 = 1260$.", "role": "entry", "step_index": 29, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "Therefore, the final answer is 1260.", "role": "entry", "step_index": 30, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 401, "metadata": { "path_label": "evaluate_sqrt_3_cdot6_cdot10_cdot15_cdot21_cdot28__2022-12-23T14-37-34.557007|chosen" } }, { "text": "I notice that the polynomial has degree 3, so it can have at most 3 roots. If they form an arithmetic progression and are not all real, then there must be exactly one real root and two complex roots that are conjugates of each other.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 402, "metadata": { "path_label": "find_all_real_numbers_a_such_that_the_roots_of_the_polynom__2022-12-23T14-33-42.946360|chosen" } }, { "text": "Let me call the real root $r$ and the complex roots $s \\pm ti$, where $s$ and $t$ are real numbers and $i$ is the imaginary unit. Then the arithmetic progression means that $s \\pm ti = r \\pm d$, where $d$ is the common difference.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 402, "metadata": { "path_label": "find_all_real_numbers_a_such_that_the_roots_of_the_polynom__2022-12-23T14-33-42.946360|chosen" } }, { "text": "Expanding and equating real and imaginary parts, I get $s = r$ and $t = \\pm d$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 402, "metadata": { "path_label": "find_all_real_numbers_a_such_that_the_roots_of_the_polynom__2022-12-23T14-33-42.946360|chosen" } }, { "text": "So the roots are $r - di$, $r$, and $r + di$. By the sum and product of roots formulas, I can write two equations involving $r$ and $d$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 402, "metadata": { "path_label": "find_all_real_numbers_a_such_that_the_roots_of_the_polynom__2022-12-23T14-33-42.946360|chosen" } }, { "text": "The sum of roots is $r - di + r + r + di = 3r$, which is equal to the opposite of the coefficient of $x^2$, so $3r = 6$ or $r = 2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 402, "metadata": { "path_label": "find_all_real_numbers_a_such_that_the_roots_of_the_polynom__2022-12-23T14-33-42.946360|chosen" } }, { "text": "This is a problem of partial fraction decomposition, where we want to write a rational function as a sum of simpler rational functions with linear denominators.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "One way to approach this problem is to clear the denominators on both sides of the equation and then equate the coefficients of the same powers of $x.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "So, multiplying both sides by $x(x + 1)(x + 3),$ we get \\[x + 2 = A(x + 1)(x + 3) + Bx(x + 3) + Cx(x + 1).\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "Expanding the right-hand side, we get \\[x + 2 = A(x^2 + 4x + 3) + Bx^2 + 3Bx + Cx^2 + Cx.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "Combining like terms, we get \\[x + 2 = (A + B + C)x^2 + (4A + 3B + C)x + 3A.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "Now, we can equate the coefficients of $x^2,$ $x,$ and the constant terms on both sides.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "For the constant terms, we have $2 = 3A,$ so $A = \\frac{2}{3}.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "For the linear terms, we have $1 = 4A + 3B + C,$ so plugging in $A = \\frac{2}{3},$ we get $1 = \\frac{8}{3} + 3B + C,$ so $3B + C = -\\frac{5}{3}.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "For the quadratic terms, we have $0 = A + B + C,$ so plugging in $A = \\frac{2}{3},$ we get $0 = \\frac{2}{3} + B + C,$ so $B + C = -\\frac{2}{3}.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "Now, we have two equations involving $B$ and $C,$ so we can solve for them by elimination or substitution.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "For example, subtracting the second equation from the first, we get $2B = -1,$ so $B = -\\frac{1}{2}.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "Then, plugging in $B = -\\frac{1}{2}$ into either equation, we get $C = -\\frac{1}{6}.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "So, we have found $A,$ $B,$ and $C,$ and their product is $ABC = \\frac{2}{3} \\cdot -\\frac{1}{2} \\cdot -\\frac{1}{6} = \\frac{1}{9}.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 403, "metadata": { "path_label": "there_exist_constants_a_b_and_c_so_that_frac_x__2022-12-23T14-30-57.399378|chosen" } }, { "text": "To solve for $wx + yz$, I need to find the values of $w$, $x$, $y$, and $z$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "I notice that the given equations have a lot of symmetry: each one involves three of the four variables, and each one has a different constant term.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "This suggests that I might be able to use elimination to reduce the system of equations to simpler ones.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "For example, if I subtract the first equation from the second, I get $(w+x+z) - (w+x+y) = z - y = 4 - (-2) = 6$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Similarly, if I subtract the first equation from the third, I get $(w+y+z) - (w+x+y) = z - x = 19 - (-2) = 21$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "And if I subtract the first equation from the fourth, I get $(x+y+z) - (w+x+y) = z - w = 12 - (-2) = 14$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Now I have three equations with two variables each: \\begin{align*} z - y &= 6, \\\\ z - x &= 21, \\text{ and} \\\\ z - w &= 14. \\end{align*}", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "I can use these equations to express $y$, $x$, and $w$ in terms of $z$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "For example, adding $y$ to both sides of the first equation, I get $z = y + 6$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Substituting this into the second equation, I get $y + 6 - x = 21$, or $x = y - 15$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Substituting $z = y + 6$ into the third equation, I get $y + 6 - w = 14$, or $w = y - 8$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "So now I have $y$, $x$, $w$, and $z$ all in terms of $y$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "I can plug these expressions into $wx + yz$ and simplify: \\begin{align*} wx + yz &= (y - 15)(y - 8) + y(y + 6) \\\\ &= y^2 - 23y + 120 + y^2 + 6y \\\\ &= 2y^2 - 17y + 120. \\end{align*}", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "To find the value of this expression, I need to find the value of $y$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "I can use any of the original equations to do this.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "For example, using the first equation, I get $w + x + y = -2$, or $(y - 8) + (y - 15) + y = -2$, or $3y - 23 = -2$, or $y = 7$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Plugging this into $2y^2 - 17y + 120$, I get $2(7)^2 - 17(7) + 120 = 98 - 119 + 120 = 99$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "Therefore, the answer is 99.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 404, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-29-16.715564|chosen" } }, { "text": "To solve for $wx + yz$, I need to find the values of $w$, $x$, $y$, and $z$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "I notice that the given equations have a lot of symmetry: each one involves three of the four variables, and each one has a different constant term.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "This suggests that I might be able to use elimination to reduce the system of equations to simpler ones.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "For example, if I subtract the first equation from the second, I get $(w+x+z) - (w+x+y) = z - y = 4 - (-2) = 6$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Similarly, if I subtract the first equation from the third, I get $(w+y+z) - (w+x+y) = z - x = 19 - (-2) = 21$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "And if I subtract the first equation from the fourth, I get $(x+y+z) - (w+x+y) = z - w = 12 - (-2) = 14$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Now I have three equations with two variables each: \\begin{align*} z - y &= 6, \\\\ z - x &= 21, \\text{ and} \\\\ z - w &= 14. \\end{align*}", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "I can use these equations to express $y$, $x$, and $w$ in terms of $z$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "For example, adding $y$ to both sides of the first equation, I get $z = y + 6$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Substituting this into the second equation, I get $y + 6 - x = 21$, or $x = y - 15$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Substituting $z = y + 6$ into the third equation, I get $y + 6 - w = 14$, or $w = y - 8$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "So now I have $y$, $x$, $w$, and $z$ all in terms of $y$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "I can plug these expressions into $wx + yz$ and simplify: \\begin{align*} wx + yz &= (y - 15)(y - 8) + y(y + 6) \\\\ &= y^2 - 23y + 120 + y^2 + 6y \\\\ &= 2y^2 - 17y + 120. \\end{align*}", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "To find the value of this expression, I need to find the value of $y$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "I can use any of the original equations to do this.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "For example, using the first equation, I get $w + x + y = -2$, or $(y - 8) + (y - 15) + y = -2$, or $3y - 23 = -2$, or $y = 7$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Plugging this into $2y^2 - 17y + 120$, I get $2(7)^2 - 17(7) + 120 = 98 - 119 + 120 = 99$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "Therefore, the answer is 99.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 405, "metadata": { "path_label": "if_w_x_y_and_z_are_real_numbers_satisfying_begi__2022-12-23T14-26-52.944954|chosen" } }, { "text": "I know that the cosine function is positive when the angle is between $ 0$ and $ \\frac{\\pi}{2}$ or between $ \\frac{3 \\pi}{2}$ and $ 2 \\pi$, and negative otherwise.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 406, "metadata": { "path_label": "let_x_be_a_real_number_such_that_the_five_numbers_cos__2022-12-23T14-21-26.716856|chosen" } }, { "text": "So, I want to find a value of $ x$ such that all five angles are in the second or third quadrants.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 406, "metadata": { "path_label": "let_x_be_a_real_number_such_that_the_five_numbers_cos__2022-12-23T14-21-26.716856|chosen" } }, { "text": "I also know that the cosine function has a period of $ 2 \\pi$, so if I add or subtract multiples of $ \\frac{1}{2}$ to $ x$, the value of the cosine function will not change.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 406, "metadata": { "path_label": "let_x_be_a_real_number_such_that_the_five_numbers_cos__2022-12-23T14-21-26.716856|chosen" } }, { "text": "I know that the graph of $r = \\sin 2 \\theta$ is a four-petaled rose that is symmetric about the origin.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 407, "metadata": { "path_label": "find_the_maximum_y_coordinate_of_a_point_on_the_graph_of__2022-12-23T14-14-46.105672|chosen" } }, { "text": "To find the maximum $y$-coordinate, I need to find the maximum value of $r$ when $\\theta$ is in the first or second quadrant, since that's where $y$ is positive.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 407, "metadata": { "path_label": "find_the_maximum_y_coordinate_of_a_point_on_the_graph_of__2022-12-23T14-14-46.105672|chosen" } }, { "text": "I recall that the sine function has a maximum value of $1$, so the maximum value of $r$ is also $1$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 407, "metadata": { "path_label": "find_the_maximum_y_coordinate_of_a_point_on_the_graph_of__2022-12-23T14-14-46.105672|chosen" } }, { "text": "This happens when $\\sin 2 \\theta = 1$, or when $2 \\theta = \\frac{\\pi}{2}$ or $\\frac{3 \\pi}{2}$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 407, "metadata": { "path_label": "find_the_maximum_y_coordinate_of_a_point_on_the_graph_of__2022-12-23T14-14-46.105672|chosen" } }, { "text": "Dividing by $2$, I get $\\theta = \\frac{\\pi}{4}$ or $\\frac{3 \\pi}{4}$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 407, "metadata": { "path_label": "find_the_maximum_y_coordinate_of_a_point_on_the_graph_of__2022-12-23T14-14-46.105672|chosen" } }, { "text": "I know that a regular hexagon can be divided into six equilateral triangles, and that the area of an equilateral triangle with side length $s$ is $\\frac{\\sqrt{3}}{4}s^2$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "If Fido's leash is long enough to reach the midpoint of each side, then it is also long enough to form the circumcircle of the hexagon, which passes through all the vertices.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "The radius of this circle is equal to the side length of the hexagon, since each vertex forms a right angle with the center and the midpoint of the opposite side.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "Therefore, the area of the circle is $\\pi s^2$, where $s$ is the side length of the hexagon.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "The area of the hexagon is $6 \\cdot \\frac{\\sqrt{3}}{4}s^2 = \\frac{3\\sqrt{3}}{2}s^2$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "The fraction of the area of the hexagon that Fido can reach is the ratio of the area of the circle to the area of the hexagon, which is $\\frac{\\pi s^2}{\\frac{3\\sqrt{3}}{2}s^2} = \\frac{2\\pi}{3\\sqrt{3}}$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "To simplify this fraction, I need to rationalize the denominator by multiplying the numerator and denominator by $\\sqrt{3}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "This gives me $\\frac{2\\pi}{3\\sqrt{3}} \\cdot \\frac{\\sqrt{3}}{\\sqrt{3}} = \\frac{2\\sqrt{3}\\pi}{9}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "Comparing this to the given form $\\frac{\\sqrt{a}}{b}\\pi$, I can see that $a = 12$ and $b = 9$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "Therefore, the value of the product $ab$ is $12 \\cdot 9 = 108$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 408, "metadata": { "path_label": "fido_s_leash_is_tied_to_a_stake_at_the_center_of_his_yard_w__2022-12-23T14-04-34.689216|chosen" } }, { "text": "This problem is asking me to count how many numbers in a certain range have a specific property.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "One way to approach this is to list all the possible numbers that meet the criteria and then count them.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "However, that might take a long time and be prone to errors.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "I wonder if there is a faster or more systematic way to do this.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "I notice that the numbers have to be between $500$ and $1000$, so they must have three digits.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "That means the first digit has to be either $5, 6, 7, 8,$ or $9$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "The second and third digits have to contain both $3$ and $4$, but not necessarily in that order.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "So, for each possible first digit, how many ways can I arrange $3$ and $4$ in the second and third digits?", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "Well, I can either put $3$ in the second digit and $4$ in the third digit, or vice versa.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "So, for each first digit, there are two possible ways to make a number that contains both $3$ and $4$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "Since there are five possible first digits, the total number of numbers that contain both $3$ and $4$ is $5 \\times 2 = 10$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "To check my answer, I can list the numbers: $534, 543, 634, 643, 734, 743, 834, 843, 934, 943$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "Indeed, there are $10$ of them.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 409, "metadata": { "path_label": "how_many_integers_between_500_and_1000_contain_both_the__2022-12-23T14-01-47.709290|chosen" } }, { "text": "I notice that the equation is a quadratic, so I can use the discriminant to determine when the roots are real.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "The discriminant is the part under the square root in the quadratic formula: $b^2 - 4ac.$", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "In this case, $a = 1, b = -2k, c = k^2 + k - 5,$ so the discriminant is $(-2k)^2 - 4(1)(k^2 + k - 5).$", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "I simplify this to get $4k^2 - 4k^2 - 4k + 20 = -4k + 20.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "Since the roots are real, the discriminant must be nonnegative, so I have $-4k + 20 \\geq 0.$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "I rearrange this to get $k \\leq 5.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "This means that any value of $k$ that is less than or equal to 5 will make the roots real.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "However, I also need to make sure that both roots are less than 5, not just real.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "To do that, I can use the fact that the sum of the roots is equal to $-b/a,$ which in this case is $2k/1 = 2k.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 410, "metadata": { "path_label": "let_k_be_a_real_number_such_that_both_roots_of_x_2_2k__2022-12-23T13-56-30.298027|chosen" } }, { "text": "I notice that the denominators of the fractions have the form $1 - a^2$, which is a difference of squares and can be factored as $(1 - a)(1 + a)$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "So I rewrite the fractions as \\[\\frac{1}{(1 - \\tan x)(1 + \\tan x)} + \\frac{1}{(1 - \\cot x)(1 + \\cot x)}.\\]", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "Next, I recall that $\\tan x = \\frac{\\sin x}{\\cos x}$ and $\\cot x = \\frac{\\cos x}{\\sin x}$, so I can substitute these expressions into the fractions and get \\[\\frac{1}{(\\cos x - \\sin x)(\\cos x + \\sin x)} + \\frac{1}{(\\sin x - \\cos x)(\\sin x + \\cos x)}.\\]", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "I notice that the second fraction has a negative sign in the numerator and the denominator, so I can multiply the fraction by $-1/-1$ to get rid of the negative signs and obtain \\[\\frac{1}{(\\cos x - \\sin x)(\\cos x + \\sin x)} - \\frac{1}{(\\cos x - \\sin x)(\\cos x + \\sin x)}.\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "Now I see that the fractions have the same denominator, so I can combine them into one fraction by adding the numerators and get \\[\\frac{1 - 1}{(\\cos x - \\sin x)(\\cos x + \\sin x)} = \\frac{0}{(\\cos x - \\sin x)(\\cos x + \\sin x)}.\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "Finally, I simplify the fraction by dividing the numerator and the denominator by the same nonzero constant and get \\[\\frac{0}{(\\cos x - \\sin x)(\\cos x + \\sin x)} = 0.\\]", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 411, "metadata": { "path_label": "simplify_frac_1_1_tan_2_x_frac_1_1_cot_2_x__2022-12-23T13-47-34.599559|chosen" } }, { "text": "I need to find a list of five positive integers that satisfies all three given conditions.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "Since the list has an odd number of elements, the median is the middle one when they are arranged in order.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "So the list must have $9$ as the third element.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "That means there are two elements below $9$ and two elements above $9$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "The problem also tells me that $8$ is the only integer that occurs more than once, so I can assume that both elements below $9$ are $8$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "That gives me the first three elements of the list: $8, 8, 9$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "Now I need to find the last two elements, which are above $9$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "The problem also tells me that the average of the list is $10$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "That means the sum of the five elements is $10 \\times 5 = 50$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "I already know the sum of the first three elements is $8 + 8 + 9 = 25$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "So the sum of the last two elements must be $50 - 25 = 25$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "I want to find the largest possible integer that could appear in the list, so I should make the fourth element as small as possible, while still being above $9$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "The smallest integer above $9$ is $10$, so I can make the fourth element $10$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "That means the fifth element must be $25 - 10 = 15$, which is the largest possible integer that could appear in the list.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "So the final list is $8, 8, 9, 10, 15$, and the answer is $15$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 412, "metadata": { "path_label": "a_list_of_five_positive_integers_has_all_of_the_following_pr__2022-12-23T13-40-40.400491|chosen" } }, { "text": "To find a five-digit perfect cube with an 8 in the ten-thousands place, I need to think of what cube roots are possible.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "Since the number has an 8 in the ten-thousands place, it must be at least 80,000 and at most 89,999.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "So the cube root of the number must be at least 43 and at most 44, since 43^3 = 79507 and 44^3 = 85184.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "I can eliminate 43 as a candidate, since its cube does not have an 8 in the ten-thousands place.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "So the only possible cube root is 44.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "Therefore, the number I'm looking for is 44^3 = 85184.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 413, "metadata": { "path_label": "what_positive_five_digit_whole_number_with_an_8_in_the_ten_t__2022-12-23T10-04-49.943023|chosen" } }, { "text": "I know that 23 is a prime number, so I can try dividing 2323 by 23 and see what I get.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "Using long division, I get 2323 = 23 * 101.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "Now I need to check if 101 is a prime number or not.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "I know that 101 is not divisible by 2, 3, 5, 7, or 11, which are the first five prime numbers.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "The next prime number is 13, and 13 * 13 = 169, which is larger than 101.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "This means that 101 has no factors between 1 and 101, except itself, so it is a prime number.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "Therefore, the largest prime factor of 2323 is 101.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 414, "metadata": { "path_label": "what_is_the_largest_prime_factor_of_2323__2022-12-23T09-51-41.289420|chosen" } }, { "text": "I notice that the function f is piecewise, so I wonder if its inverse will also be piecewise.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "I also notice that the function f is strictly increasing, so it is one-to-one and its inverse will also be strictly increasing.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "To find the inverse of f, I need to switch the roles of x and y and solve for y in terms of x.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "For the piece where f(x) = x - 2, I have y = x - 2, so swapping x and y gives x = y - 2 and solving for y gives y = x + 2.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "This means that for this piece, the inverse function is f^{-1}(x) = x + 2.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "For the piece where f(x) = \\sqrt{x}, I have y = \\sqrt{x}, so swapping x and y gives x = \\sqrt{y} and squaring both sides gives x^2 = y.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "This means that for this piece, the inverse function is f^{-1}(x) = x^2.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "However, I have to be careful with the domain and range of the inverse functions.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 415, "metadata": { "path_label": "the_function_f_x_left_begin_aligned_x_2_quad_t__2022-12-23T09-28-30.816662|chosen" } }, { "text": "I know that the product of two integers is equal to the product of their greatest common divisor and their least common multiple, so I can write an equation: $24 \\cdot n = (x+2) \\cdot x(x+2)$, where $n$ is the other integer.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "I want to find the smallest possible value of $n$, so I need to minimize $x$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "I also know that $x+2$ is a divisor of 24, since it is the greatest common divisor of 24 and $n$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "The possible divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "I can check each of these values and see which one makes $x(x+2)$ a multiple of 24.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 1$, then $x = -1$, which is not positive, so this is not valid.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 2$, then $x = 0$, which makes $x(x+2) = 0$, which is not a multiple of 24, so this is not valid either.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 3$, then $x = 1$, which makes $x(x+2) = 3$, which is not a multiple of 24, so this is not valid.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 4$, then $x = 2$, which makes $x(x+2) = 8$, which is not a multiple of 24, so this is not valid.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 6$, then $x = 4$, which makes $x(x+2) = 24$, which is a multiple of 24, so this is valid.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 8$, then $x = 6$, which makes $x(x+2) = 48$, which is a multiple of 24, but larger than 24, so this is not optimal.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 12$, then $x = 10$, which makes $x(x+2) = 120$, which is a multiple of 24, but larger than 48, so this is not optimal.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "If $x+2 = 24$, then $x = 22$, which makes $x(x+2) = 528$, which is a multiple of 24, but larger than 120, so this is not optimal.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "Therefore, the smallest possible value of $x$ that satisfies the given conditions is $x = 4$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "To find the corresponding value of $n$, I can plug in $x = 4$ into the equation $24 \\cdot n = (x+2) \\cdot x(x+2)$ and solve for $n$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "I get $24 \\cdot n = (4+2) \\cdot 4(4+2)$, which simplifies to $24 \\cdot n = 6 \\cdot 24$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "Dividing both sides by 24, I get $n = 6$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "This means that the smallest possible value of the other integer is 6.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 416, "metadata": { "path_label": "the_greatest_common_divisor_of_two_integers_is_x_2_and_t__2022-12-23T09-16-30.361565|chosen" } }, { "text": "The sum of the numbers on Katie's list is equal to the sum of the squares of the numbers on her list, so I can write an equation: x_1 + x_2 + ... + x_n = x_1^2 + x_2^2 + ... + x_n^2, where x_i are the numbers on her list and n is the number of numbers on her list.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "I want to find the largest possible value of the arithmetic mean of her numbers, which is the sum of her numbers divided by the number of numbers: (x_1 + x_2 + ... + x_n) / n.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "To simplify the equation, I can divide both sides by n: (x_1 + x_2 + ... + x_n) / n = (x_1^2 + x_2^2 + ... + x_n^2) / n.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "Now I notice that the right-hand side is the mean of the squares of her numbers, and the left-hand side is the mean of her numbers.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "I recall that there is an inequality that relates the mean of the squares of some numbers and the square of the mean of those numbers: the mean of the squares is always greater than or equal to the square of the mean, and equality holds if and only if all the numbers are equal.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "This means that in Katie's case, the mean of the squares of her numbers is equal to the square of the mean of her numbers, since they are both equal to the same value.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "So I can write: (x_1^2 + x_2^2 + ... + x_n^2) / n = ((x_1 + x_2 + ... + x_n) / n)^2.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "Taking the square root of both sides, I get: sqrt((x_1^2 + x_2^2 + ... + x_n^2) / n) = (x_1 + x_2 + ... + x_n) / n.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "This means that the arithmetic mean of her numbers is equal to the positive square root of the mean of the squares of her numbers.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "To maximize the arithmetic mean of her numbers, I need to maximize the mean of the squares of her numbers, since the square root function is increasing.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "The mean of the squares of her numbers is maximized when all the numbers are equal to the largest possible value, since the square function is increasing.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "The largest possible value of the numbers on her list is 1, since if any of them were larger than 1, their square would be larger than themselves, and the sum of the squares would exceed the sum of the numbers.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "Therefore, the largest possible value of the arithmetic mean of her numbers is 1, which occurs when all the numbers on her list are 1.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 417, "metadata": { "path_label": "katie_has_a_list_of_real_numbers_such_that_the_sum_of_the_nu__2022-12-23T09-15-39.398900|chosen" } }, { "text": "I notice that the sum of the four products can be rewritten as $(a+c)(b+d)$, which is the product of two sums of two numbers.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 418, "metadata": { "path_label": "the_values_of_a_b_c_and_d_are_1_2_3_and_4_but_n__2022-12-23T09-07-05.488668|chosen" } }, { "text": "To maximize this product, I want to make both factors as large as possible, and as close to each other as possible.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 418, "metadata": { "path_label": "the_values_of_a_b_c_and_d_are_1_2_3_and_4_but_n__2022-12-23T09-07-05.488668|chosen" } }, { "text": "This means that I want to pair the largest and smallest numbers together, and the middle two numbers together.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 418, "metadata": { "path_label": "the_values_of_a_b_c_and_d_are_1_2_3_and_4_but_n__2022-12-23T09-07-05.488668|chosen" } }, { "text": "So, the optimal values for $a$ and $c$ are 4 and 1, and for $b$ and $d$ are 3 and 2.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 418, "metadata": { "path_label": "the_values_of_a_b_c_and_d_are_1_2_3_and_4_but_n__2022-12-23T09-07-05.488668|chosen" } }, { "text": "Then, the sum of the four products is $(4+1)(3+2) = 25$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 418, "metadata": { "path_label": "the_values_of_a_b_c_and_d_are_1_2_3_and_4_but_n__2022-12-23T09-07-05.488668|chosen" } }, { "text": "To solve this problem, I need to consider all the possible ways to choose two or more distinct members of the set and multiply them together.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "One way to organize this is to group the choices by the number of factors involved.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "For example, if I choose two factors, I can get $1\\cdot 2=2$, $1\\cdot 3=3$, $1\\cdot 5=5$, $1\\cdot 11=11$, $2\\cdot 3=6$, $2\\cdot 5=10$, $2\\cdot 11=22$, $3\\cdot 5=15$, $3\\cdot 11=33$, $5\\cdot 11=55$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's 10 numbers so far.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "If I choose three factors, I can get $1\\cdot 2\\cdot 3=6$, $1\\cdot 2\\cdot 5=10$, $1\\cdot 2\\cdot 11=22$, $1\\cdot 3\\cdot 5=15$, $1\\cdot 3\\cdot 11=33$, $1\\cdot 5\\cdot 11=55$, $2\\cdot 3\\cdot 5=30$, $2\\cdot 3\\cdot 11=66$, $2\\cdot 5\\cdot 11=110$, $3\\cdot 5\\cdot 11=165$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's another 10 numbers, but some of them are repeats from the previous group. For example, $6$ and $10$ were already obtained by multiplying two factors.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "So I need to eliminate the duplicates and count only the new numbers.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "The new numbers are $30$, $66$, $110$, and $165$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's 4 new numbers, so I have 10 + 4 = 14 numbers so far.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "If I choose four factors, I can get $1\\cdot 2\\cdot 3\\cdot 5=30$, $1\\cdot 2\\cdot 3\\cdot 11=66$, $1\\cdot 2\\cdot 5\\cdot 11=110$, $1\\cdot 3\\cdot 5\\cdot 11=165$, $2\\cdot 3\\cdot 5\\cdot 11=330$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's 5 numbers, but again, some of them are repeats from the previous groups. For example, $30$ and $66$ were already obtained by multiplying three factors.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "So I need to eliminate the duplicates and count only the new numbers.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "The new number is $330$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's 1 new number, so I have 14 + 1 = 15 numbers so far.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "If I choose five factors, I can get $1\\cdot 2\\cdot 3\\cdot 5\\cdot 11=330$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "That's 1 number, but it is a repeat from the previous group. So I have no new numbers.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "Therefore, the total number of numbers I can get by multiplying two or more distinct members of the set $\\{1,2,3,5,11\\}$ together is 15.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 419, "metadata": { "path_label": "how_many_numbers_can_you_get_by_multiplying_two_or_more_dist__2022-12-23T09-05-41.009784|chosen" } }, { "text": "I know that the sum of any two sides of a triangle must be greater than the third side, by the triangle inequality.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "So, if the perimeter is 7, then each side must be less than 7/2 = 3.5.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "That means the only possible integer side lengths are 1, 2, and 3.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "I also know that the order of the sides does not matter for congruence, so I can assume that the first side is the smallest or equal to the others.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "If the first side is 1, then the other two sides must add up to 6, and the only way to do that with integers is 2 and 4 or 3 and 3.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "But 1, 2, and 4 do not form a triangle, because 1 + 2 is not greater than 4.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "So, 1, 3, and 3 is one possible non-congruent triangle with perimeter 7.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "If the first side is 2, then the other two sides must add up to 5, and the only way to do that with integers is 2 and 3.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "So, 2, 2, and 3 is another possible non-congruent triangle with perimeter 7.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "If the first side is 3, then the other two sides must add up to 4, and the only way to do that with integers is 1 and 3.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "But 1, 3, and 3 is the same as 3, 1, and 3, which I already counted.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "So, there are no new non-congruent triangles with perimeter 7 and first side 3.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "Therefore, there are only two non-congruent triangles with perimeter 7 and integer side lengths: 1, 3, and 3; and 2, 2, and 3.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 420, "metadata": { "path_label": "how_many_non_congruent_triangles_with_perimeter_7_have_integ__2022-12-23T08-59-04.608058|chosen" } }, { "text": "I want to find the point that is symmetric to $(3,6,15)$ with respect to $L.$", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "One way to do that is to find the projection of $(3,6,15)$ onto $L,$ and then move twice as far along $L$ in the same direction.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "To find the projection of $(3,6,15)$ onto $L,$ I need to use the dot product formula.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "If $u$ is a vector along $L,$ and $v$ is the vector from the origin to $(3,6,15),$ then the projection of $v$ onto $u$ is given by $\\frac{u\\cdot v}{u\\cdot u}u.$", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "I can choose any vector along $L,$ but a convenient one is the vector from the origin to $(2,1,-2),$ which is also $(2,1,-2).$", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "So $u=(2,1,-2)$ and $v=(3,6,15).$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "The dot product of $u$ and $v$ is $u\\cdot v = 2\\cdot 3 + 1\\cdot 6 + (-2)\\cdot 15 = 6 + 6 - 30 = -18.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "The dot product of $u$ and $u$ is $u\\cdot u = 2^2 + 1^2 + (-2)^2 = 4 + 1 + 4 = 9.$", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "So the projection of $v$ onto $u$ is $\\frac{-18}{9}(2,1,-2) = -2(2,1,-2) = (-4,-2,4).$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "This is the vector from the origin to the point where $(3,6,15)$ meets $L.$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "To find the reflection of $(3,6,15)$ across $L,$ I need to add this vector to the vector from $(3,6,15)$ to the point of intersection.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "The vector from $(3,6,15)$ to the point of intersection is $(-4,-2,4) - (3,6,15) = (-7,-8,-11).$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "So the reflection of $(3,6,15)$ across $L$ is $(-4,-2,4) + (-7,-8,-11) = (-11,-10,-7).$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 421, "metadata": { "path_label": "let_l_be_the_line_in_space_that_passes_through_the_origin__2022-12-23T08-53-01.507662|chosen" } }, { "text": "I want to find the numbers that satisfy the equation 47 = qn + 5, where q is the quotient and n is the divisor.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 422, "metadata": { "path_label": "how_many_different_counting_numbers_will_each_leave_a_remain__2022-12-23T08-47-12.553713|chosen" } }, { "text": "I can rewrite this equation as 42 = qn, so I need to find the factors of 42 that are greater than 5.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 422, "metadata": { "path_label": "how_many_different_counting_numbers_will_each_leave_a_remain__2022-12-23T08-47-12.553713|chosen" } }, { "text": "The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 422, "metadata": { "path_label": "how_many_different_counting_numbers_will_each_leave_a_remain__2022-12-23T08-47-12.553713|chosen" } }, { "text": "Out of these, only 6, 7, 14, 21, and 42 are greater than 5.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 422, "metadata": { "path_label": "how_many_different_counting_numbers_will_each_leave_a_remain__2022-12-23T08-47-12.553713|chosen" } }, { "text": "Therefore, there are five different counting numbers that will leave a remainder of 5 when divided into 47.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 422, "metadata": { "path_label": "how_many_different_counting_numbers_will_each_leave_a_remain__2022-12-23T08-47-12.553713|chosen" } }, { "text": "I notice that the function $f(x)$ has two parts: an exponent of $(-1)$ and a square root.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I wonder what happens when I plug in $x$ and $-x$ into the function, since that's the usual way to test for evenness and oddness.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I start with $x$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I see that $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$, so it gives me the integer part of $x$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I also see that $x - \\lfloor x \\rfloor$ is the fractional part of $x$, so it's always between $0$ and $1$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I write $x = \\lfloor x \\rfloor + \\{x\\}$, where $\\{x\\}$ is the fractional part of $x$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I plug this into the function and simplify.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I get \\[f(x) = (-1)^{\\lfloor x \\rfloor} \\sqrt{\\frac{1}{4} - \\left( \\{x\\} - \\frac{1}{2} \\right)^2}.\\]", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I notice that the square root part is always positive, since it's the distance from $\\frac{1}{2}$ to $\\{x\\}$ squared and then subtracted from $\\frac{1}{4}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I also notice that the exponent of $(-1)$ depends on the parity of $\\lfloor x \\rfloor$, so it's either $1$ or $-1$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I conclude that $f(x)$ is either positive or negative, depending on whether $\\lfloor x \\rfloor$ is even or odd.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I now plug in $-x$ into the function.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I see that $\\lfloor -x \\rfloor$ is the greatest integer less than or equal to $-x$, so it gives me the negative of the ceiling of $x$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I write $\\lfloor -x \\rfloor = -\\lceil x \\rceil$, where $\\lceil x \\rceil$ is the smallest integer greater than or equal to $x$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I plug this into the function and simplify.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I get \\[f(-x) = (-1)^{-\\lceil x \\rceil} \\sqrt{\\frac{1}{4} - \\left( \\{-x\\} - \\frac{1}{2} \\right)^2}.\\]", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I notice that the square root part is still positive, since it's the distance from $\\frac{1}{2}$ to $\\{-x\\}$ squared and then subtracted from $\\frac{1}{4}$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I also notice that the exponent of $(-1)$ depends on the parity of $-\\lceil x \\rceil$, so it's either $1$ or $-1$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I realize that $-\\lceil x \\rceil$ has the opposite parity of $\\lfloor x \\rfloor$, since they are the negative of each other and one of them is an integer.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I conclude that $f(-x)$ has the opposite sign of $f(x)$, since the exponent of $(-1)$ switches sign.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I remember that a function is odd if $f(-x) = -f(x)$ for all $x$.", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I check that this is true for $f(x)$ by multiplying both sides by $-1$ and seeing that they are equivalent.", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I answer that $f(x)$ is an odd function.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 423, "metadata": { "path_label": "is_f_x_1_lfloor_x_rfloor_sqrt_frac_1_4_le__2022-12-23T08-43-37.893413|chosen" } }, { "text": "I know that a regular hexagon can be divided into six equilateral triangles, each with the same side length as the hexagon.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "If I draw a perpendicular from an interior point to one of these sides, it will form a right triangle with half the side length as the hypotenuse and the perpendicular as one of the legs.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "I also know that the area of an equilateral triangle is $\\frac{s^2 \\sqrt{3}}{4}$, where $s$ is the side length.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "So, if I add up the areas of all six triangles, I should get the total area of the hexagon.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "The area of each triangle is $\\frac{s^2 \\sqrt{3}}{4} = \\frac{sh}{2}$, where $h$ is the length of the perpendicular.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "Therefore, the area of the hexagon is $\\frac{s}{2} \\sum_{i=1}^6 h_i$, where $h_i$ are the given lengths of the perpendiculars.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "This means that $\\frac{s}{2} \\sum_{i=1}^6 h_i = \\frac{s^2 \\sqrt{3}}{4} \\cdot 6$, or $\\sum_{i=1}^6 h_i = \\frac{3s \\sqrt{3}}{2}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "Plugging in the given values, I get $4 + 5 + 6 + 8 + 9 + 10 = \\frac{3s \\sqrt{3}}{2}$, or $42 = \\frac{3s \\sqrt{3}}{2}$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "Solving for $s$, I get $s = \\frac{2 \\cdot 42}{3 \\sqrt{3}}$, or $s = \\frac{28 \\sqrt{3}}{3}$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "This is the length of a side of the hexagon in simplest radical form.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 424, "metadata": { "path_label": "the_lengths_of_the_perpendiculars_drawn_to_the_sides_of_a_re__2022-12-23T08-32-46.288808|chosen" } }, { "text": "This problem involves finding the minimum value of a function of two variables, subject to a constraint.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "One possible approach is to use calculus, but that might be messy and tedious.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Another possible approach is to use some algebra and inequality techniques, which might be faster and more elegant.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "I will try the second approach.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "To simplify the notation, let me write $f(a, b) = \\frac{1}{a} + \\frac{2}{b}$ for the function we want to minimize.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Then the problem is to find the minimum value of $f(a, b)$ for $a, b > 0$ and $a + 2b = 1.$", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "One useful tool for minimizing functions is the arithmetic mean-geometric mean (AM-GM) inequality, which says that for any nonnegative numbers $x$ and $y,$ we have $\\frac{x + y}{2} \\geq \\sqrt{xy}.$", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This inequality is useful because it relates the arithmetic mean of two numbers, which is easy to compute, to their geometric mean, which is harder to compute but often more relevant to optimization problems.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "The equality case of the AM-GM inequality occurs when $x = y.$", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Now, let me apply the AM-GM inequality to the terms $\\frac{1}{a}$ and $\\frac{2}{b}$ in $f(a, b).$", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "I get $\\frac{\\frac{1}{a} + \\frac{2}{b}}{2} \\geq \\sqrt{\\frac{1}{a} \\cdot \\frac{2}{b}} = \\frac{\\sqrt{2}}{\\sqrt{ab}}.$", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This means that $f(a, b) \\geq 2 \\cdot \\frac{\\sqrt{2}}{\\sqrt{ab}}.$", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This is a lower bound for $f(a, b),$ but it is not necessarily the minimum value, because it depends on $ab,$ which is not fixed by the constraint $a + 2b = 1.$", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "However, I can use the constraint to eliminate one of the variables and express $ab$ in terms of the other.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "For example, I can solve for $b$ and get $b = \\frac{1 - a}{2}.$", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Then I can substitute this into $ab$ and get $ab = a \\cdot \\frac{1 - a}{2}.$", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Now, I can plug this into the lower bound for $f(a, b)$ and get $f(a, b) \\geq 2 \\cdot \\frac{\\sqrt{2}}{\\sqrt{a \\cdot \\frac{1 - a}{2}}}.$", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This is a function of $a$ only, which I can simplify by multiplying both the numerator and the denominator by $\\sqrt{2}$ and get $f(a, b) \\geq \\frac{4}{\\sqrt{a(1 - a)}}.$", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Now, I want to find the minimum value of this function of $a.$", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "One way to do this is to use calculus and find the critical points, but I can also use some common sense and intuition.", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "I notice that the denominator is the square root of a product of two factors, $a$ and $1 - a.$", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "These factors are both positive for $0 < a < 1,$ and they add up to $1.$", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "I recall that the product of two positive factors that add up to a constant is maximized when the factors are equal.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This means that $a(1 - a)$ is maximized when $a = 1 - a,$ or $a = \\frac{1}{2}.$", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This also means that the denominator $\\sqrt{a(1 - a)}$ is maximized when $a = \\frac{1}{2},$ and the whole fraction $\\frac{4}{\\sqrt{a(1 - a)}}$ is minimized when $a = \\frac{1}{2}.$", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "Therefore, the minimum value of $f(a, b)$ occurs when $a = \\frac{1}{2}.$", "role": "entry", "step_index": 25, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "To find the corresponding value of $b,$ I can use the constraint $a + 2b = 1$ and get $b = \\frac{1}{4}.$", "role": "entry", "step_index": 26, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "So the minimum value of $f(a, b)$ is $f(\\frac{1}{2}, \\frac{1}{4}) = \\frac{1}{\\frac{1}{2}} + \\frac{2}{\\frac{1}{4}} = 2 + 8 = 10.$", "role": "entry", "step_index": 27, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 425, "metadata": { "path_label": "let_a_and_b_be_positive_real_numbers_such_that_a_2b__2022-12-23T08-31-23.738913|chosen" } }, { "text": "This is a problem of counting circular permutations, where the order of the beads matters, but not their orientation.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "One way to approach this problem is to fix the position of the teal bead, and then count how many ways we can arrange the remaining 6 beads around it.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "Since the teal bead is fixed, we can think of the bracelet as a line segment with 6 spots for the orange and black beads.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "If we ignore the fact that some of the beads are identical, we can count 6! ways to arrange 6 beads in a line.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "However, this overcounts by a factor of 3! for the orange beads and another factor of 3! for the black beads, since swapping any of them does not change the bracelet.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "Therefore, we need to divide 6! by 3!3! to get the number of distinct arrangements.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "This gives us 6!/(3!3!) = 20 ways to arrange the orange and black beads around the teal bead.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "However, this still overcounts by a factor of 2, because flipping the bracelet over does not change it.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "To account for this, we need to divide 20 by 2, which gives us 10 ways to arrange the beads on the bracelet without regard to orientation.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 426, "metadata": { "path_label": "how_many_different_bracelets_have_3_identical_orange_beads__2022-12-23T20-07-23.116087|chosen" } }, { "text": "This problem involves fractions, decimals, and digits, so I want to think about how to convert between them.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "I know that $0.abc$ is the same as $\\frac{abc}{1000}$, where $abc$ is a three-digit number.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "So, if I want $0.abc$ to be equal to $\\frac{1}{y}$, then I need to find a value of $y$ that makes $abc$ a factor of $1000$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "The factors of $1000$ are $1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500$, and $1000$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "But, the problem says that $y$ has to be between $0$ and $9$, so I can eliminate most of these factors and only focus on $1, 2, 4, 5$, and $8$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "Now, I need to find the largest possible value of $abc$ that is a multiple of one of these factors.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 427, "metadata": { "path_label": "if_a_b_and_c_are_digits_and_0_abc_can_be_expressed__2022-12-23T14-52-19.857474|chosen" } }, { "text": "I notice that each term in the sum is a perfect square, so I can use the formula (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd) to simplify each term.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "For example, the first term is (-\\sqrt{3} + \\sqrt{5} + \\sqrt{6} + \\sqrt{7})^2, so I can write it as (-\\sqrt{3})^2 + (\\sqrt{5})^2 + (\\sqrt{6})^2 + (\\sqrt{7})^2 + 2(-\\sqrt{3}\\sqrt{5} - \\sqrt{3}\\sqrt{6} - \\sqrt{3}\\sqrt{7} + \\sqrt{5}\\sqrt{6} + \\sqrt{5}\\sqrt{7} + \\sqrt{6}\\sqrt{7}).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "Similarly, I can write the second term as (\\sqrt{3})^2 - (\\sqrt{5})^2 + (\\sqrt{6})^2 + (\\sqrt{7})^2 + 2(\\sqrt{3}\\sqrt{5} - \\sqrt{3}\\sqrt{6} - \\sqrt{3}\\sqrt{7} - \\sqrt{5}\\sqrt{6} - \\sqrt{5}\\sqrt{7} + \\sqrt{6}\\sqrt{7}).", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "And the third term as (\\sqrt{3})^2 + (\\sqrt{5})^2 - (\\sqrt{6})^2 + (\\sqrt{7})^2 + 2(\\sqrt{3}\\sqrt{5} - \\sqrt{3}\\sqrt{6} + \\sqrt{3}\\sqrt{7} - \\sqrt{5}\\sqrt{6} + \\sqrt{5}\\sqrt{7} - \\sqrt{6}\\sqrt{7}).", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "And the fourth term as (\\sqrt{3})^2 + (\\sqrt{5})^2 + (\\sqrt{6})^2 - (\\sqrt{7})^2 + 2(\\sqrt{3}\\sqrt{5} + \\sqrt{3}\\sqrt{6} - \\sqrt{3}\\sqrt{7} + \\sqrt{5}\\sqrt{6} - \\sqrt{5}\\sqrt{7} - \\sqrt{6}\\sqrt{7}).", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "Now I can add up all these terms and see what cancels out.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "I notice that the first and second terms have opposite signs for each pair of square roots, so they will cancel out when added. For example, -\\sqrt{3}\\sqrt{5} + \\sqrt{3}\\sqrt{5} = 0.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "Similarly, the third and fourth terms have opposite signs for each pair of square roots, so they will also cancel out when added.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "So the only terms that will not cancel out are the ones that are squared, which are the same for each term.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "Therefore, the sum is equal to 4((-\\sqrt{3})^2 + (\\sqrt{5})^2 + (\\sqrt{6})^2 + (\\sqrt{7})^2) = 4(3 + 5 + 6 + 7) = 4(21) = 84.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 428, "metadata": { "path_label": "compute_sqrt_3_sqrt_5_sqrt_6_sqrt_7_2__2022-12-23T05-42-58.136544|chosen" } }, { "text": "I know that $\\frac{3}{26}$ is a repeating decimal, because $26$ has a factor of $13$, which is a prime that does not divide $10$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "So, I need to find the length of the repeating cycle and the position of the $100$th digit within that cycle.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "To find the length of the cycle, I can use long division and look for the first time that the remainder repeats.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I start by dividing $3$ by $26$, which gives me $0.1$ with a remainder of $4$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I bring down a zero and divide $40$ by $26$, which gives me $1.5$ with a remainder of $14$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I bring down another zero and divide $140$ by $26$, which gives me $5.3$ with a remainder of $8$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I bring down one more zero and divide $80$ by $26$, which gives me $3.0$ with a remainder of $2$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I bring down a fifth zero and divide $20$ by $26$, which gives me $0.7$ with a remainder of $18$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I bring down a sixth zero and divide $180$ by $26$, which gives me $6.9$ with a remainder of $4$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "This is where I see that the remainder has repeated, so the cycle is $6$ digits long: $0.115384$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "To find the position of the $100$th digit, I can use modular arithmetic.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 429, "metadata": { "path_label": "what_is_the_100_th_digit_after_the_decimal_point_when_fra__2022-12-23T05-40-34.955696|chosen" } }, { "text": "I notice that both numbers have the form $83^9+a$, where $a$ is a constant.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "So I can use the Euclidean algorithm to find their greatest common divisor (gcd).", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "The Euclidean algorithm works by repeatedly applying the division algorithm, which says that for any integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a=bq+r$ and $0\\leq r < b$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "So I start by applying the division algorithm to the two numbers, where I take $a=83^9+83^2+1$ and $b=83^9+1$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "I want to find $q$ and $r$ such that $83^9+83^2+1=(83^9+1)q+r$.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "To do this, I can try to estimate $q$ by dividing the leading terms of both sides.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "That is, I divide $83^9$ by $83^9$ and get $q=1$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "This means that $r$ is the difference between $83^9+83^2+1$ and $83^9+1$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "That is, $r=83^2$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "So I have $83^9+83^2+1=(83^9+1)+83^2$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "Now I repeat the division algorithm, but this time I take $a=83^9+1$ and $b=83^2$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "I want to find $q$ and $r$ such that $83^9+1=83^2q+r$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "Again, I can try to estimate $q$ by dividing the leading terms of both sides.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "That is, I divide $83^9$ by $83^2$ and get $q=83^7$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "This means that $r$ is the difference between $83^9+1$ and $83^2\\cdot 83^7$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "That is, $r=1$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "So I have $83^9+1=83^2\\cdot 83^7+1$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "Now I have found the gcd, because the division algorithm says that the gcd is the last nonzero remainder in the process.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "That is, $\\gcd(83^9+1,83^9+83^2+1)=1$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 430, "metadata": { "path_label": "compute_gcd_83_9_1_83_9_83_2_1__2022-12-23T05-32-03.613938|chosen" } }, { "text": "I need to find all the two-digit primes less in the range $[10,39]$.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "I can use the fact that any prime greater than 3 must end with 1, 3, 7 or 9, since otherwise it would be divisible by 2 or 5.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "So I can start by listing all the numbers ending with 1, 3, 7 or 9 in that range and then checking if they are prime.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "For 1, I have 11, 21, 31.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "11 is prime, 21 is not (divisible by 3), and 31 is prime.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "For 3, I have 13, 23, 33.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "13 and 23 are prime, but 33 is not (divisible by 3).", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "For 7, I have 17, 27, 37.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "17 and 37 are prime, but 27 is not (divisible by 3 and 9).", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "For 9, I have 19, 29, 39.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "19 and 29 are prime, but 39 is not (divisible by 3 and 13).", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "So the only two-digit primes in that range are 11, 13, 17, 19, 23, 29, 31, 37.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "Now I need to count how many ways I can form a four-digit house number using two of these primes.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "I can't repeat the same prime, because that would make the house number divisible by that prime.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "So I need to choose two different primes from the list of 8.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "I can use the combination formula to find the number of ways to do that: $\\binom{8}{2} = \\frac{8 \\times 7}{2 \\times 1} = 28$.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "However, this only counts the number of ways to choose the primes, not the order in which they appear in the house number.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "For each pair of primes, I can arrange them in two ways: $ABCD$ or $CDAB$.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "So I need to multiply the number of pairs by 2 to get the total number of house numbers: $28 \\times 2 = 56$.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 431, "metadata": { "path_label": "dr_math_s_four_digit_house_number_abcd_contains_no_zeroes__2022-12-23T05-26-22.707761|chosen" } }, { "text": "I want to count the number of ways to assign the friends to the classes, but I also need to account for Manoj's condition.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "I can start by considering the cases where Manoj is in the chemistry class or the biology class.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "If Manoj is in the chemistry class, then I need to assign at least one more friend to the chemistry class, and the rest can go to either class.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "So this is like choosing a non-empty subset of the remaining 5 friends to join Manoj in the chemistry class, and then assigning the rest to the biology class.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "The number of ways to choose a non-empty subset of 5 elements is 2^5 - 1, since I can either include or exclude each element, but I can't exclude all of them.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "So the number of ways to assign the friends if Manoj is in the chemistry class is 2^5 - 1.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "Similarly, if Manoj is in the biology class, then I need to assign at least one more friend to the biology class, and the rest can go to either class.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "The number of ways to choose a non-empty subset of 5 elements is still 2^5 - 1, so the number of ways to assign the friends if Manoj is in the biology class is also 2^5 - 1.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "Therefore, the total number of ways to assign the friends to the classes, accounting for Manoj's condition, is 2(2^5 - 1).", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 432, "metadata": { "path_label": "how_many_ways_are_there_to_assign_each_of_6_friends_to_eithe__2022-12-23T04-56-55.215899|chosen" } }, { "text": "To find $AB$, I will use the fact that the area of an equilateral triangle is $\\frac{s^2\\sqrt{3}}{4}$, where $s$ is the side length.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 433, "metadata": { "path_label": "point_p_is_inside_equilateral_triangle_abc_points_q__2022-12-23T04-49-41.944603|chosen" } }, { "text": "I will also use the fact that the sum of the areas of $\\triangle PQB$, $\\triangle PRC$, and $\\triangle PSA$ is equal to the area of $\\triangle ABC$, since they are all contained in it.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 433, "metadata": { "path_label": "point_p_is_inside_equilateral_triangle_abc_points_q__2022-12-23T04-49-41.944603|chosen" } }, { "text": "So, I can write the equation $\\frac{AB^2\\sqrt{3}}{4}=\\frac{1\\cdot AB}{2}+\\frac{2\\cdot AB}{2}+\\frac{3\\cdot AB}{2}$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 433, "metadata": { "path_label": "point_p_is_inside_equilateral_triangle_abc_points_q__2022-12-23T04-49-41.944603|chosen" } }, { "text": "To solve this problem, I need to find a way to relate the candle's height and the time elapsed.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "One way to do that is to find a formula for the height of the candle after $t$ seconds.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "I notice that the time it takes to burn down each centimeter is an arithmetic sequence: $10, 20, 30, \\dots, 10k, \\dots$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "This means that the total time it takes to burn down $k$ centimeters is the sum of the first $k$ terms of this sequence.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "I recall that the sum of an arithmetic series is given by the formula $S_n = \\frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "In this case, $n = k$, $a_1 = 10$, and $a_k = 10k$.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "Plugging these values into the formula, I get $S_k = \\frac{k}{2}(10 + 10k) = 5k(k + 1)$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "This means that the candle burns down $k$ centimeters in $5k(k + 1)$ seconds.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "To find the height of the candle after $t$ seconds, I need to find the largest value of $k$ such that $5k(k + 1) \\leq t$.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "This is equivalent to finding the largest integer that satisfies the inequality $k^2 + k - \\frac{t}{5} \\leq 0$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "I can use the quadratic formula to find the roots of the equation $k^2 + k - \\frac{t}{5} = 0$.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "The roots are $k = \\frac{-1 \\pm \\sqrt{1 + \\frac{4t}{5}}}{2}$.", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "Since $k$ has to be an integer, I need to round down the positive root to get the largest value of $k$.", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "The positive root is $k = \\frac{-1 + \\sqrt{1 + \\frac{4t}{5}}}{2}$.", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "Rounding down, I get $k = \\left\\lfloor \\frac{-1 + \\sqrt{1 + \\frac{4t}{5}}}{2} \\right\\rfloor$.", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "This means that the candle burns down $\\left\\lfloor \\frac{-1 + \\sqrt{1 + \\frac{4t}{5}}}{2} \\right\\rfloor$ centimeters in $t$ seconds.", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "Therefore, the height of the candle after $t$ seconds is $119 - \\left\\lfloor \\frac{-1 + \\sqrt{1 + \\frac{4t}{5}}}{2} \\right\\rfloor$.", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "Now I can plug in $t = \\frac{T}{2}$ to get the height of the candle $\\frac{T}{2}$ seconds after it is lit.", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 434, "metadata": { "path_label": "a_large_candle_is_119_centimeters_tall_it_is_designed_to__2022-12-23T03-25-15.427273|chosen" } }, { "text": "I need to find a pattern or a formula for this sequence.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "I start by computing the first few terms: $2005, 133, 55, 250, 133, 55, \\dots$.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "I notice that the sequence repeats after $250$, so it has a cycle of length $3$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "That means that every third term is $250$, and the terms between them are $133$ and $55$.", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "To find the ${2005}^{\\text{th}}$ term, I need to divide $2005$ by $3$ and see what the remainder is.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "I can use long division or a calculator to do this.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "I get $2005 = 3 \\times 668 + 1$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "That means that the ${2005}^{\\text{th}}$ term is the same as the first term of the cycle, which is $250$.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 435, "metadata": { "path_label": "the_first_term_of_a_sequence_is_2005_each_succeeding_term__2022-12-23T03-19-32.316778|chosen" } }, { "text": "This problem involves complex numbers and trigonometry.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I notice that the expression inside the parentheses is the sum of two complex numbers in polar form.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I recall that the polar form of a complex number is \\[r(\\cos\\theta+i\\sin\\theta),\\] where r is the modulus and \\(\\theta\\) is the argument.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I also recall that to add two complex numbers in polar form, I can use the formula \\[r_1(\\cos\\theta_1+i\\sin\\theta_1)+r_2(\\cos\\theta_2+i\\sin\\theta_2)=(r_1\\cos\\theta_1+r_2\\cos\\theta_2)+i(r_1\\sin\\theta_1+r_2\\sin\\theta_2).\\]", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "Applying this formula to the expression inside the parentheses, I get \\[(\\cos12^\\circ+i\\sin12^\\circ+\\cos48^\\circ+i\\sin48^\\circ)=\\] \\[(\\cos12^\\circ+\\cos48^\\circ)+i(\\sin12^\\circ+\\sin48^\\circ).\\]", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I notice that the argument of this complex number is not obvious, so I need to find a way to simplify the real and imaginary parts.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I recall that there is a trigonometric identity that says \\[\\cos\\alpha+\\cos\\beta=2\\cos\\frac{\\alpha+\\beta}{2}\\cos\\frac{\\alpha-\\beta}{2}.\\]", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I apply this identity to the real part of the expression, using \\(\\alpha=48^\\circ\\) and \\(\\beta=12^\\circ\\).", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I get \\[\\cos12^\\circ+\\cos48^\\circ=2\\cos30^\\circ\\cos18^\\circ.\\]", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I also recall that \\(\\cos30^\\circ=\\frac{\\sqrt{3}}{2}\\), so I can write \\[\\cos12^\\circ+\\cos48^\\circ=\\sqrt{3}\\cos18^\\circ.\\]", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I do a similar thing for the imaginary part of the expression, using another trigonometric identity that says \\[\\sin\\alpha+\\sin\\beta=2\\sin\\frac{\\alpha+\\beta}{2}\\cos\\frac{\\alpha-\\beta}{2}.\\]", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I apply this identity to the imaginary part of the expression, using the same values of \\(\\alpha\\) and \\(\\beta\\).", "role": "entry", "step_index": 11, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I get \\[\\sin12^\\circ+\\sin48^\\circ=2\\sin30^\\circ\\cos18^\\circ.\\]", "role": "entry", "step_index": 12, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I also recall that \\(\\sin30^\\circ=\\frac{1}{2}\\), so I can write \\[\\sin12^\\circ+\\sin48^\\circ=\\cos18^\\circ.\\]", "role": "entry", "step_index": 13, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "Now I have simplified the expression inside the parentheses to \\[\\sqrt{3}\\cos18^\\circ+i\\cos18^\\circ.\\]", "role": "entry", "step_index": 14, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I can factor out \\(\\cos18^\\circ\\) from both terms and get \\[\\cos18^\\circ(\\sqrt{3}+i).\\]", "role": "entry", "step_index": 15, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I can now see that this is a complex number in polar form, where the modulus is \\(\\cos18^\\circ\\) and the argument is \\(\\frac{\\pi}{6}\\), since \\(\\tan\\frac{\\pi}{6}=\\frac{1}{\\sqrt{3}}\\).", "role": "entry", "step_index": 16, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I can write this as \\[\\cos18^\\circ(\\cos\\frac{\\pi}{6}+i\\sin\\frac{\\pi}{6}).\\]", "role": "entry", "step_index": 17, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I need to find the sixth power of this complex number.", "role": "entry", "step_index": 18, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I recall that to raise a complex number in polar form to a power, I can use the formula \\[r(\\cos\\theta+i\\sin\\theta)^n=r^n(\\cos n\\theta+i\\sin n\\theta).\\]", "role": "entry", "step_index": 19, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "Applying this formula to the expression, I get \\[(\\cos18^\\circ(\\cos\\frac{\\pi}{6}+i\\sin\\frac{\\pi}{6}))^6=\\] \\[(\\cos18^\\circ)^6(\\cos\\pi+i\\sin\\pi).\\]", "role": "entry", "step_index": 20, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I know that \\(\\cos\\pi=-1\\) and \\(\\sin\\pi=0\\), so I can simplify this further to \\[(\\cos18^\\circ)^6(-1).\\]", "role": "entry", "step_index": 21, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "This is the final expression for the sixth power of the original complex number.", "role": "entry", "step_index": 22, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I need to find the imaginary part of this expression, which is the coefficient of \\(i\\).", "role": "entry", "step_index": 23, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I see that there is no \\(i\\) in the expression, so the imaginary part is zero.", "role": "entry", "step_index": 24, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 436, "metadata": { "path_label": "find_the_imaginary_part_of_cos12_circ_i_sin12_circ_co__2022-12-23T03-19-26.817776|chosen" } }, { "text": "I need to find the last digit of a very large power of 2.", "role": "entry", "step_index": 0, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I wonder if there is a pattern or a cycle in the units digits of powers of 2.", "role": "entry", "step_index": 1, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I can start by listing some powers of 2 and their units digits: $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128, 2^8 = 256, 2^9 = 512, 2^{10} = 1024, \\dots$.", "role": "entry", "step_index": 2, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I notice that the units digits repeat every four powers: 2, 4, 8, 6, 2, 4, 8, 6, \\dots", "role": "entry", "step_index": 3, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "This means that the units digit of $2^n$ depends only on the remainder of $n$ when divided by 4.", "role": "entry", "step_index": 4, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I can write this as $2^n \\equiv 2^{n \\mod 4} \\pmod{10}$, where $\\equiv$ means congruent modulo 10, $\\mod$ means the remainder operation, and $\\pmod{10}$ means modulo 10.", "role": "entry", "step_index": 5, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "Now I can use this formula to find the units digit of $2^{2010}$.", "role": "entry", "step_index": 6, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I need to find the remainder of 2010 when divided by 4.", "role": "entry", "step_index": 7, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "I can do this by long division or by noticing that 2010 is even, so it is divisible by 2, but not by 4, so the remainder is 2.", "role": "entry", "step_index": 8, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "Therefore, $2^{2010} \\equiv 2^{2010 \\mod 4} \\equiv 2^2 \\equiv 4 \\pmod{10}$.", "role": "entry", "step_index": 9, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } }, { "text": "This means that the units digit of $2^{2010}$ is 4.", "role": "entry", "step_index": 10, "session_id": "896deaa4", "model_name": "text-embedding-3-large", "timestamp": null, "path_id": 437, "metadata": { "path_label": "what_is_the_units_digit_of_2_2010__2022-12-23T03-18-31.980553|chosen" } } ], "source_format": "multi_path", "llm_model": null, "created_at": 1776256592.1130815, "path_scores": { "0": { "path_length": 12.0, "solved": 0.0, "mean_prm_rating": 0.75, "min_prm_rating": 0.0 }, "1": { "path_length": 10.0, "solved": 0.0, "mean_prm_rating": 1.0, "min_prm_rating": 1.0 }, "2": { "path_length": 9.0, "solved": 0.0, "mean_prm_rating": 0.7777777777777778, "min_prm_rating": 0.0 }, "3": { "path_length": 5.0, "solved": 0.0, "mean_prm_rating": 0.4, "min_prm_rating": 0.0 }, "4": 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