idx int64 6 7.5k | question stringclasses 574
values | gt_cot stringclasses 574
values | gt stringclasses 476
values | level stringclasses 5
values | type stringclasses 7
values | solution stringclasses 574
values | my_solu sequencelengths 1 1 | code sequencelengths 1 1 | pred sequencelengths 1 1 | report sequencelengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|---|
6 | What are all values of $p$ such that for every $q>0$, we have $$\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. | First we'll simplify that complicated expression. We attempt to factor the numerator of the left side: \begin{align*}
pq^2+p^2q+3q^2+3pq &= q(pq + p^2 + 3q + 3p) \\
&= q[ p(q+p) + 3(q+p) ] \\
&= q(p+3)(q+p).
\end{align*}Substituting this in for the numerator in our inequality gives $$\frac{3q(p+3)(p+q)}{p+q}>2p^2q.$$We... | [0,3) | Level 5 | Algebra | First we'll simplify that complicated expression. We attempt to factor the numerator of the left side: \begin{align*}
pq^2+p^2q+3q^2+3pq &= q(pq + p^2 + 3q + 3p) \\
&= q[ p(q+p) + 3(q+p) ] \\
&= q(p+3)(q+p).
\end{align*}Substituting this in for the numerator in our inequality gives $$\frac{3q(p+3)(p+q)}{p+q}>2p^2q.$$We... | [
"User: What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form.\nPlease integrate natural language reasoning with programs to solve the problem above, and put your final answer within \\boxed{}.\n\nAssista... | [
"```python\nfrom sympy import symbols, simplify\nfrom sympy.solvers.inequalities import solve_univariate_inequality\nfrom sympy import Interval, oo\n\ndef solve_inequality():\n \"\"\"What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer ... | [
"(0,1.787)"
] | [
null
] |
8 | The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | Since $18 - 14 = 4$, the common difference in the first column of squares is 4, so the number above 14 is $14 - 4 = 10$, and the number above 10 is $10 - 4 = 6$. This is also the fourth number in the row, so the common difference in the row is $(6 - 21)/3 = -5$.
Then the seventh (and last) number in the row is $21 - ... | -7 | Level 2 | Algebra | Since $18 - 14 = 4$, the common difference in the first column of squares is 4, so the number above 14 is $14 - 4 = 10$, and the number above 10 is $10 - 4 = 6$. This is also the fourth number in the row, so the common difference in the row is $(6 - 21)/3 = -5$.
Then the seventh (and last) number in the row is $21 - ... | [
"User: The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)--(1,1));\ndraw((2,0)--(2,1));\ndraw((3,0)--(3,1));\ndraw((4,0)--(4,1... | [
"```python\nfrom sympy import symbols, Eq, solve\n\ndef find_n():\n \"\"\"The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?\n\n[asy]\nunitsize(0.35inch);\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0... | [
"2"
] | [
null
] |
334 | Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | First, set the two equations equal to each other to get $2x^2-10x-10=x^2-4x+6$. Combine like terms to get $x^2-6x=16$. To complete the square, we need to add $\left(\dfrac{6}{2}\right)^2=9$ to both sides, giving $(x-3)^2=16+9=25$.
So we have $x-3=\pm5$. Solving for $x$ gives us $x=-2$ or $8$. Using these in our origin... | (8,38) | Level 5 | Algebra | First, set the two equations equal to each other to get $2x^2-10x-10=x^2-4x+6$. Combine like terms to get $x^2-6x=16$. To complete the square, we need to add $\left(\dfrac{6}{2}\right)^2=9$ to both sides, giving $(x-3)^2=16+9=25$.
So we have $x-3=\pm5$. Solving for $x$ gives us $x=-2$ or $8$. Using these in our origin... | [
"User: Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons.\nPlease integrate natural language reasoning with programs to solve the problem above, and put your final answe... | [
"```python\nfrom sympy import symbols, Eq, solve\n\ndef intersection_points():\n \"\"\"Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons.\"\"\"\n x = symbols('x')\... | [
"(-2,-18)(8,26)"
] | [
null
] |
484 | The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
re... | The vertex of the parabola is $(5,-4)$, so the equation of the parabola is of the form \[x = a(y + 4)^2 + 5.\] The parabola passes through the point $(3,-2)$. Substituting these values into the equation above, we get \[3 = a(-2 + 4)^2 + 5.\] Solving for $a$, we find $a = -1/2$. Hence, the equation of the parabola is ... | -\frac{15}{2} | Level 5 | Algebra | The vertex of the parabola is $(5,-4)$, so the equation of the parabola is of the form \[x = a(y + 4)^2 + 5.\] The parabola passes through the point $(3,-2)$. Substituting these values into the equation above, we get \[3 = a(-2 + 4)^2 + 5.\] Solving for $a$, we find $a = -1/2$. Hence, the equation of the parabola is ... | [
"User: The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownlength=-0.15inch;\nr... | [
"```python\ndef coefficients_from_parabola():\n \"\"\"The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal... | [
"0.5"
] | [
null
] |
544 | The graph of $y=f(x)$ for $-3\le x\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0... | We can superimpose the graph of $y=x$ on the graph of $y=f(x)$: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown... | (-1,0] | Level 5 | Algebra | We can superimpose the graph of $y=x$ on the graph of $y=f(x)$: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown... | [
"User: The graph of $y=f(x)$ for $-3\\le x\\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=... | [
"```python\nfrom sympy import symbols, Interval\n\ndef range_of_f_minus_x():\n \"\"\"The graph of $y=f(x)$ for $-3\\le x\\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.... | [
"21"
] | [
null
] |
629 | The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)
What is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsi... | We can superimpose the graph of $y=1.8$ on the same axes as the original graph:
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real whole... | 4.5 | Level 5 | Algebra | We can superimpose the graph of $y=1.8$ on the same axes as the original graph:
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real whole... | [
"User: The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tickspace=2;\n\nreal ticklength=0... | [
"```python\ndef x_coordinates_with_f_equals_1_8():\n \"\"\"The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?\n\n[asy]\n\nsize(150);\n\nre... | [
"0"
] | [
null
] |
809 | Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$, and write your answer in the form \[
\frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F},
\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$? | Since 2, 3, and 7 are all primes, the denominator is in simplest radical form and we can't simplify it further. We attack this problem by getting rid of the square roots one step at a time. First we group the first two terms, and multiply numerator and denominator by the conjugate: \begin{align*}
\frac{1}{(\sqrt{2} + \... | 57 | Level 5 | Algebra | Since 2, 3, and 7 are all primes, the denominator is in simplest radical form and we can't simplify it further. We attack this problem by getting rid of the square roots one step at a time. First we group the first two terms, and multiply numerator and denominator by the conjugate: \begin{align*}
\frac{1}{(\sqrt{2} + \... | [
"User: Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{2} + \\sqrt{3} + \\sqrt{7}}$, and write your answer in the form \\[\n\\frac{A\\sqrt{2} + B\\sqrt{3} + C\\sqrt{7} + D\\sqrt{E}}{F},\n\\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is... | [
"```python\nfrom sympy import sqrt, simplify\n\ndef rationalize_denominator():\n \"\"\"Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{2} + \\sqrt{3} + \\sqrt{7}}$, and write your answer in the form \\[\n\\frac{A\\sqrt{2} + B\\sqrt{3} + C\\sqrt{7} + D\\sqrt{E}}{F},\n\\]where everything is in simp... | [
"\\frac{1}{(sqrt(2)+sqrt(3)+sqrt(7))}"
] | [
null
] |
837 | The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction. | Since the difference of the first two terms is $-2y$, the third and fourth terms of the sequence must be $x-3y$ and $x-5y$. Thus \[
x-3y = xy \quad\text{and}\quad x-5y = \frac{x}{y},
\]so $xy - 5y^{2} = x.$ Combining these equations we obtain \[
(x - 3y) - 5y^{2}= x\quad\text{and, therefore, }\quad -3y - 5y^{2} = 0.
\]... | \frac{123}{40} | Level 5 | Algebra | Since the difference of the first two terms is $-2y$, the third and fourth terms of the sequence must be $x-3y$ and $x-5y$. Thus \[
x-3y = xy \quad\text{and}\quad x-5y = \frac{x}{y},
\]so $xy - 5y^{2} = x.$ Combining these equations we obtain \[
(x - 3y) - 5y^{2}= x\quad\text{and, therefore, }\quad -3y - 5y^{2} = 0.
\]... | [
"User: The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction.\nPlease integrate natural language reasoning with programs to solve the problem above, and put your final answer within \\boxed{}.\n\nAssistant: ... | [
"```python\nfrom sympy import symbols, Eq, solve, simplify\nfrom sympy import Rational\n\ndef fifth_term_arithmetic_sequence():\n \"\"\"The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common difference.\"\"\"\n x... | [
"TypeError:listindicesmustbeintegersorslices,notSymbol"
] | [
null
] |
927 | A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?
[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,linewidth(1)... | We'll start with rods. The first row has 3 rods, the second row has 6 rods, and continuing down, we see that the next rows have 9, 12, 15, and so on rods. So the total number of rods in an eight-row triangle is $$
3 + 6 + 9 + \cdots + 24 = 3(1+2+3+\cdots+8) = 3(36) = 108.
$$For the connectors, note that in an $n$-row t... | 153 | Level 5 | Algebra | We'll start with rods. The first row has 3 rods, the second row has 6 rods, and continuing down, we see that the next rows have 9, 12, 15, and so on rods. So the total number of rods in an eight-row triangle is $$
3 + 6 + 9 + \cdots + 24 = 3(1+2+3+\cdots+8) = 3(36) = 108.
$$For the connectors, note that in an $n$-row t... | [
"User: A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?\n\n[asy]\ndraw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));\ndraw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cyc... | [
"```python\ndef total_pieces_created():\n \"\"\"A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?\"\"\"\n # Define the constants\n unit_rods_per_row = 3\n connector_... | [
"32"
] | [
null
] |
960 | Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \max\{f(x),g(x),h(x)\},$$$$k(x) = \min\{f(x),g(x),h(x)\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)$ is the least of these thre... | The graphs of $f(x),g(x),h(x)$ are all lines, and we have a segment of each, so we can extend these segments to form the superimposed graphs of $f(x),$ $g(x),$ and $h(x)$ on one set of axes:
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
r... | 245 | Level 5 | Algebra | The graphs of $f(x),g(x),h(x)$ are all lines, and we have a segment of each, so we can extend these segments to form the superimposed graphs of $f(x),$ $g(x),$ and $h(x)$ on one set of axes:
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
r... | [
"User: Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \\max\\{f(x),g(x),h(x)\\},$$$$k(x) = \\min\\{f(x),g(x),h(x)\\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)$ is the le... | [
"```python\nfrom sympy import sqrt\n\ndef compute_length_squared():\n \"\"\"Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \\max\\{f(x),g(x),h(x)\\},$$$$k(x) = \\min\\{f(x),g(x),h(x)\\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x)... | [
"22"
] | [
null
] |
1,177 | There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\neq 0$, find the equation of the line. Enter your answer in the form "$y = mx + b$". | The line $x=k$ intersects $y=x^2+6x+5$ at the point $(k, k^2+6k+5)$ and the line $y=mx+b$ at the point $(k,mk+b)$. Since these two points have the same $x$-coordinate, the distance between them is the difference of their $y$-coordinates, so we have $$|(k^2+6k+5)-(mk+b)|=5.$$ Simplifying, this gives us two quadratic equ... | 10x-4 | Level 5 | Algebra | The line $x=k$ intersects $y=x^2+6x+5$ at the point $(k, k^2+6k+5)$ and the line $y=mx+b$ at the point $(k,mk+b)$. Since these two points have the same $x$-coordinate, the distance between them is the difference of their $y$-coordinates, so we have $$|(k^2+6k+5)-(mk+b)|=5.$$ Simplifying, this gives us two quadratic equ... | [
"User: There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\\neq 0$, find the equation of the line. Enter your answer in the form \"$y = mx + b$\".\nP... | [
"```python\nfrom sympy import symbols, Eq, solve, sqrt, simplify\n\ndef find_line():\n \"\"\"There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\\n... | [
"TimeoutError"
] | [
null
] |
1,195 | Let $p(x)$ be defined on $2 \le x \le 10$ such that $$p(x) = \begin{cases} x + 1 &\quad \lfloor x \rfloor\text{ is prime} \\ p(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $p$ in interval notation. | By definition of $p$, for any prime number $x$ such that $2 \le x \le 10$, then $[x+1,x+2) \subset \text{range}\,(p)$. It follows that $[3,4) \cup [4,5) \cup [6,7) \cup [8,9) \subset \text{range}\,(p)$. Since the largest prime factor of a composite number less than or equal to $10$ is $5$, then the largest possible val... | [3,7]\cup[8,9) | Level 5 | Algebra | By definition of $p$, for any prime number $x$ such that $2 \le x \le 10$, then $[x+1,x+2) \subset \text{range}\,(p)$. It follows that $[3,4) \cup [4,5) \cup [6,7) \cup [8,9) \subset \text{range}\,(p)$. Since the largest prime factor of a composite number less than or equal to $10$ is $5$, then the largest possible val... | [
"User: Let $p(x)$ be defined on $2 \\le x \\le 10$ such that $$p(x) = \\begin{cases} x + 1 &\\quad \\lfloor x \\rfloor\\text{ is prime} \\\\ p(y) + (x + 1 - \\lfloor x \\rfloor) &\\quad \\text{otherwise} \\end{cases}$$ where $y$ is the greatest prime factor of $\\lfloor x\\rfloor.$ Express the range of $p$ in inter... | [
"```python\nfrom sympy import primefactors, floor\n\ndef find_range():\n \"\"\"Let $p(x)$ be defined on $2 \\le x \\le 10$ such that $$p(x) = \\begin{cases} x + 1 &\\quad \\lfloor x \\rfloor\\text{ is prime} \\\\ p(y) + (x + 1 - \\lfloor x \\rfloor) &\\quad \\text{otherwise} \\end{cases}$$ where $y$ is the great... | [
"[3,8]"
] | [
null
] |
1,556 | Laura is training for a triathlon, but she doesn't feel like swimming. She bikes $20$ miles at $2x+1$ miles per hour, spends five minutes parking her bike, changing to her running shoes, and catching a drink of water, and then runs $5$ miles at $x$ miles per hour. Her total workout lasts $110$ minutes. How fast did Lau... | Since Laura spends $5$ minutes in transition, a total of $110-5=105$ minutes are spent in motion. This is equivalent to $\frac{105}{60}=1.75$ hours. We know that $\text{distance}=\text{rate}\cdot\text{time}$, so $\text{time}=\frac{\text{distance}}{\text{rate}}$. Thus the time Laura spends biking is $\frac{20\text{ mile... | \approx8.24 | Level 5 | Algebra | Since Laura spends $5$ minutes in transition, a total of $110-5=105$ minutes are spent in motion. This is equivalent to $\frac{105}{60}=1.75$ hours. We know that $\text{distance}=\text{rate}\cdot\text{time}$, so $\text{time}=\frac{\text{distance}}{\text{rate}}$. Thus the time Laura spends biking is $\frac{20\text{ mile... | [
"User: Laura is training for a triathlon, but she doesn't feel like swimming. She bikes $20$ miles at $2x+1$ miles per hour, spends five minutes parking her bike, changing to her running shoes, and catching a drink of water, and then runs $5$ miles at $x$ miles per hour. Her total workout lasts $110$ minutes. How f... | [
"1. To solve this problem, we first need to set up an equation based on the information given. The time Laura spent biking $20$ miles at a speed of $2x+1$ miles per hour is $\\frac{20}{2x+1}$ hours. The time she spent running $5$ miles at a speed of $x$ miles per hour is $\\frac{5}{x}$ hours.\n\n2. Adding five minu... | [
"-0.17"
] | [
null
] |
1,596 | The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In thr... | The fact that 60 workers produce 240 widgets and 300 whoosits in two hours implies that 100 workers produce 400 widgets and 500 whoosits in two hours, or 200 widgets and 250 whoosits in one hour. Let $a$ be the time required for a worker to produce a widget, and let $b$ be the time required for a worker to produce a wh... | 450 | Level 5 | Algebra | The fact that 60 workers produce 240 widgets and 300 whoosits in two hours implies that 100 workers produce 400 widgets and 500 whoosits in two hours, or 200 widgets and 250 whoosits in one hour. Let $a$ be the time required for a worker to produce a widget, and let $b$ be the time required for a worker to produce a wh... | [
"User: The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoos... | [
"```python\nfrom sympy import symbols, Eq, solve\n\ndef find_m():\n \"\"\"The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 wh... | [
"TypeError:listindicesmustbeintegersorslices,notSymbol"
] | [
null
] |
1,763 | Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?
[asy]
draw((-75,0)--(-45,0)--(-60,26)--cycle);
draw((0,0)--(30,0)--(15,26)--cycle);
draw((75,0)--(10... | There are six ways to color the equilateral triangle on the left. Without loss of generality, assume it is colored as below.
[asy]
draw((-75,0)--(-45,0)--(-60,26)--cycle);
draw((0,0)--(30,0)--(15,26)--cycle);
draw((75,0)--(105,0)--(90,26)--cycle);
draw((-60,26)--(90,26));
draw((-45,0)--(75,0));
dot("B", (-75,0), S);... | 54 | Level 5 | Counting & Probability | There are six ways to color the equilateral triangle on the left. Without loss of generality, assume it is colored as below.
[asy]
draw((-75,0)--(-45,0)--(-60,26)--cycle);
draw((0,0)--(30,0)--(15,26)--cycle);
draw((75,0)--(105,0)--(90,26)--cycle);
draw((-60,26)--(90,26));
draw((-45,0)--(75,0));
dot("B", (-75,0), S);... | [
"User: Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)--(-45,0)--(-60,26)--cycle);\ndraw((0,0)--(30,0)--(15,26)--cycle);\n... | [
"```python\nfrom sympy import factorial\n\ndef coloring_ways():\n \"\"\"Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)... | [
"243"
] | [
null
] |
1,765 | An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minutes, 30... | We will calculate the probability of her hearing every second of her favorite song and then subtract that from 1 to get the probability that we're looking for. There are a total of $10!$ ways in which the 10 songs can be ordered. If her favorite song is the first song, she obviously hears the whole thing, and then ther... | \frac{79}{90} | Level 5 | Counting & Probability | We will calculate the probability of her hearing every second of her favorite song and then subtract that from 1 to get the probability that we're looking for. There are a total of $10!$ ways in which the 10 songs can be ordered. If her favorite song is the first song, she obviously hears the whole thing, and then ther... | [
"User: An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 ... | [
"```python\nfrom sympy import Rational\n\ndef song_probability():\n \"\"\"An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent son... | [
"\\frac{61}{269}"
] | [
null
] |
1,769 | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | If a sequence contains no more than one 0, there are $7\cdot 6\cdot
5\cdot 4\cdot 3 = 2520$ sequences formed from the characters A, I, M, E, 2, 0, and 7. If a sequence contains two 0's, the 0's can be placed in $\binom{5}{2} = 10$ ways, the remaining characters can be chosen in $\binom{6}{3} = 20$ ways, and those remai... | 372 | Level 5 | Counting & Probability | If a sequence contains no more than one 0, there are $7\cdot 6\cdot
5\cdot 4\cdot 3 = 2520$ sequences formed from the characters A, I, M, E, 2, 0, and 7. If a sequence contains two 0's, the 0's can be placed in $\binom{5}{2} = 10$ ways, the remaining characters can be chosen in $\binom{6}{3} = 20$ ways, and those remai... | [
"User: A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digit... | [
"```python\nfrom sympy import factorial\n\ndef license_plates():\n \"\"\"A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more t... | [
"840"
] | [
null
] |
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