idx int64 0 7.5k | prompt stringlengths 203 10k | first_round bool 2
classes | gt stringlengths 1 139 | rewards sequencelengths 4 8 ⌀ | my_solu sequencelengths 1 1 | flag bool 1
class | turn int64 3 4 | conversations listlengths 2 4 | level stringclasses 6
values | type stringclasses 7
values | solution stringlengths 29 6.77k ⌀ | pred sequencelengths 1 1 ⌀ | my_prompt_conversations listlengths 3 3 ⌀ |
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746 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st t... | true | 1000 | [
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true
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nTwo arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the... | true | 4 | [
{
"content": "Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$?\nPlease reason s... | null | null | null | null | null |
5,545 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.)
Please reason step by step, and put your final answer within... | false | 249 | [
true,
false,
false,
false,
false,
false,
false,
false
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.)\nPlease reason step by step, and put your final a... | true | 4 | [
{
"content": "The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... | null | null | null | null | null |
6,117 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The longest side of a right triangle is 5 meters and the shortest side is 3 meters. What is the area of the triangle in square meters?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
... | false | 6 | [
true,
true,
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false,
true,
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true,
true
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe longest side of a right triangle is 5 meters and the shortest side is 3 meters. What is the area of the triangle in square meters?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_hea... | true | 4 | [
{
"content": "The longest side of a right triangle is 5 meters and the shortest side is 3 meters. What is the area of the triangle in square meters?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To find the area of the triangle, we need to... | null | null | null | null | null |
5,991 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Twenty-seven increased by twice a number is 39. What is the number?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
| true | 6 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nTwenty-seven increased by twice a number is 39. What is the number?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nLet's call the number $x$.\n\nThe problem states that \... | null | 3 | [
{
"content": "Twenty-seven increased by twice a number is 39. What is the number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Let's call the number $x$.\n\nThe problem states that \"Twenty-seven increased by twice a number is 39.\" This ... | Level 1 | Prealgebra | We are asked to solve $27+2x=39$ for $x$. We subtract 27 from both sides and then multiply both sides by $\frac{1}{2}$: \begin{align*}
27+2x&=39 \\
2x &= 12 \\
x &= \boxed{6}.
\end{align*} | [
""
] | [
{
"content": "Twenty-seven increased by twice a number is 39. What is the number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Let's call the number $x$.\n\nThe problem states that \"Twenty-seven increased by twice a number is 39.\" This ... |
6,911 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $a$ and $b$ be angles such that $\sin (a + b) = \frac{3}{4}$ and $\sin (a - b) = \frac{1}{2}.$ Find $\frac{\tan a}{\tan b}.$
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
To f... | false | 5 | [
true,
false,
false,
false,
false,
false,
true,
false
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nLet $a$ and $b$ be angles such that $\\sin (a + b) = \\frac{3}{4}$ and $\\sin (a - b) = \\frac{1}{2}.$ Find $\\frac{\\tan a}{\\tan b}.$\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_h... | true | 4 | [
{
"content": "Let $a$ and $b$ be angles such that $\\sin (a + b) = \\frac{3}{4}$ and $\\sin (a - b) = \\frac{1}{2}.$ Find $\\frac{\\tan a}{\\tan b}.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To find $\\frac{\\tan a}{\\tan b}$, we use... | null | null | null | null | null |
6,507 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Jordan wants to divide his $\frac{48}{5}$ pounds of chocolate into $4$ piles of equal weight. If he gives one of these piles to his friend Shaina, how many pounds of chocolate will Shaina get?
Please reason step by step, and put your final answer within \boxed... | true | \frac{12}{5} | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nJordan wants to divide his $\\frac{48}{5}$ pounds of chocolate into $4$ piles of equal weight. If he gives one of these piles to his friend Shaina, how many pounds of chocolate will Shaina get?\nPlease reason step by step, and put your final answer with... | null | 3 | [
{
"content": "Jordan wants to divide his $\\frac{48}{5}$ pounds of chocolate into $4$ piles of equal weight. If he gives one of these piles to his friend Shaina, how many pounds of chocolate will Shaina get?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... | Level 2 | Prealgebra | We need to divide the amount of chocolate Jordan has by the the number of piles, so our expression is $\frac{48}{5} \div 4$. Recall that dividing is the same as multiplying by the reciprocal. Therefore, $\frac{48}{5} \div 4$ is the same thing as $\frac{48}{5} \cdot \frac{1}{4}.$ We can rewrite $\frac{48}{5} \cdot \fr... | [
""
] | [
{
"content": "Jordan wants to divide his $\\frac{48}{5}$ pounds of chocolate into $4$ piles of equal weight. If he gives one of these piles to his friend Shaina, how many pounds of chocolate will Shaina get?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
4,259 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
| true | -37 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nIf $x - 5... | null | 3 | [
{
"content": "What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "If $x - 5$ is a factor of $P(x)$, then $P(5) = 0$. Substitute $x ... | Level 2 | Intermediate Algebra | From the Factor Theorem, if $x-5$ is a factor of $P$, then $P(5) = 0$. Using this we have
$$5^3+2(5^2)+5c+10 = 0.$$Solving for $c$ gives us $c = \boxed{-37}$. | [
"-37"
] | [
{
"content": "What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "If $x - 5$ is a factor of $P(x)$, then $P(5) = 0$. Substitute $x ... |
4,435 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Let $0 \le a,$ $b,$ $c,$ $d \le 1.$ Find the possible values of the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}.\]
Please reason step by step, and put your final answer within \boxed{}.<|eot_id... | false | [2\sqrt{2},4] | [
false,
false,
false,
false,
false,
true,
false,
false
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nLet $0 \\le a,$ $b,$ $c,$ $d \\le 1.$ Find the possible values of the expression\n\\[\\sqrt{a^2 + (1 - b)^2} + \\sqrt{b^2 + (1 - c)^2} + \\sqrt{c^2 + (1 - d)^2} + \\sqrt{d^2 + (1 - a)^2}.\\]\nPlease reason step by step, and put your final answer within ... | true | 4 | [
{
"content": "Let $0 \\le a,$ $b,$ $c,$ $d \\le 1.$ Find the possible values of the expression\n\\[\\sqrt{a^2 + (1 - b)^2} + \\sqrt{b^2 + (1 - c)^2} + \\sqrt{c^2 + (1 - d)^2} + \\sqrt{d^2 + (1 - a)^2}.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... | null | null | null | null | null |
6,010 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header... | true | 90 | [
false,
false,
true,
false
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nA rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|e... | true | 4 | [
{
"content": "A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Let's denote the lengths of the diagon... | null | null | null | null | null |
7,187 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
If $\sin x = 3 \cos x,$ then what is $\sin x \cos x$?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
We can solve the given equation by squaring both sides to eliminate the square r... | true | \frac{3}{10} | [
true,
true,
true,
true
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nIf $\\sin x = 3 \\cos x,$ then what is $\\sin x \\cos x$?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nWe can solve the given equation by squaring both sides to elimina... | true | 4 | [
{
"content": "If $\\sin x = 3 \\cos x,$ then what is $\\sin x \\cos x$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We can solve the given equation by squaring both sides to eliminate the square root:\n\\[ \\sin^2 x = 9 \\cos^2 x \\]\n\n... | null | null | null | null | null |
1,630 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
| true | 6 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the length of the segment of the number line whose endpoints satisfy $|x-\\sqrt[5]{16}|=3$?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nWe solve for $x$ step-b... | null | 3 | [
{
"content": "What is the length of the segment of the number line whose endpoints satisfy $|x-\\sqrt[5]{16}|=3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We solve for $x$ step-by-step:\n\\begin{align*}\n|x-\\sqrt[5]{16}|&=3\\\\\n\\Ri... | Level 5 | Algebra | We have $x-\sqrt[5]{16}=3$ or $x-\sqrt[5]{16}=-3$. Our two solutions are $x=\sqrt[5]{16}+3$ and $x=\sqrt[5]{16}-3$. These are the endpoints of the segment, and we need to find the length, so take the larger minus the smaller: $(\sqrt[5]{16}+3)-(\sqrt[5]{16}-3)=\boxed{6}$. | [
""
] | [
{
"content": "What is the length of the segment of the number line whose endpoints satisfy $|x-\\sqrt[5]{16}|=3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We solve for $x$ step-by-step:\n\\begin{align*}\n|x-\\sqrt[5]{16}|&=3\\\\\n\\Ri... |
3,829 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Suppose $z$ is a complex number such that $z^2 = 24-32i$. Find $|z|$.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
Given that $z^2 = 24 - 32i$, we are asked to find the magnitude... | false | 2\sqrt{10} | [
false,
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false,
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false,
false,
false
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nSuppose $z$ is a complex number such that $z^2 = 24-32i$. Find $|z|$.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nGiven that $z^2 = 24 - 32i$, we are asked to find th... | true | 4 | [
{
"content": "Suppose $z$ is a complex number such that $z^2 = 24-32i$. Find $|z|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Given that $z^2 = 24 - 32i$, we are asked to find the magnitude of $z$, denoted by $|z|$, which is the squar... | null | null | null | null | null |
2,429 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
In the geoboard shown, the points are evenly spaced vertically and horizontally. Segment $AB$ is drawn using two points, as shown. Point $C$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $ABC$ being isosce... | false | 6 | [
false,
false,
false,
false,
false,
false,
false,
true
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nIn the geoboard shown, the points are evenly spaced vertically and horizontally. Segment $AB$ is drawn using two points, as shown. Point $C$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $ABC$ being... | true | 4 | [
{
"content": "In the geoboard shown, the points are evenly spaced vertically and horizontally. Segment $AB$ is drawn using two points, as shown. Point $C$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $ABC$ being isosceles? [asy]\ndraw((0,0)--(0,6)--(6,6)-... | null | null | null | null | null |
3,486 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the greatest integer value of $x$ such that $\frac{x^2 + 2x + 5}{x-3}$ is an integer?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
We can write $x^2 + 2x + 5 = (x-3)(x+5)$... | false | 23 | [
false,
false,
false,
false,
true,
true,
false,
false
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the greatest integer value of $x$ such that $\\frac{x^2 + 2x + 5}{x-3}$ is an integer?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nWe can write $x^2 + 2x + 5 =... | true | 4 | [
{
"content": "What is the greatest integer value of $x$ such that $\\frac{x^2 + 2x + 5}{x-3}$ is an integer?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We can write $x^2 + 2x + 5 = (x-3)(x+5)$ because $x^2 + 2x + 5 - (x-3)(x+5) = x^2 + ... | null | null | null | null | null |
3,236 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
How many different triangles can be formed having a perimeter of 7 units if each side must have integral length?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
To find the different... | false | 2 | [
true,
false,
false,
false,
false,
false,
false,
false
] | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nHow many different triangles can be formed having a perimeter of 7 units if each side must have integral length?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nTo find th... | true | 4 | [
{
"content": "How many different triangles can be formed having a perimeter of 7 units if each side must have integral length?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To find the different triangles with integral sides and a perimete... | null | null | null | null | null |
2,916 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}... | true | 10\sqrt{2} | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form.\nPlease reason step by step, and put your final answer within ... | null | 3 | [
{
"content": "The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... | Level 4 | Geometry | The length of the median to the hypotenuse is half the length of the hypotenuse, so the hypotenuse is $10\cdot2=20$ units long. Since the right triangle is isosceles, the length of a leg is $20/\sqrt{2}=\boxed{10\sqrt{2}}$ units. | [
""
] | [
{
"content": "The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
... |
5,799 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Calculate $[(12^{12} \div 12^{11})^2 \cdot 4^2] \div 2^4$.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
| true | 144 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nCalculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nTo calculate the expression, let's follow the order of operat... | null | 3 | [
{
"content": "Calculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To calculate the expression, let's follow the order of operations (PEMDAS):\n\nThe given expression is $[(12^{12} \\d... | Level 2 | Prealgebra | Remembering proper order of operations, first simplify the terms in the parentheses using the quotient of powers rule:
$12^{12} \div 12^{11} = 12^{12-11} = 12$ so that the expression becomes \[(12^2 \cdot 4^2) \div 2^4 = 12^2 \cdot 4^2 \div 2^4.\] Since $4^2 = 4 \cdot 4 = 2 \cdot 2 \cdot 2 \cdot 2 = 2^4$, we have \[1... | [
""
] | [
{
"content": "Calculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To calculate the expression, let's follow the order of operations (PEMDAS):\n\nThe given expression is $[(12^{12} \\d... |
2,560 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symmetry, resulting in the third square ... | true | {CBAD} | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symmetry, resulting in the third ... | null | 3 | [
{
"content": "The first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symmetry, resulting in the third square in position CBAD. If the pattern of a... | Level 3 | Geometry | If we extend the pattern, we note that the rearrangements of the vertices return to the original order after four steps: ABCD $\rightarrow$ DABC $\rightarrow$ CBAD $\rightarrow$ DCBA $\rightarrow$ ABCD. Thus, since the sequence repeats, we know that every fourth rearrangement will be of the form DCBA. The 2007th square... | [
""
] | [
{
"content": "The first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symmetry, resulting in the third square in position CBAD. If the pattern of a... |
5,099 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Marcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder does he get when he divides $a+b$ by 15?
Please reason step by step, and put your final answer with... | false | 1 | [
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nMarcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder does he get when he divides $a+b$ by 15?\nPlease reason step by step, and put your final ans... | true | 4 | [
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"content": "Marcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder does he get when he divides $a+b$ by 15?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... | null | null | null | null | null |
3,125 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
In $\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is
[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=... | false | 9 | [
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nIn $\\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=... | true | 4 | [
{
"content": "In $\\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B); draw(A-... | null | null | null | null | null |
6,640 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find $4^4 \cdot 5^4$
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>
| true | 160000 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nFind $4^4 \\cdot 5^4$\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_id|>\n\nWe need to find the value of $4^4 \\cdot 5^4$.\n\nStep 1: Calculate the value of $4^4$ and $5^4$ sepa... | null | 3 | [
{
"content": "Find $4^4 \\cdot 5^4$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We need to find the value of $4^4 \\cdot 5^4$.\n\nStep 1: Calculate the value of $4^4$ and $5^4$ separately.\n\nTo find $4^4$, we can rewrite it as $(2^2)^4 ... | Level 2 | Prealgebra | Finding $4^4$ and $5^4$ individually, then multiplying the result, is quite hard and time-consuming. So, let's list out the full product, and see if we can regroup anything: $$ 4^4 \cdot 5^4 = (4 \cdot 4 \cdot 4 \cdot 4) \cdot (5 \cdot 5 \cdot 5 \cdot 5) $$Now, multiplication is associative, so we can group a $5$ with... | [
"6"
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"content": "Find $4^4 \\cdot 5^4$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We need to find the value of $4^4 \\cdot 5^4$.\n\nStep 1: Calculate the value of $4^4$ and $5^4$ separately.\n\nTo find $4^4$, we can rewrite it as $(2^2)^4 ... |
6,714 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
The 5 a.m. temperatures for seven consecutive days were $-7^{\circ}$, $-4^{\circ}$, $-4^{\circ}$, $-5^{\circ}$, $1^{\circ}$, $3^{\circ}$ and $2^{\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?
Please reason step by step, an... | true | -2 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nThe 5 a.m. temperatures for seven consecutive days were $-7^{\\circ}$, $-4^{\\circ}$, $-4^{\\circ}$, $-5^{\\circ}$, $1^{\\circ}$, $3^{\\circ}$ and $2^{\\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?\nPlease reason s... | null | 3 | [
{
"content": "The 5 a.m. temperatures for seven consecutive days were $-7^{\\circ}$, $-4^{\\circ}$, $-4^{\\circ}$, $-5^{\\circ}$, $1^{\\circ}$, $3^{\\circ}$ and $2^{\\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?\nPlease reason step by step, and put your final answer withi... | Level 2 | Prealgebra | To add these up quickly, a trick is to simply cancel out one of the $-4$ and the 1 and 3, because those will sum to 0, and then add up 2 and $-5$ to make $-3$. Pairing that with $-7$ to make $-10$, and then $-14$. Dividing by 7, we get an average of $\boxed{-2}$. | [
"10"
] | [
{
"content": "The 5 a.m. temperatures for seven consecutive days were $-7^{\\circ}$, $-4^{\\circ}$, $-4^{\\circ}$, $-5^{\\circ}$, $1^{\\circ}$, $3^{\\circ}$ and $2^{\\circ}$ Celsius. What is the mean 5 a.m. temperature for the week in degrees Celsius?\nPlease reason step by step, and put your final answer withi... |
3,989 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the maximum number of real roots to a polynomial of the form
\[x^n + x^{n - 1} + \dots + x + 1 = 0,\]where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_header_i... | true | 1 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nFind the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer.\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>assistant<... | null | 3 | [
{
"content": "Find the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To find the maximum number of real r... | Level 4 | Intermediate Algebra | If $x^n + x^{n - 1} + \dots + x + 1 = 0,$ then
\[(x - 1)(x^n + x^{n - 1} + \dots + x + 1) = 0,\]which expands as $x^{n + 1} - 1 = 0.$ Then $x^{n + 1} = 1.$ The only possible real roots of this equation are $x = 1$ and $x = -1.$
Note that $x = 1$ cannot be a real root of
\[x^n + x^{n - 1} + \dots + x + 1 = 0,\]but $x... | [
""
] | [
{
"content": "Find the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To find the maximum number of real r... |
3,885 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
What is the largest integer less than $\log_2 \frac{2}{1} + \log_2 \frac{3}{2} + \cdots + \log_2 \frac{2009}{2008} + \log_2 \frac{2010}{2009}$?
Please reason step by step, and put your final answer within \boxed{}.<|eot_id|><|start_header_id|>assistant<|end_hea... | true | 10 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWhat is the largest integer less than $\\log_2 \\frac{2}{1} + \\log_2 \\frac{3}{2} + \\cdots + \\log_2 \\frac{2009}{2008} + \\log_2 \\frac{2010}{2009}$?\nPlease reason step by step, and put your final answer within \\boxed{}.<|eot_id|><|start_header_id|>... | null | 3 | [
{
"content": "What is the largest integer less than $\\log_2 \\frac{2}{1} + \\log_2 \\frac{3}{2} + \\cdots + \\log_2 \\frac{2009}{2008} + \\log_2 \\frac{2010}{2009}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "\\log_2 \\frac{2}{1} + \\l... | Level 4 | Intermediate Algebra | Recall that $\log_2 \frac{x}{y} = \log_2 x - \log_2 y$. Applying this identity to each term in the sum, we find that the sum is equal to to $(\log_2 2 - \log_2 1) + (\log_2 3 - \log_2 2) + \cdots + (\log_2 2010 - \log_2 2009)$. Most of the intermediate terms cancel out; the expression eventually evaluates to
\[\log_2 2... | [
""
] | [
{
"content": "What is the largest integer less than $\\log_2 \\frac{2}{1} + \\log_2 \\frac{3}{2} + \\cdots + \\log_2 \\frac{2009}{2008} + \\log_2 \\frac{2010}{2009}$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "\\log_2 \\frac{2}{1} + \\l... |
3,601 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\]
Please reason step by step, and put your fina... | false | 2 | [
false,
false,
false,
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false,
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false,
false
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"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nFind the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$\n\\[\\sqrt{\\frac{a}{b + c + d}} + \\sqrt{\\frac{b}{a + c + d}} + \\sqrt{\\frac{c}{a + b + d}} + \\sqrt{\\frac{d}{a + b + c}} > m.\\]\nPlease reason step by step... | true | 4 | [
{
"content": "Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$\n\\[\\sqrt{\\frac{a}{b + c + d}} + \\sqrt{\\frac{b}{a + c + d}} + \\sqrt{\\frac{c}{a + b + d}} + \\sqrt{\\frac{d}{a + b + c}} > m.\\]\nPlease reason step by step, and put your final answer within \\boxed{}... | null | null | null | null | null |
2,306 | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum o... | true | 5 | null | [
"<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWe are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that th... | null | 3 | [
{
"content": "We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 10?\nPlea... | Level 2 | Counting & Probability | For each integer $x$ in the list besides 5, the integer $10-x$ is also in the list. So, for each of these integers, removing $x$ reduces the number of pairs of distinct integers whose sum is 10. However, there is no other integer in list that can be added to 5 to give 10, so removing 5 from the list will not reduce t... | [
"indeed5.Pleaseprovideastep-by-stepsolutionforthefirstquestionputyourfinalanswerintheonlyboxprovided.bf{Pleasedonotincludethelistofpairsofnumbersthatsumto10,asitwasprovidedinthepreviousresponse.}"
] | [
{
"content": "We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 10?\nPlea... |
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