id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-8f37a0eb341851f0928594a9710b7d787f3adcbb351ef1944e426581ab595c4c | A unit square is cut by $n$ straight lines. What is the minimum side length $s$ of a square that can be completely fit into at least one of these parts? Express your answer in terms of $n$. | [
"\\dfrac{1}{n+1}"
] | [
"A unit square is cut by $n$ straight lines. What is the minimum side length $s$ of a square that can be completely fit into at least one of these parts? Express your answer in terms of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-18dbceeda8bc5f9751aed7b1f4be7a78b860aa73066df87c72efa8fac7b621bc | A unit square is dissected into $n>1$ rectangles with sides parallel to the sides of the original square. Any line parallel to one of the sides of the square that also intersects its interior also intersects the interior of some rectangle. Let $A$ be the area of the largest rectangle that does not have any of its point... | [
"\\dfrac{1}{5}"
] | [
"A unit square is dissected into $n>1$ rectangles with sides parallel to the sides of the original square. Any line parallel to one of the sides of the square that also intersects its interior also intersects the interior of some rectangle. Let $A$ be the area of the largest rectangle that does not have any of its ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2b711049cb7c15198497fc4c0b5fd06b7bced0af4096826647fbd8d6797efb37 | A unit square is divided into \( m \) quadrilaterals \( Q_1, \ldots, Q_m \). For each \( i = 1, \ldots, m \), let \( S_i \) be the sum of the squares of the four sides of \( Q_i \). If \( S_1 + \ldots + S_m = 4 + k \), find the minimum possible value of \( k \). | [
"0"
] | [
"A unit square is divided into \\( m \\) quadrilaterals \\( Q_1, \\ldots, Q_m \\). For each \\( i = 1, \\ldots, m \\), let \\( S_i \\) be the sum of the squares of the four sides of \\( Q_i \\). If \\( S_1 + \\ldots + S_m = 4 + k \\), find the minimum possible value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3c32dba6c1a9bd10f6524c5704e958b698be1b9d8476f2a5a31e53d3fa778087 | A unit square is divided into polygons such that all sides of a polygon are parallel to the sides of the given square. If the total length of the segments inside the square (excluding the square's sides) is \( 2n \) (where \( n \) is a positive real number), find the minimum possible value of \( n \) such that there ex... | [
"\\dfrac{7}{2}"
] | [
"A unit square is divided into polygons such that all sides of a polygon are parallel to the sides of the given square. If the total length of the segments inside the square (excluding the square's sides) is \\( 2n \\) (where \\( n \\) is a positive real number), find the minimum possible value of \\( n \\) such th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-6fa9b2d96da5d3ec5f6e3c35281cfbe99e8cc36782a3c10f95b95de66158f0c7 | A unit square is divided into polygons such that all sides of each polygon are parallel to the sides of the square. If the total length of the segments inside the square (excluding the boundary of the square) is $2n$ (where $n$ is a positive real number), find the minimum value of $k$ such that there is guaranteed to e... | [
"1"
] | [
"A unit square is divided into polygons such that all sides of each polygon are parallel to the sides of the square. If the total length of the segments inside the square (excluding the boundary of the square) is $2n$ (where $n$ is a positive real number), find the minimum value of $k$ such that there is guaranteed... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-977fa6c426dc3fad69960160c7d397bcd5f8e0548490ea2a4cb579ae467cf207 | A unit square is divided into two rectangles such that the smaller rectangle can be placed on the larger rectangle with each vertex of the smaller rectangle touching exactly one edge of the larger rectangle. Calculate the dimensions of the smaller rectangle. | [
"\\( \\frac{1}{2} \\times \\frac{1}{2} \\)"
] | [
"A unit square is divided into two rectangles such that the smaller rectangle can be placed on the larger rectangle with each vertex of the smaller rectangle touching exactly one edge of the larger rectangle. Calculate the dimensions of the smaller rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-645d2e94b383af59c0f01a30fc0097c87dfe548aa70c71b5482a5d51a0e045bc | A unit square is partitioned into $n$ triangles. Determine the minimum side length $s$ of a square that can be covered by at least one of these triangles. Find $s$ when $n=12$. | [
"\\dfrac{1}{7}"
] | [
"A unit square is partitioned into $n$ triangles. Determine the minimum side length $s$ of a square that can be covered by at least one of these triangles. Find $s$ when $n=12$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-16346c06948c8238c5820e78bb6bf12c5099e101f279d084afff9dfa3c057e64 | A unit square is partitioned into two sets $A$ and $B$. Let $d_A$ and $d_B$ be the diameters of sets $A$ and $B$, respectively. Determine the maximum possible value of $\min(d_A, d_B)$. | [
"\\sqrt{2}"
] | [
"A unit square is partitioned into two sets $A$ and $B$. Let $d_A$ and $d_B$ be the diameters of sets $A$ and $B$, respectively. Determine the maximum possible value of $\\min(d_A, d_B)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-4eaf89893bfd2d1acc363ff90b91b943703c580fab52df9a263d632f5b219f7b | A unit square is placed across two parallel lines a unit distance apart, such that two triangular areas of the square lie outside the lines. If $P_1$ and $P_2$ are the perimeters of these two triangles, find the value of $P_1 + P_2$. | [
"4 + 2\\sqrt{2}"
] | [
"A unit square is placed across two parallel lines a unit distance apart, such that two triangular areas of the square lie outside the lines. If $P_1$ and $P_2$ are the perimeters of these two triangles, find the value of $P_1 + P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-8a7087c3affe50b2cb5faca6412a3a6291e01f1ac17d798107ee6cb23fccb460 | A unit square is rotated $30^\circ$ counterclockwise about one of its vertices. Determine the area of the intersection of the original square with the rotated one. | [
"\\dfrac{\\sqrt{3}}{3}"
] | [
"A unit square is rotated $30^\\circ$ counterclockwise about one of its vertices. Determine the area of the intersection of the original square with the rotated one."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-285ea2a04e196b58ca5ba20a771cb3a7d493e345a9517b1bf2cc9ef57679c61f | A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square? | [
"\\( 2 - \\sqrt{2} + \\frac{\\pi}{4} \\)",
"\\(2 - \\sqrt{2} + \\frac{\\pi}{4}\\)"
] | [
"A unit square is rotated $45^\\circ$ about its center. What is the area of the region swept out by the interior of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-d0a4a1e58a097e9751c9e8293da31cd886e7fff8e372216e4aae2cfc93f4f72d | A unit square is rotated about a vertex by an acute angle \( a \) such that \( \cos(a) = \frac{4}{5} \). What is the area of the shaded region formed by the overlap of the original and rotated squares? | [
"\\(\\frac{1}{2}\\)"
] | [
"A unit square is rotated about a vertex by an acute angle \\( a \\) such that \\( \\cos(a) = \\frac{4}{5} \\). What is the area of the shaded region formed by the overlap of the original and rotated squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-2fe0c2c291840ff01fdeda4a1aba1bad5570201153f8c7a51583062f1ab75121 | A unit square is subdivided into a grid composed of 9 squares each with side length \(\frac{1}{3}\). A circle is drawn through the centers of the 4 squares in the outermost corners of the grid. Compute the area of this circle. | [
"\\dfrac{2\\pi}{9}"
] | [
"A unit square is subdivided into a grid composed of 9 squares each with side length \\(\\frac{1}{3}\\). A circle is drawn through the centers of the 4 squares in the outermost corners of the grid. Compute the area of this circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-397514bbd5162ee22a19f0c1040716e043ecbc052d338fd684f831fa1a93ec1d | A university has \(n = 10001\) students. Some students form teams, and some teams form societies. Each pair of students belongs to exactly one society. For each student and each society, the student belongs to at most one team within the society. Each team has an odd number of students, and each team with \(2m+1\) stud... | [
"5000"
] | [
"A university has \\(n = 10001\\) students. Some students form teams, and some teams form societies. Each pair of students belongs to exactly one society. For each student and each society, the student belongs to at most one team within the society. Each team has an odd number of students, and each team with \\(2m+... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-84a404d4d0bfbc89dbf1282f84738e57456af89e927d19590a77bbd174edf01a | A university has an undergraduate enrollment of 22,000 students. There are 366 possible birthdays (including February 29). What is the largest number of these undergraduates that must share a birthday? | [
"61"
] | [
"A university has an undergraduate enrollment of 22,000 students. There are 366 possible birthdays (including February 29). What is the largest number of these undergraduates that must share a birthday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-8b5bf18b52419aff4541102c4fb75511a0426aa9d8e7ea0842ea491eb20d5a48 | A used car dealer sells two cars for the same price. He loses 10% on the first car but breaks even for both cars combined. Calculate the profit percentage for the second car. | [
"11.\\overline{1}\\%"
] | [
"A used car dealer sells two cars for the same price. He loses 10% on the first car but breaks even for both cars combined. Calculate the profit percentage for the second car."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ebde21927736ccc25f22ed16b85832302592ec2e47ea71a47bb0180717955b9b | A user for a secure internet service has a 20-digit key phrase, where each digit can be any value from 0 to 9. To log in, the user is asked for the values of 3 random digits from the key phrase. A keylogger records the values of these 3 digits each time the user logs in. How many observations would the keylogger need t... | [
"10"
] | [
"A user for a secure internet service has a 20-digit key phrase, where each digit can be any value from 0 to 9. To log in, the user is asked for the values of 3 random digits from the key phrase. A keylogger records the values of these 3 digits each time the user logs in. How many observations would the keylogger n... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-61596f36e0926612a05da0582c859e364c6b1bd9750f2ad2f56411930586f69b | A user has a post rating of 4.65 stars. After receiving 6-star ratings from a friend on less than 15 posts, the user's post rating increases to 5.09 stars. How many total times have the user's posts been rated? | [
"43"
] | [
"A user has a post rating of 4.65 stars. After receiving 6-star ratings from a friend on less than 15 posts, the user's post rating increases to 5.09 stars. How many total times have the user's posts been rated?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-35fa483b8755b06b103f27d14107b9271cb6b1514c543119d6f8da1aad1800b2 | A vacuum cleaner in the shape of a circle moves across a flat floor. For every point on its edge, there is a line that contains that point at all times. Let $d$ be the distance from the center of the circle to the closest such line. If the radius of the vacuum cleaner is $r$, find the value of $d$ in terms of $r$. | [
"r"
] | [
"A vacuum cleaner in the shape of a circle moves across a flat floor. For every point on its edge, there is a line that contains that point at all times. Let $d$ be the distance from the center of the circle to the closest such line. If the radius of the vacuum cleaner is $r$, find the value of $d$ in terms of $r$.... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-493fbd194e29989c6edafb868a4c0baa3da160ab5aa3bd85ec102eac90fd3d80 | A valid license plate in Xanadu consists of two letters followed by three digits. How many valid license plates are possible? | [
"676,\\!000"
] | [
"A valid license plate in Xanadu consists of two letters followed by three digits. How many valid license plates are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-edf18cf957e9756f5dad7dcfa0ef4fd50257f13d492c6a50adde9990cec4d770 | A valuable treasure worth $ \$100,000$ fell out of an airplane and landed in a large swamp. If it is not found within a day, it will sink to the bottom and never be found again. Each helicopter costs $ \$1000$ to hire for one day and has a 90% chance of finding the treasure. How many helicopters should be hired to maxi... | [
"97000"
] | [
"A valuable treasure worth $ \\$100,000$ fell out of an airplane and landed in a large swamp. If it is not found within a day, it will sink to the bottom and never be found again. Each helicopter costs $ \\$1000$ to hire for one day and has a 90% chance of finding the treasure. How many helicopters should be hired ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3241d505991969ecb9170db0584b9cccd5336a217b40bbc7d90dcea70f920f68 | A variable circle cuts each of the two circles \( x^{2} + y^{2} - 2x = 0 \) and \( x^{2} + y^{2} - 4x - 5 = 0 \) orthogonally. The variable circle passes through two fixed points. Find the coordinates of these fixed points. | [
"\\(\\left( \\dfrac{-5 + 3\\sqrt{5}}{2}, 0 \\right)\\) and \\(\\left( \\dfrac{-5 - 3\\sqrt{5}}{2}, 0 \\right)\\)"
] | [
"A variable circle cuts each of the two circles \\( x^{2} + y^{2} - 2x = 0 \\) and \\( x^{2} + y^{2} - 4x - 5 = 0 \\) orthogonally. The variable circle passes through two fixed points. Find the coordinates of these fixed points."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 21
} | 25 | 0 | 0 |
omr-ws-683ed481aed4a8c05f72e0324709e9d3aaeecbc2a7bc5a7db9f012e1a02ed606 | A variable circle is tangent to a fixed line \( d \) at a fixed point \( A \). Let \( B \) and \( C \) be two fixed points on \( d \). Let \( D \) be the projection of \( C \) on the polar of \( B \) with respect to the circle. Find the locus of the circumcenter of triangle \( OAD \), where \( O \) is the center of the... | [
"x = \\dfrac{c - b}{2}"
] | [
"A variable circle is tangent to a fixed line \\( d \\) at a fixed point \\( A \\). Let \\( B \\) and \\( C \\) be two fixed points on \\( d \\). Let \\( D \\) be the projection of \\( C \\) on the polar of \\( B \\) with respect to the circle. Find the locus of the circumcenter of triangle \\( OAD \\), where \\( O... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-0bec4ecc1b6786ea00b6664ba87fb7791b3f8aeaf2566bb6b51a3054511fa3cf | A variable circle passes through the intersection point $O$ of any two straight lines and cuts off distances $OP$ and $OQ$ such that $m \cdot OP + n \cdot OQ = 1$. Let $R$ be the fixed point through which the circle always passes. If the coordinates of $O$ are $(0,0)$ and the coordinates of $R$ are $(a,b)$, find the va... | [
"\\dfrac{1}{13}"
] | [
"A variable circle passes through the intersection point $O$ of any two straight lines and cuts off distances $OP$ and $OQ$ such that $m \\cdot OP + n \\cdot OQ = 1$. Let $R$ be the fixed point through which the circle always passes. If the coordinates of $O$ are $(0,0)$ and the coordinates of $R$ are $(a,b)$, find... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-94d72036e4c7b9cd726e4df2d9b31401dc1230031728ed6828585a73a6b97194 | A variable circle touches the circle \(x^2 + y^2 = 1\) externally and also touches the x-axis. The center of this circle is in the first or second quadrant. Find the locus of the center of this circle. | [
"\\((\\pm \\sqrt{2b+1}, b), b > 0\\)"
] | [
"A variable circle touches the circle \\(x^2 + y^2 = 1\\) externally and also touches the x-axis. The center of this circle is in the first or second quadrant. Find the locus of the center of this circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-223171f913b31222fcf51661bbfe28cb1a122621f25dea214a6dd330273f6f09 | A variable line \( L \) passes through the incenter of a given triangle \( ABC \) and intersects \( BC, AC, AB \) at points \( M, N, P \) respectively. Calculate the value of \( S \) for any such line \( L \): \[ S = \frac{AB}{PA \cdot PB} + \frac{AC}{NA \cdot NC} + \frac{BC}{MB \cdot MC} \] | [
"0"
] | [
"A variable line \\( L \\) passes through the incenter of a given triangle \\( ABC \\) and intersects \\( BC, AC, AB \\) at points \\( M, N, P \\) respectively. Calculate the value of \\( S \\) for any such line \\( L \\):\n\\[ S = \\frac{AB}{PA \\cdot PB} + \\frac{AC}{NA \\cdot NC} + \\frac{BC}{MB \\cdot MC} \\]"
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3d938e21ec0475ef713ed3e7cae1e3526cf5d88255f5f5b9ce168291e2971a8 | A variable line \( L \) passing through the point \( B (2, 5) \) intersects the lines \( 2x^2 - 5xy + 2y^2 = 0 \) at points \( P \) and \( Q \). Find the locus of the point \( R \) on \( L \) such that the distances \( BP \), \( BR \), and \( BQ \) are in harmonic progression. | [
"\\( 17x - 10y = 0 \\) and \\( 17x - 10y + 32 = 0 \\)"
] | [
"A variable line \\( L \\) passing through the point \\( B (2, 5) \\) intersects the lines \\( 2x^2 - 5xy + 2y^2 = 0 \\) at points \\( P \\) and \\( Q \\). Find the locus of the point \\( R \\) on \\( L \\) such that the distances \\( BP \\), \\( BR \\), and \\( BQ \\) are in harmonic progression."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fd83d65b2e9ea62cc02ddfc470f0f0d568c73a2d30841cba99abd8c958280ee | A variable line \( L \) passing through the point \( B(2,5) \) intersects the line \( 2x^2 - 5xy + 2y^2 = 0 \) at points \( P \) and \( Q \). Find a point \( R \) on \( L \) such that the distances \( BP \), \( BR \), and \( BQ \) are in harmonic progression. | [
"\\( \\left(2 + \\frac{19}{5m-1}, 5 + \\frac{19m}{5m-1}\\right) \\)"
] | [
"A variable line \\( L \\) passing through the point \\( B(2,5) \\) intersects the line \\( 2x^2 - 5xy + 2y^2 = 0 \\) at points \\( P \\) and \\( Q \\). Find a point \\( R \\) on \\( L \\) such that the distances \\( BP \\), \\( BR \\), and \\( BQ \\) are in harmonic progression."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8972bc59c304f8848fb50f63681b37033b3609905f3ac8dcd666cdab530d053 | A variable line moves in such a way that the product of the perpendiculars from $(a,0)$ and $(0,0)$ is equal to a constant $k^2$. Determine the locus of the foot of the perpendicular from $(0,0)$ upon the variable line. Find the square of the radius of this circle. | [
"$\\frac{1}{4}a^2 + k^2$",
"\\( \\frac{1}{4}a^2 + k^2 \\)"
] | [
"A variable line moves in such a way that the product of the perpendiculars from $(a,0)$ and $(0,0)$ is equal to a constant $k^2$. Determine the locus of the foot of the perpendicular from $(0,0)$ upon the variable line. Find the square of the radius of this circle."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 14,
"tir": 9
} | 45 | 0 | 0 |
omr-ws-5fca206cf5122e7e866c33efb8f17639c27fe3f0ccea5eb5d13417e21661c619 | A variable parabola $\mathcal{P}$ is tangent to two fixed perpendicular lines, and its axis passes through another fixed point. Find the locus of the focus of the parabola. | [
"\\( x^2 - y^2 - px + qy = 0 \\)"
] | [
"A variable parabola $\\mathcal{P}$ is tangent to two fixed perpendicular lines, and its axis passes through another fixed point. Find the locus of the focus of the parabola."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-030b328f553f57a232da9f758c7701e1801faf4de3564371694e7f27ea6e9dda | A variable plane has intercepts on the coordinate axes such that the sum of the squares of these intercepts is a constant \( K^2 \). Let \( P(x, y, z) \) be the foot of the perpendicular from the origin to the plane. Calculate the value of \((x^2 + y^2 + z^2)^2 \left( \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \righ... | [
"K^2"
] | [
"A variable plane has intercepts on the coordinate axes such that the sum of the squares of these intercepts is a constant \\( K^2 \\). Let \\( P(x, y, z) \\) be the foot of the perpendicular from the origin to the plane. Calculate the value of \\((x^2 + y^2 + z^2)^2 \\left( \\frac{1}{x^2} + \\frac{1}{y^2} + \\frac... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14f5b7e4e930104372eb17daac1cf06d1291eaab8b296808db5d57d56d077108 | A variable plane passes through a fixed point $(a, b, c)$ and cuts the coordinate axes at points $P$, $Q$, and $R$. Find the locus of the center $(x, y, z)$ of the sphere passing through $P$, $Q$, $R$, and the origin. | [
"\\(\\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 2\\)"
] | [
"A variable plane passes through a fixed point $(a, b, c)$ and cuts the coordinate axes at points $P$, $Q$, and $R$. Find the locus of the center $(x, y, z)$ of the sphere passing through $P$, $Q$, $R$, and the origin."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-cb70cd4d256a7ee9b66871be67e8e839475f0a2267d3a127f29ef4aa1493b3d0 | A variable plane passes through a fixed point \((\alpha, \beta, \gamma)\) and intersects the coordinate axes at points \(A\), \(B\), and \(C\) respectively. Find the locus of the point common to the planes through \(A\), \(B\), and \(C\) that are parallel to the coordinate planes. | [
"\\(\\frac{\\alpha}{x} + \\frac{\\beta}{y} + \\frac{\\gamma}{z} = 1\\)"
] | [
"A variable plane passes through a fixed point \\((\\alpha, \\beta, \\gamma)\\) and intersects the coordinate axes at points \\(A\\), \\(B\\), and \\(C\\) respectively. Find the locus of the point common to the planes through \\(A\\), \\(B\\), and \\(C\\) that are parallel to the coordinate planes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-44438e3839e5ff742b83022de476551f1c10893d11924f69f82665d38591772e | A variable straight line $AB$ intersects the $x$-axis at $A$ and the $y$-axis at $B$ and always passes through a fixed point $(\alpha, \beta)$. Find the locus of the point that divides the line segment $AB$ in the ratio $1:2$. | [
"\\(\\frac{2\\alpha}{3x} + \\frac{\\beta}{3y} = 1\\)"
] | [
"A variable straight line $AB$ intersects the $x$-axis at $A$ and the $y$-axis at $B$ and always passes through a fixed point $(\\alpha, \\beta)$. Find the locus of the point that divides the line segment $AB$ in the ratio $1:2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 12
} | 19 | 0 | 0 |
omr-ws-324a6f744cfa7bd77f80aabd55a6f67f5a79b994882b9d93363a5afc71118fbd | A variable tangent $t$ to the circle $C_1$ with radius $r_1$ intersects the circle $C_2$ with radius $r_2$ at points $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect at point $P$. Find the distance between the centers of $C_1$ and $C_2$ as a function of $r_1$ and $r_2$ such that the locus of $P$ when $t... | [
"\\( d = r_1 \\sqrt{2} \\)"
] | [
"A variable tangent $t$ to the circle $C_1$ with radius $r_1$ intersects the circle $C_2$ with radius $r_2$ at points $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect at point $P$. Find the distance between the centers of $C_1$ and $C_2$ as a function of $r_1$ and $r_2$ such that the locus of $P$ wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 2
} | 30 | 0 | 0 |
omr-ws-dc8b9023445e495c274374568cfb662c76f31a5077a5c839d4c683ff245abf90 | A variable tangent \( t \) to the circle \( C_1 \) of radius \( r_1 \) intersects the circle \( C_2 \) of radius \( r_2 \) in points \( A \) and \( B \). The tangents to \( C_2 \) through \( A \) and \( B \) intersect in point \( P \). Find, as a function of \( r_1 \) and \( r_2 \), the distance between the centers of ... | [
"\\( \\sqrt{2} (r_1 + r_2) \\)"
] | [
"A variable tangent \\( t \\) to the circle \\( C_1 \\) of radius \\( r_1 \\) intersects the circle \\( C_2 \\) of radius \\( r_2 \\) in points \\( A \\) and \\( B \\). The tangents to \\( C_2 \\) through \\( A \\) and \\( B \\) intersect in point \\( P \\). Find, as a function of \\( r_1 \\) and \\( r_2 \\), the d... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4e971f642e941b021ed7ccdd2c29ae901098d96bf6ea635dce35d285cc82e72 | A variable tangent to an ellipse intersects the tangents from vertices \( A \) and \( A' \) at points \( T \) and \( T' \), respectively. Determine the locus of the intersection point of lines \( TA' \) and \( T'A \). | [
"An ellipse tangent to the tangents through \\( A \\) and \\( A' \\) at those points."
] | [
"A variable tangent to an ellipse intersects the tangents from vertices \\( A \\) and \\( A' \\) at points \\( T \\) and \\( T' \\), respectively. Determine the locus of the intersection point of lines \\( TA' \\) and \\( T'A \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7d4386b6bc6fb4137e8b6589a7f193cece5c8d72bb0baf99e4883c54421546b5 | A variable tetrahedron $ABCD$ has the properties that its opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$, and vertices $A$, $B$, and $C$ lie on three fixed spheres centered at $P$ with radii $3$, $4$, and $12$, respectively. What is the maximal length of $PD$? | [
"19"
] | [
"A variable tetrahedron $ABCD$ has the properties that its opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$, and vertices $A$, $B$, and $C$ lie on three fixed spheres centered at $P$ with radii $3$, $4$, and $12$, respectively. What is the maximal length of $PD$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cecad9c4b8a1d70f9a5ebcf3177bf3d99506692499c3d892e9f9d4376ee1416 | A variable triangle ABC in the xy plane has its orthocenter at vertex 'B', a fixed vertex 'A' at the origin, and the third vertex 'C' restricted to lie on the parabola $y = 1 + \frac{7x^2}{36}$. The point B starts at the point (0,1) at time $t = 0$ and moves upward along the y-axis at a constant velocity of 2 cm/s. How... | [
"\\(\\frac{66}{7}\\)"
] | [
"A variable triangle ABC in the xy plane has its orthocenter at vertex 'B', a fixed vertex 'A' at the origin, and the third vertex 'C' restricted to lie on the parabola $y = 1 + \\frac{7x^2}{36}$. The point B starts at the point (0,1) at time $t = 0$ and moves upward along the y-axis at a constant velocity of 2 cm/... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ed92b61aab3d313bc7167b3845cddc25ab3c4f7ec161c83000c86ec00b5359e5 | A variation of Pascal's triangle is constructed by writing the numbers $2$ and $3$ in the top row and writing each subsequent term as the sum of the two terms above it. Find the fifth term from the left in the thirteenth row. | [
"1882"
] | [
"A variation of Pascal's triangle is constructed by writing the numbers $2$ and $3$ in the top row and writing each subsequent term as the sum of the two terms above it. Find the fifth term from the left in the thirteenth row."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22063721eaeef50822eed973339c8839a3f9d55d26ecc41c395f00c9567797e2 | A vault can be opened by \( n \) number of keys. Five people, A, B, C, D, E are given some of the keys. Each key can be duplicated an arbitrary number of times. Find the smallest number \( n \) and the distribution of the keys per person so that the vault can be opened if and only if one of the following situations hap... | [
"\\( n = 5 \\)"
] | [
"A vault can be opened by \\( n \\) number of keys. Five people, A, B, C, D, E are given some of the keys. Each key can be duplicated an arbitrary number of times. Find the smallest number \\( n \\) and the distribution of the keys per person so that the vault can be opened if and only if one of the following situa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-96228fed45de99856ae398fde8f2fc40eff8dfeb5e8ced006608eb07fad51418 | A vector $\overrightarrow{AB} ={\bf a}$ is laid off from the point $A$. Find the coordinates of the point $B$ in each of the following cases: (a) $A (0, 0)$, ${\bf a} = (-2, 1)$, (b) $A ( -1 , 5)$, ${\bf a} = (1, -3 ) $, (c) $A (2, 7)$, ${\bf a} = ( - 2 , -5 ) $. | [
"$(-2,1)$; (b) $(0,2)$; (c) $(0,2)$"
] | [
"A vector $\\overrightarrow{AB} ={\\bf a}$ is laid off from the point $A$. Find the coordinates of the point $B$ in each of the following cases: (a) $A (0, 0)$, ${\\bf a} = (-2, 1)$, (b) $A ( -1 , 5)$, ${\\bf a} = (1, -3 ) $, (c) $A (2, 7)$, ${\\bf a} = ( - 2 , -5 ) $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b14b0cb408807514d8f2d7197002c3b95f2c290ff18b552ae2155fc88da4816 | A vector $\vec{r}$ has projections along the vectors $\frac{2\hat{i}+\hat{j}}{\sqrt{5}}$, $\frac{-2\hat{i}+\hat{j}}{\sqrt{5}}$, and $\hat{k}$ which are equal. Find the unit vector along $\vec{r}$. | [
"\\(\\begin{bmatrix} 0\\\\ \\pm \\frac{\\sqrt{5}}{\\sqrt{6}} \\\\ \\frac{1}{\\sqrt{6}} \\end{bmatrix}\\) and \\(\\begin{bmatrix} \\pm \\frac{\\sqrt{5}}{3}\\\\ 0 \\\\ \\frac{2}{3} \\end{bmatrix}\\)"
] | [
"A vector $\\vec{r}$ has projections along the vectors $\\frac{2\\hat{i}+\\hat{j}}{\\sqrt{5}}$, $\\frac{-2\\hat{i}+\\hat{j}}{\\sqrt{5}}$, and $\\hat{k}$ which are equal. Find the unit vector along $\\vec{r}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9898b85c53767cc190f6dca0a51d3abfbb323d753c4fb365607ac27186f19f6 | A vendor had 4 meters of yellow cloth in stock and bought additional lengths from a wholesaler for PhP 1120. The vendor then sold all the cloth (original 4 meters plus the additional lengths) to Duchess for a total of PhP 1600. If the vendor's price per meter is at least PhP 10 more than the wholesaler's price per mete... | [
"\\( y = 28 \\) or \\( y = 16 \\)"
] | [
"A vendor had 4 meters of yellow cloth in stock and bought additional lengths from a wholesaler for PhP 1120. The vendor then sold all the cloth (original 4 meters plus the additional lengths) to Duchess for a total of PhP 1600. If the vendor's price per meter is at least PhP 10 more than the wholesaler's price per... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-35c71325654c5ca42081e75f0d7b16250789e812531fbfdf22ca8986b4ddc3c2 | A vertical dam has the shape of an isosceles trapezoid with a height of 30 m, a bottom width of 40 m, and a top width of 60 m. The dam holds 25 m of water. Let \( y \) be the height measured from the bottom of the dam. The hydrostatic force acting on the portion of the dam between \( y \) m and \( y + \Delta y \) m is ... | [
"\\rho g (25 - y) \\left(40 + \\frac{2}{3}y\\right)"
] | [
"A vertical dam has the shape of an isosceles trapezoid with a height of 30 m, a bottom width of 40 m, and a top width of 60 m. The dam holds 25 m of water. Let \\( y \\) be the height measured from the bottom of the dam. The hydrostatic force acting on the portion of the dam between \\( y \\) m and \\( y + \\Delta... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c4bd95db1b30d975fa6f742b74de8a54294283496ab201dff3ccd53d662b09e | A vertical line divides a triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ into two areas of equal area. What is the equation of the line? | [
"x=3"
] | [
"A vertical line divides a triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ into two areas of equal area. What is the equation of the line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-24fff079bf414ab97fb13915b7b351aba5cc16701443a030263772511c425c25 | A vertical line divides a triangle with vertices (0, 0), (1, 1), and (9, 1) into two regions of equal area. What is the equation of this line? | [
"\\( x = 3 \\)"
] | [
"A vertical line divides a triangle with vertices (0, 0), (1, 1), and (9, 1) into two regions of equal area. What is the equation of this line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-c4fdcbff6ef6f74438a41fcb0eaf75d514ddf1881848e326d41b935fcd5aa1d3 | A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy$-plane into two regions of equal area. Find the equation of the line in the form $x = \text{constant}$. | [
"\\( x = 3 \\)"
] | [
"A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy$-plane into two regions of equal area. Find the equation of the line in the form $x = \\text{constant}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-375a7469cf704b2c25b8dc018f18279a9d6b41ed5248a73861e0c7e9bfcc0133 | A vertical line divides the triangle with vertices (0,0), (1,1), and (9,1) in the xy-plane into two regions of equal area. Find the equation of the vertical line. | [
"\\( x = 3 \\)"
] | [
"A vertical line divides the triangle with vertices (0,0), (1,1), and (9,1) in the xy-plane into two regions of equal area. Find the equation of the vertical line."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-49b72c3ac8d212afaadbb1683aa9bbc3119c95b4c9e6a2cf4570fcc0f15bda5e | A vertical pin is stuck into each knot of an endless checkered paper. A needle of length $\ell$ lies on the paper parallel to the grid lines. At what $\ell$ can the needle be rotated $90^\circ$, without leaving the plane of the paper, while ensuring it passes between the pins? Ignore the thickness of pins and needles. | [
"\\(\\ell < 1\\)"
] | [
"A vertical pin is stuck into each knot of an endless checkered paper. A needle of length $\\ell$ lies on the paper parallel to the grid lines. At what $\\ell$ can the needle be rotated $90^\\circ$, without leaving the plane of the paper, while ensuring it passes between the pins? Ignore the thickness of pins and n... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97bc6aaad7b7482f54f19b7eee694cc7a71cbfd835711525e137d444ea4de7d1 | A very bizarre weighted coin comes up heads with probability $\frac12$, tails with probability $\frac13$, and rests on its edge with probability $\frac16$. If it comes up heads, I win 1 dollar. If it comes up tails, I win 3 dollars. But if it lands on its edge, I lose 5 dollars. What is the expected winnings from flipp... | [
"\\$\\dfrac23 \\approx \\$0.67"
] | [
"A very bizarre weighted coin comes up heads with probability $\\frac12$, tails with probability $\\frac13$, and rests on its edge with probability $\\frac16$. If it comes up heads, I win 1 dollar. If it comes up tails, I win 3 dollars. But if it lands on its edge, I lose 5 dollars. What is the expected winning... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-afbfb70661fa6998bf27605fb9d54a0e74110fb15e8e1bff760eaf7526c1b5bb | A very bouncy ball with negligible size is shot from the top left-hand corner of a unit square. At the instant the ball is fired, the angle the ball makes with the top of the unit square is \( \theta \). If the ball rebounds off the sides 2020 times before reaching a corner of the square, find the number of different p... | [
"800"
] | [
"A very bouncy ball with negligible size is shot from the top left-hand corner of a unit square. At the instant the ball is fired, the angle the ball makes with the top of the unit square is \\( \\theta \\). If the ball rebounds off the sides 2020 times before reaching a corner of the square, find the number of dif... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-01669ee9cd9fef978e535fc5910e03a9a91742b1017ee1a6e39695d443defaf2 | A very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied with $x$, produces a product that is a perfect square? | [
"105"
] | [
"A very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied with $x$, produces a product that is a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fec453644cf52393f658e469f863fc061cdb68fa9a6c551b1fcbb0b72f39e501 | A very large rectangle is divided into small squares. In the 1998 small squares on the edge of the rectangle, one writes the alphabet, one letter per square, without leaving an empty square between the individual letters. There are so many small squares that one can write the alphabet more than 10 times, always leaving... | [
"28"
] | [
"A very large rectangle is divided into small squares. In the 1998 small squares on the edge of the rectangle, one writes the alphabet, one letter per square, without leaving an empty square between the individual letters. There are so many small squares that one can write the alphabet more than 10 times, always le... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-c36948b5e8dceef4452c219b5f889cdba346d79c5ff45201eaed887121a722e9 | A very rich guy has \(2016^{2016}\) boxes, each initially containing 1 pound. At each step, he arbitrarily chooses 2016 boxes and replaces the number of pounds in each of these boxes with the sum of the pounds in all 2016 chosen boxes. After \(2016^{2016}\) steps, let \(S\) be the sum of the pounds in all the boxes. Fi... | [
"2016^{2016}"
] | [
"A very rich guy has \\(2016^{2016}\\) boxes, each initially containing 1 pound. At each step, he arbitrarily chooses 2016 boxes and replaces the number of pounds in each of these boxes with the sum of the pounds in all 2016 chosen boxes. After \\(2016^{2016}\\) steps, let \\(S\\) be the sum of the pounds in all th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-9f72011860e0784aaf504af8708afd893a2aca3ec4786d3ff5efb8712161e98d | A very wild ant is moving in a plane with a fixed Cartesian (orthogonal) coordinate system. (a) Suppose the ant is moving parallel either to the $x$-axis or the $y$-axis. Suppose the ant travels $3$ centimeters per second parallel to the $y$-axis direction (in either direction) and $2$ centimeters per second when movin... | [
"\\left\\{(x, y) \\left| \\, |x| \\leq 2, \\, |y| \\leq \\frac{4}{3} \\text{ or } 16(y - 3)^2 - 25x^2 \\leq 20 \\text{ if } y \\geq \\frac{4}{3}, \\, 16(y + 3)^2 - 25x^2 \\leq 20 \\text{ if } y \\leq -\\frac{4}{3} \\right. \\right\\}"
] | [
"A very wild ant is moving in a plane with a fixed Cartesian (orthogonal) coordinate system.\n(a) Suppose the ant is moving parallel either to the $x$-axis or the $y$-axis. Suppose the ant travels $3$ centimeters per second parallel to the $y$-axis direction (in either direction) and $2$ centimeters per second when... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ea0ee9537a6c901beb0ce3e3f6d4b7f92a347f61566d4f8cab644a31ca7906e | A vessel contains \( x \) gallons of wine and another contains \( y \) gallons of water. From each vessel, \( z \) gallons are taken out and transferred to the other. From the resulting mixture in each vessel, \( z \) gallons are again taken out and transferred to the other. If after the second transfer, the quantity o... | [
"6"
] | [
"A vessel contains \\( x \\) gallons of wine and another contains \\( y \\) gallons of water. From each vessel, \\( z \\) gallons are taken out and transferred to the other. From the resulting mixture in each vessel, \\( z \\) gallons are again taken out and transferred to the other. If after the second transfer, t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-8e7f8b976f3897d29dbe300e1cad74d2f9af5686b1e93bd74c2aebf308d96b79 | A vessel contains a $p\%$ solution of an acid. $a$ liters of the solution are then poured out and the same quantity of a $q\%$ solution of the acid is added $(q < p)$. After mixing this operation is repeated $k-1$ times which results in a $r\%$ solution. Find the volume of the vessel. | [
"$v=\\displaystyle\\frac{a}{1-\\sqrt [k]{\\displaystyle\\frac{r-q}{p-q}}}$"
] | [
"A vessel contains a $p\\%$ solution of an acid. $a$ liters of the solution are then poured out and the same quantity of a $q\\%$ solution of the acid is added $(q < p)$. After mixing this operation is repeated $k-1$ times which results in a $r\\%$ solution. Find the volume of the vessel."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe1a130d43149b0499010e8edbbe169e3a649b9d7911b30421a22de213a954c2 | A vessel initially contains \( V \) liters of water with \( x \) kg of salt dissolved in it. Pure water is added to it from the top at the rate of \( m \) L/s and the solution is allowed to flow out from an outlet at the bottom at the rate of \( n \) L/s such that \( m > n \) and the salt concentration is uniformly dis... | [
"x \\left( \\frac{V}{V + (m - n)t} \\right)^{\\frac{n}{m - n}}"
] | [
"A vessel initially contains \\( V \\) liters of water with \\( x \\) kg of salt dissolved in it. Pure water is added to it from the top at the rate of \\( m \\) L/s and the solution is allowed to flow out from an outlet at the bottom at the rate of \\( n \\) L/s such that \\( m > n \\) and the salt concentration i... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74d9c7506626457a7ce0f4c31aa59149db7976dbe1706fb68cf043605a604fee | A vessel is filled completely with water. 27 liters of water is removed and replaced with 27 liters of milk. This process is repeated once more. After these operations, the ratio of milk to water in the vessel becomes 9:16. Find the capacity of the vessel. | [
"135"
] | [
"A vessel is filled completely with water. 27 liters of water is removed and replaced with 27 liters of milk. This process is repeated once more. After these operations, the ratio of milk to water in the vessel becomes 9:16. Find the capacity of the vessel."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-ccb22447a157612a4881a80fc28e05491469d37a3c8a398dba422f1f422e9eab | A village $M$ is on the road between villages $A$ and $B$. The distance between $A$ and $M$ is twice as long as the distance between $B$ and $M$. Ann, Bob, and Tom live in $A$, $B$, and $M$ respectively. Ann and Bob walk towards $M$ at the same constant speed, starting at the same time. Tom has a motorbike that is nine... | [
"12"
] | [
"A village $M$ is on the road between villages $A$ and $B$. The distance between $A$ and $M$ is twice as long as the distance between $B$ and $M$. Ann, Bob, and Tom live in $A$, $B$, and $M$ respectively. Ann and Bob walk towards $M$ at the same constant speed, starting at the same time. Tom has a motorbike that is... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-8367f1a1e04eec5fcfc96d7c57439154a1acdf267f05b126a0ac9f0d7836b53c | A village green is ringed by a fence with \( N \) fenceposts, labeled \( 0, 1, \ldots, N-1 \). The village idiot starts at post \( 0 \) and paints it. He then moves to one of the two nearest neighbors (either \( 1 \) or \( N-1 \)) with equal probability, paints it, and continues this process. Determine the distribution... | [
"\\(\\frac{1}{N-1}\\)"
] | [
"A village green is ringed by a fence with \\( N \\) fenceposts, labeled \\( 0, 1, \\ldots, N-1 \\). The village idiot starts at post \\( 0 \\) and paints it. He then moves to one of the two nearest neighbors (either \\( 1 \\) or \\( N-1 \\)) with equal probability, paints it, and continues this process. Determine ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-4ddfa1157077c15400e72f23078317b67715241d448783e19ddea7f60a6fcf5f | A village has a circular wall with four gates pointing north, south, east, and west. A tree stands 16 meters north of the north gate and can be just seen appearing on the horizon from a point 48 meters east of the south gate. What is the diameter in meters of the wall that surrounds the village? | [
"48"
] | [
"A village has a circular wall with four gates pointing north, south, east, and west. A tree stands 16 meters north of the north gate and can be just seen appearing on the horizon from a point 48 meters east of the south gate. What is the diameter in meters of the wall that surrounds the village?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-22ba0530e0c86a58bb848a80b5c8d052a9c122c76a1327bdb42fbb926776d11d | A village is located on a river bank and a school on a highway which crosses the river at right angles. In winter the schoolboy skies from the village to the school taking a straight road across the river and it takes him 40 min to get to the school. In spring, when the roads are bad, he walks along the river bank to t... | [
"The village is farther away from the highway than the school is from the river"
] | [
"A village is located on a river bank and a school on a highway which crosses the river at right angles. In winter the schoolboy skies from the village to the school taking a straight road across the river and it takes him 40 min to get to the school. In spring, when the roads are bad, he walks along the river bank... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f308d116b0fed2062f3273aabb0498ee0eac3cabd9f84bdb12a6dbad5ad16e61 | A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there original... | [
"14 boys and 12 girls"
] | [
"A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there ori... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 30
} | 41 | 0 | 0 |
omr-ws-3214d448273c19cd920c931f25aa73be3ef17e8455ac6da13327d3a25b19e528 | A viper had a length of 30 centimeters at 6 months of age and a length of 1.3 meters at 15 years old. Assuming that its growth was linear, what is its length in meters at 10 years old? | [
"1"
] | [
"A viper had a length of 30 centimeters at 6 months of age and a length of 1.3 meters at 15 years old. Assuming that its growth was linear, what is its length in meters at 10 years old?"
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5bd1d7c9ea0cbb28e3e2e927dc0c76c3a8252097cc579ea6acea71a7ffed86d5 | A virus can either die, split into two, or split into three. The probabilities are as follows: dying with probability 1/4, splitting into two with probability 1/2, and splitting into three with probability 1/4. What is the probability that the virus survives indefinitely? | [
"\\dfrac{5 - \\sqrt{13}}{2}"
] | [
"A virus can either die, split into two, or split into three. The probabilities are as follows: dying with probability 1/4, splitting into two with probability 1/2, and splitting into three with probability 1/4. What is the probability that the virus survives indefinitely?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc17a79ae1d324e6a6d28d438cb01a13e7529b949e19c4712491721b68aab020 | A visitor walks through the Planar National Park, which consists of several trails meeting at junctions, with each junction being the endpoint of exactly three trails. The visitor starts at a junction and alternates between turning left and right at each junction until she returns to her starting point. What is the lar... | [
"3"
] | [
"A visitor walks through the Planar National Park, which consists of several trails meeting at junctions, with each junction being the endpoint of exactly three trails. The visitor starts at a junction and alternates between turning left and right at each junction until she returns to her starting point. What is th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-f217f62d02ce28d3f00b3673deceda648b3f3566437dbe7f49ee9fc574ba790b | A walker and a jogger travel along the same straight line in the same direction. The walker walks at one meter per second, while the jogger runs at two meters per second. The jogger starts one meter in front of the walker. A dog starts with the walker and runs back and forth between the walker and the jogger with a con... | [
"\\( f(n) = \\frac{7n}{3} \\)"
] | [
"A walker and a jogger travel along the same straight line in the same direction. The walker walks at one meter per second, while the jogger runs at two meters per second. The jogger starts one meter in front of the walker. A dog starts with the walker and runs back and forth between the walker and the jogger with ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-214bd8e71790c271cf23ee1e37f2d779efc882763b977d9760a75cb3d4745691 | A wall in Victoria's room is 83 inches wide. She wants to cover the wall exactly with wood panels that come in either $2\frac{1}{2}$-inch or $1\frac{3}{4}$-inch widths. What is the fewest number of panels she will need to cover the width of the wall without cutting any of the panels? | [
"35"
] | [
"A wall in Victoria's room is 83 inches wide. She wants to cover the wall exactly with wood panels that come in either $2\\frac{1}{2}$-inch or $1\\frac{3}{4}$-inch widths. What is the fewest number of panels she will need to cover the width of the wall without cutting any of the panels?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d1b3ee03fbe79cd5a56cb7994a25f3ea7512dde952153892ebb208bf2782663 | A wall-to-wall carpet costs $15 per square yard. Rita paid $330 to carpet a room that is 6 yards long. What is the width of the room? | [
"\\(\\frac{11}{3}\\)"
] | [
"A wall-to-wall carpet costs $15 per square yard. Rita paid $330 to carpet a room that is 6 yards long. What is the width of the room?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-7861edfa3987091250205c72bd18cf1f4ea5e12e99d8e997a023ea6db37a136d | A walnut-salesman knows that 20% of the nuts are empty. He has found a test that discards 20% of the nuts. However, when cracking the discarded nuts, one fourth of them were not empty after all. What proportion of the nuts that passed the test are then empty? | [
"$\\frac{3}{16}$"
] | [
"A walnut-salesman knows that 20% of the nuts are empty. He has found a test that discards 20% of the nuts. However, when cracking the discarded nuts, one fourth of them were not empty after all. What proportion of the nuts that passed the test are then empty?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 14
} | 15 | 0 | 0 |
omr-ws-dcc0bd6d0281aed94e61b908b7349e1156c8468ece317ba41125a1bd93af195a | A walrus starts at the origin and walks 1 mile due east. He then turns 60° counterclockwise and walks ½ mile. He continues turning 60° counterclockwise each time and walks one-half the previous distance. Find the exact simplified value of the x-coordinate of the limiting location of the walrus. | [
"1"
] | [
"A walrus starts at the origin and walks 1 mile due east. He then turns 60° counterclockwise and walks ½ mile. He continues turning 60° counterclockwise each time and walks one-half the previous distance. Find the exact simplified value of the x-coordinate of the limiting location of the walrus."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8c5e66abc40e8d6f7f7debe2ef0a4f92d7aef99ab18184e59787bd0042caa62f | A waste management site has its waste amount modeled by the function \( W(t) = t^3 - 12t^2 + 48t + 10 \) for \( t \) in the interval \([0, 8]\), where \( t \) is the time in days. Find the value of \( W(t) \) at the time \( t \) when the amount of waste is minimized. | [
"10"
] | [
"A waste management site has its waste amount modeled by the function \\( W(t) = t^3 - 12t^2 + 48t + 10 \\) for \\( t \\) in the interval \\([0, 8]\\), where \\( t \\) is the time in days. Find the value of \\( W(t) \\) at the time \\( t \\) when the amount of waste is minimized."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c15735bd89652bb65398fc256da6cafa61777192ff285d98d0b9463cc175230 | A watch is stopped for 15 minutes every hour on the hour. According to this watch, how many hours elapse between 12 noon and 12 midnight (actual time)? | [
"9 hours"
] | [
"A watch is stopped for 15 minutes every hour on the hour. According to this watch, how many hours elapse between 12 noon and 12 midnight (actual time)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-0abc1acbdeffb435c399990ec2534811b1b81446da5d3321bd4f3da26f7124b9 | A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. When the watch shows $9$ A.M. on March 21, what is the positive correction, in minutes, to be added to the time shown by the watch at that time? | [
"\\( 14\\frac{14}{23} \\)"
] | [
"A watch loses $2\\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. When the watch shows $9$ A.M. on March 21, what is the positive correction, in minutes, to be added to the time shown by the watch at that time?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 8,
"tir": 26
} | 61 | 0 | 0 |
omr-ws-b8f48ec7ab9a1705f4def83a7e45349b4e48e041d7ac18c042d4e20a95572f15 | A water bottle (cylindrical in shape) stands on a horizontal table. From a point on this plane, a man stares at the cap of the water bottle, from which four of its corner points are visible, with their angular elevations from the eye of the observer being \(30^\circ\), \(\alpha^\circ\), \(30^\circ\), and \(60^\circ\). ... | [
"8"
] | [
"A water bottle (cylindrical in shape) stands on a horizontal table. From a point on this plane, a man stares at the cap of the water bottle, from which four of its corner points are visible, with their angular elevations from the eye of the observer being \\(30^\\circ\\), \\(\\alpha^\\circ\\), \\(30^\\circ\\), and... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-67289616ebc00cd47be70cabc6282b4790bb8b2366119d27930d27f5000a1652 | A water cup in the shape of a right circular cone is made from a circular sheet of radius \( R = 6 \) by cutting out a sector and gluing the cut edges of the remaining piece together. What is the maximum volume of the cone? Give exact answers. | [
"\\( 16\\pi \\sqrt{3} \\)"
] | [
"A water cup in the shape of a right circular cone is made from a circular sheet of radius \\( R = 6 \\) by cutting out a sector and gluing the cut edges of the remaining piece together. What is the maximum volume of the cone? Give exact answers."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-38ac4f26a74cc16c5564be9001b841d3999bf667a975814060a092e4b74f8f6d | A water jet flows out horizontally from a point $A$, which is 4 meters above the ground, and follows a parabolic path that opens downwards, hitting the ground at a point $S$ that is 1 meter away from the point $P$ directly below $A$. A ball is to be placed on the ground at point $P$. Determine the maximum radius of the... | [
"1"
] | [
"A water jet flows out horizontally from a point $A$, which is 4 meters above the ground, and follows a parabolic path that opens downwards, hitting the ground at a point $S$ that is 1 meter away from the point $P$ directly below $A$. A ball is to be placed on the ground at point $P$. Determine the maximum radius o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ba37ca82d38ef715f423d6f1805fdfcc63ffc226d5f6b8206e215d5a4d86b68 | A water reservoir has the shape of a paraboloid with equation \( f(x,y) = x^2 + y^2 \), with the height of the reservoir equal to \( 10 \, \mathrm{m} \). An open faucet pours \( 0.2 \, \mathrm{m}^3 \) of water per minute into the reservoir. What is the rate of change of the height of the water level when the height is ... | [
"\\(\\frac{0.05}{\\pi}\\)"
] | [
"A water reservoir has the shape of a paraboloid with equation \\( f(x,y) = x^2 + y^2 \\), with the height of the reservoir equal to \\( 10 \\, \\mathrm{m} \\). An open faucet pours \\( 0.2 \\, \\mathrm{m}^3 \\) of water per minute into the reservoir. What is the rate of change of the height of the water level when... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b1e5969f03a2b951ce26a5e60ff22c3f22d5c0bacc299a9083f78a5da906fef3 | A water tank is 150 cm long, 40 cm high, and 40 cm deep. How many mL of water will it hold? | [
"240000"
] | [
"A water tank is 150 cm long, 40 cm high, and 40 cm deep. How many mL of water will it hold?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d372afe9b05d50e11b3fb87fb62e86292fce1b903dbb89479912a4beff219ef5 | A water tank is in the shape of a hemisphere surmounted by a right-circular cylinder. The radius of both the hemisphere and the cylinder is 4 ft, and the altitude of the cylinder is 8 ft. If the tank is full of water, find the work necessary to empty the tank by pumping it through an outlet at the top of the tank. | [
"\\dfrac{286208}{5} \\pi"
] | [
"A water tank is in the shape of a hemisphere surmounted by a right-circular cylinder. The radius of both the hemisphere and the cylinder is 4 ft, and the altitude of the cylinder is 8 ft. If the tank is full of water, find the work necessary to empty the tank by pumping it through an outlet at the top of the tank.... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-7b2745c185e2d5218e8d86d6ef2fd60bb8370150ce8866553d742c02672bb16a | A water tank is obtained by revolving the curve $y=\frac{25}{4096}x^4$ from $x=0$ to $x=8$ about the y-axis (x and y measured in feet). Water flows into the tank at a constant rate of 250 cubic feet per minute. How fast is the depth of the water in the tank increasing when the depth is 4 feet? | [
"\\(\\frac{1250}{128 \\pi}\\)"
] | [
"A water tank is obtained by revolving the curve $y=\\frac{25}{4096}x^4$ from $x=0$ to $x=8$ about the y-axis (x and y measured in feet). Water flows into the tank at a constant rate of 250 cubic feet per minute. How fast is the depth of the water in the tank increasing when the depth is 4 feet?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-23e99a81c79368ebf13f73e0cb94fd0d31b3af1c5fbe1eb4d46bbd25f7fd860f | A watermelon, \( 99\% \) of which is water, was placed in the sun. After a whole day, \( 98\% \) of the water evaporated. What fraction of the original watermelon is left? | [
"\\dfrac{149}{5000}"
] | [
"A watermelon, \\( 99\\% \\) of which is water, was placed in the sun. After a whole day, \\( 98\\% \\) of the water evaporated. What fraction of the original watermelon is left?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-c84051458a66895d9d4a659d23774bb5e696e0d0c9ba0fe34fb1f019f4dbe543 | A watermelon, modeled as a sphere with radius \( R \), lies on a table. \( n \) flies are flying above the table, each at a distance of \( \sqrt{2}R \) from the center of the watermelon. At some moment, no fly can see any other fly because the segment connecting any two flies intersects or touches the watermelon. Find ... | [
"6"
] | [
"A watermelon, modeled as a sphere with radius \\( R \\), lies on a table. \\( n \\) flies are flying above the table, each at a distance of \\( \\sqrt{2}R \\) from the center of the watermelon. At some moment, no fly can see any other fly because the segment connecting any two flies intersects or touches the water... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-ad3f88fce53e73a5997c3df2a6f44ecda8b714ec9d4f1c0128271fddca33025f | A weather balloon is directly over a $2000$m airstrip. The angles of elevation from the ends of the airstrip to the balloon are $62^\circ$ and $31^\circ$ respectively. Find the distance of the balloon from each end of the airstrip. | [
"1031.5 m and 1768.3 m"
] | [
"A weather balloon is directly over a $2000$m airstrip. The angles of elevation from the ends of the airstrip to the balloon are $62^\\circ$ and $31^\\circ$ respectively. Find the distance of the balloon from each end of the airstrip."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 30
} | 32 | 0 | 0 |
omr-ws-d081214cbfbe33a2ec1bc8c0f8eeca669f502ff57e4b95c09931708397c2b7b7 | A weather balloon is directly over a $2000$m airstrip. The angles of elevation from the ends of the airstrip to the balloon are $62^\circ$ and $31^\circ$ respectively. Find the height of the balloon. | [
"\\( 910.75 \\) m."
] | [
"A weather balloon is directly over a $2000$m airstrip. The angles of elevation from the ends of the airstrip to the balloon are $62^\\circ$ and $31^\\circ$ respectively. Find the height of the balloon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-adbc0909cac2b057999b714b95351512be28273f7bd4a0bc315012ad7562edf6 | A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec, how fast is the angle of the camera changing when the balloon is 100 ft. high? | [
"\\( \\frac{3}{68} \\) radians per second or approximately \\( 0.044 \\) radians per second."
] | [
"A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec, how fast is the angle of the camera changing when the balloon is 100 ft. high?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f2bb7fe77e81e9fa05bea6d474cb5dbb0f1601630a864bf39b914374fef6c3c7 | A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec., calculate the rate of change of the angle of the camera when the balloon is 80 feet above the ground. Express your answer in radians per second. | [
"\\dfrac{3}{50}"
] | [
"A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec., calculate the rate of change of the angle of the camera when the balloon is 80 feet above the ground. Express your answer in radians per second."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3ce9f6400fe0e4225d8bd960d1a278a63a9317cf0119a4993ec75af95534b3c | A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec., how fast is the angle of the camera changing when the balloon is 100 ft. high? | [
"\\dfrac{3}{68}"
] | [
"A weather balloon is released 60 feet from an observer with a camera. If the balloon is rising at 10 ft. per sec., how fast is the angle of the camera changing when the balloon is 100 ft. high?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81f6d8c534a8592226581f2e093a462f6d5734447e7b17331b4434bfa15c92b8 | A weather forecast predicts that for each day from Monday to Friday this week, there is a $40\%$ chance of sun, a $25\%$ chance of getting 4 inches of rain, and a $35\%$ chance of getting 10 inches of rain. What is the expected value of the total number of inches of rain which will fall from Monday to Friday? Give your... | [
"22.5"
] | [
"A weather forecast predicts that for each day from Monday to Friday this week, there is a $40\\%$ chance of sun, a $25\\%$ chance of getting 4 inches of rain, and a $35\\%$ chance of getting 10 inches of rain. What is the expected value of the total number of inches of rain which will fall from Monday to Friday? ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-541b735c3472c535d67e4d5affac771c33715ccc48a66645514f040fe2dde23b | A website sells 3 types of chocolate bars of equal weight: Milk (35% cocoa), Dark (50% cocoa), and Bittersweet (70% cocoa). If Serena orders 2 chocolate bars at random from the website and melts them together, what is the probability that the resulting mix contains at least 50% cocoa by weight? | [
"\\( \\frac{2}{3} \\)"
] | [
"A website sells 3 types of chocolate bars of equal weight: Milk (35% cocoa), Dark (50% cocoa), and Bittersweet (70% cocoa). If Serena orders 2 chocolate bars at random from the website and melts them together, what is the probability that the resulting mix contains at least 50% cocoa by weight?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-cc9ef61265ece629fa85439857276f9fb77212c98179a09cc058df2761288d8a | A weighted coin flips heads with probability \( p \). You flip the coin \( n \) times and it comes up heads \( k \) times. Find the expected value of \( p \). | [
"\\(\\frac{k+1}{n+2}\\)"
] | [
"A weighted coin flips heads with probability \\( p \\). You flip the coin \\( n \\) times and it comes up heads \\( k \\) times. Find the expected value of \\( p \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-046e609d02bc5aebfc1cb211d8805a335b4cf8198735ef5e62400e4d1181200d | A weighted die is constructed so that the 6 comes up 25% of the time and each of the other sides comes up 15% of the time. If this die is tossed 4 times, what is the probability of getting either one six or two sixes out of the four tosses? | [
"0.6328125"
] | [
"A weighted die is constructed so that the 6 comes up 25% of the time and each of the other sides comes up 15% of the time. If this die is tossed 4 times, what is the probability of getting either one six or two sixes out of the four tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5fbd43114d221ca04c5d01a905d4541744867cce08f8ce44265c972bb8e6c6fd | A weighted n-sided die has probability $p_k$ of landing on side $k$. It is rolled $m$ times. What is the expected value of the longest streak of consecutive rolls of the same number? | [
"\\sum_{l=1}^m \\left(1 - \\sum_{k=1}^n \\sum_{s=1}^{l-1} F(m, s, k)\\right)"
] | [
"A weighted n-sided die has probability $p_k$ of landing on side $k$. It is rolled $m$ times. What is the expected value of the longest streak of consecutive rolls of the same number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-631170d74e14403d129f83f156788e3164de3390cd84e50980af5230964c56c5 | A weightless rod is hinged at $O$ so that it can rotate without friction in a vertical plane. A mass $m$ is attached to the end of the rod $A,$ which is balanced vertically above $O.$ At time $t = 0,$ the rod moves away from the vertical with negligible initial angular velocity. Calculate the time $t$ at which the mass... | [
"\\dfrac{\\pi}{2\\sqrt{9.8}}"
] | [
"A weightless rod is hinged at $O$ so that it can rotate without friction in a vertical plane. A mass $m$ is attached to the end of the rod $A,$ which is balanced vertically above $O.$ At time $t = 0,$ the rod moves away from the vertical with negligible initial angular velocity. Calculate the time $t$ at which the... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cd8d61b2513f0902bd0ee72e67b322eeaad5647389e4d78d0f2e48f808dfc6b4 | A wheel consists of a fixed circular disk with numbers $1, 2, 3, \ldots, N$ marked on it and a mobile circular ring with $N$ integers $a_1, a_2, \ldots, a_N$ marked on it, where the sum of these integers is $1$. The ring can be turned into $N$ different positions, each corresponding to a unique pairing of the numbers o... | [
"0"
] | [
"A wheel consists of a fixed circular disk with numbers $1, 2, 3, \\ldots, N$ marked on it and a mobile circular ring with $N$ integers $a_1, a_2, \\ldots, a_N$ marked on it, where the sum of these integers is $1$. The ring can be turned into $N$ different positions, each corresponding to a unique pairing of the nu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef48d127766ada3d7ab644a9204877a326afe3cb7497a44c1bcf203654a0020f | A wheel consists of two concentric circles and five spokes, with labeled points at each intersection of a spoke with a circle. A bug starts at point \(A\) and moves to adjacent labeled points. Determine the number of 15-step paths that begin and end at point \(A\). Find the remainder when this number is divided by 1000... | [
"64"
] | [
"A wheel consists of two concentric circles and five spokes, with labeled points at each intersection of a spoke with a circle. A bug starts at point \\(A\\) and moves to adjacent labeled points. Determine the number of 15-step paths that begin and end at point \\(A\\). Find the remainder when this number is divide... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-c676b757c65358968c382dd3835ab75d1a768b3c3f03bf53f78b829285b9d409 | A wheel consists of two concentric circles with five spokes connecting corresponding points on the circles. The points where the spokes meet the circles are labeled \(A, B, C, D, E\) on the inner circle and \(F, G, H, I, J\) on the outer circle. A bug starts at point \(A\) and walks along the wheel, moving only to adja... | [
"004"
] | [
"A wheel consists of two concentric circles with five spokes connecting corresponding points on the circles. The points where the spokes meet the circles are labeled \\(A, B, C, D, E\\) on the inner circle and \\(F, G, H, I, J\\) on the outer circle. A bug starts at point \\(A\\) and walks along the wheel, moving o... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 5
} | 29 | 0 | 0 |
omr-ws-85fe5bf925bfa60b2b7e7f326d6017c8820b4c6e98ccd55b8523c49b9f15450a | A wheel has thirty equal spaces: 16 blue, 8 red, and 6 white. Landing on a blue space means you win, landing on a red space means you lose, and landing on a white space means you must spin again. What are your chances of winning and losing? | [
"\\( \\frac{2}{3} \\) (for winning) and \\( \\frac{1}{3} \\) (for losing)."
] | [
"A wheel has thirty equal spaces: 16 blue, 8 red, and 6 white. Landing on a blue space means you win, landing on a red space means you lose, and landing on a white space means you must spin again. What are your chances of winning and losing?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
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