data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The triangle shown is an equilateral triangle with side length 12 cm. A side of the triangle is the diameter of the circle. If the sum of the areas of the two small shaded regions in square centimeters in simplest radical form is $a\\pi - b\\sqrt{c}$, what is $a+b+c$? [asy]\nim... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 34700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 34701,
"solution": "1. **Identify the larger triangle and its dimensions**: The problem describes a larger triangle with vertices at points A, B, and D. The base of this triangle (AD) is along Main Street and measures 6 miles. The height of this triangle (from point B to line AD) is along Brown Road and ... |
DeepScaleR | [
{
"role": "user",
"content": "The trip from Carville to Nikpath requires $4\\frac 12$ hours when traveling at an average speed of 70 miles per hour. How many hours does the trip require when traveling at an average speed of 60 miles per hour? Express your answer as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "5.25"
} | {
"index": 34702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The truncated right circular cone has a large base radius 8 cm and a small base radius of 4 cm. The height of the truncated cone is 6 cm. How many $\\text{cm}^3$ are in the volume of this solid? [asy]\n\nimport olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;\n\nd... | math | {
"style": "rule",
"ground_truth": "224\\pi"
} | {
"index": 34703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The twelve-sided figure shown has been drawn on $1 \\text{ cm}\\times 1 \\text{ cm}$ graph paper. What is the area of the figure in $\\text{cm}^2$? \n[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34704,
"solution": "To find the area of the twelve-sided figure, we can break it down into simpler shapes whose areas we can easily calculate. The figure is composed of several unit squares and triangles on a $1 \\text{ cm} \\times 1 \\text{ cm}$ grid.\n\n1. **Count the full unit squares**: \n - There ... |
DeepScaleR | [
{
"role": "user",
"content": "The two banners below:\n中华少年 杯赛联谊 切磋勾股\n炎黄子孙 惠州弘志 振兴中华\n\nEach character represents a non-zero natural number less than 25. Different characters represent different numbers, and the same characters represent the same number. It is known that the average value of these 34 charac... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 34705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The two circles pictured have the same center $C$. Chord $\\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\\overline{AD}$ has length $16$. What is the area between the two circles?"
}
] | math | {
"style": "rule",
"ground_truth": "64 \\pi"
} | {
"index": 34706,
"solution": "1. **Identify Key Information and Setup:**\n - The circles have the same center $C$.\n - Chord $\\overline{AD}$ is tangent to the inner circle at point $B$.\n - $AC = 10$ (radius of the outer circle).\n - Length of chord $\\overline{AD} = 16$.\n\n2. **Use the Tangency Property... |
DeepScaleR | [
{
"role": "user",
"content": "The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 34707,
"solution": "1. **Define the ages**: Let Jack's age be represented as $\\overline{ab} = 10a + b$ where $a$ and $b$ are the tens and units digits respectively. Similarly, Bill's age is $\\overline{ba} = 10b + a$.\n\n2. **Future ages**: In five years, Jack's age will be $10a + b + 5$ and Bill's age ... |
DeepScaleR | [
{
"role": "user",
"content": "The two figures shown are made of unit squares. What is the positive difference of the perimeters, in units?\n\n[asy]\ndraw((0,0)--(0,1)--(5,1)--(5,0)--cycle,linewidth(1));\ndraw((1,0)--(1,2)--(4,2)--(4,0),linewidth(1));\ndraw((2,-1)--(2,3)--(3,3)--(3,-1)--cycle,linewidth(1));\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The two lines $y = 4x - 19$ and $2x+y = 95$ intersect. What is the value of $x$ at the point of intersection?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The two roots of the equation $x^{2}-9x+18=0$ are the base and one of the legs of an isosceles triangle. Find the perimeter of this isosceles triangle."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 34711,
"solution": "To solve this problem, we first need to understand the possible outcomes when spinning the two spinners. Each spinner lands on one of the numbered sectors, and we are interested in the sums of these numbers.\n\n#### Step 1: Identify the numbers on each spinner\n- Spinner 1 has sectors... |
DeepScaleR | [
{
"role": "user",
"content": "The two squares shown share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n[asy] //code taken from thread for problem r... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"index": 34712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$\n[asy] //code taken from thread for ... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"index": 34713,
"solution": "Triangles $AOB$, $BOC$, $COD$, etc. are congruent by symmetry (you can prove it rigorously by using the power of a point to argue that exactly two chords of length $1$ in the circumcircle of the squares pass through $B$, etc.), and each area is $\\frac{\\frac{43}{99}\\cdot\\frac{1}{2}... |
DeepScaleR | [
{
"role": "user",
"content": "The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 34714,
"solution": "1. **Identify the probability of even and odd outcomes for each wheel:**\n - For the first wheel, the probability of landing on an even number is $\\frac{1}{4}$ (since one out of four sections is even), and the probability of landing on an odd number is $\\frac{3}{4}$ (since three o... |
DeepScaleR | [
{
"role": "user",
"content": "The union of sets $A$ and $B$ is $A \\cup B = \\left\\{a_{1}, a_{2}, a_{3}\\right\\}$. When $A \\neq B$, the pairs $(A, B)$ and $(B, A)$ are considered different. How many such pairs $(A, B)$ are there?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\) is \\( A \\cup B = \\left\\{a_{1}, a_{2}, a_{3}\\right\\} \\). When \\( A \\neq B \\), the pairs \\((A, B)\\) and \\((B, A)\\) are considered different. How many such pairs \\((A, B)\\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\) is \\( A \\cup B = \\{a_1, a_2, a_3\\} \\). When \\( A \\neq B \\), \\((A, B)\\) and \\((B, A)\\) are considered different pairs. How many such pairs \\((A, B)\\) exist?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\), \\( A \\cup B = \\{a_1, a_2, a_3\\} \\), and \\( A \\neq B \\). When \\( (A, B) \\) and \\( (B, A) \\) are considered as different pairs, how many such pairs \\( (A, B) \\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\), \\( A \\cup B = \\{a_1, a_2, a_3\\} \\). When \\( A \\neq B \\), pairs \\((A, B)\\) and \\((B, A)\\) are considered different. How many such pairs \\((A, B)\\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 34720,
"solution": "1. **Identify the nature of the problem**: We need to find the unit's digit of the product of any six consecutive positive whole numbers.\n\n2. **Consider the factors of the product**: Let's denote the smallest of these six consecutive numbers as $n$. Therefore, the six numbers are $n... |
DeepScaleR | [
{
"role": "user",
"content": "The units digit of $3^{1001} 7^{1002} 13^{1003}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34721,
"solution": "1. **Finding the units digit of $3^{1001}$:**\n - We observe the pattern of the units digits of powers of $3$ modulo $10$:\n - $3^1 \\equiv 3 \\pmod{10}$\n - $3^2 \\equiv 9 \\pmod{10}$\n - $3^3 \\equiv 27 \\equiv 7 \\pmod{10}$\n - $3^4 \\equiv 81 \\equiv 1 \\pmod{10}... |
DeepScaleR | [
{
"role": "user",
"content": "The units of length include , and the conversion rate between two adjacent units is ."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The unknown real numbers $x, y, z$ satisfy the equations $$\\frac{x+y}{1+z}=\\frac{1-z+z^{2}}{x^{2}-x y+y^{2}} ; \\quad \\frac{x-y}{3-z}=\\frac{9+3 z+z^{2}}{x^{2}+x y+y^{2}}$$ Find $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{14}"
} | {
"index": 34723,
"solution": "$\\sqrt[3]{14}$ Cross-multiplying in both equations, we get, respectively, $x^{3}+y^{3}=$ $1+z^{3}, x^{3}-y^{3}=27-z^{3}$. Now adding gives $2 x^{3}=28$, or $x=\\sqrt[3]{14}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The updated stem-and-leaf plot shows the duration of rides for each of the $21$ top-rated roller coasters. Each entry in the plot represents ride time, where, for example, $3 \\ 05$ means $3$ minutes, $5$ seconds. Convert this time to seconds to find the median of the data set.... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 34724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The user has three computer disks from companies $\\mathrm{K}$, $\\mathrm{L}$, and $\\mathrm{M}$, one disk from each of these companies, but the company stamps on the disks are absent. Two out of the three disks are defective. What is the probability that the defective disks ar... | math | {
"style": "rule",
"ground_truth": "0.4821"
} | {
"index": 34725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 34726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 34727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value $b^n$ has both $b$ and $n$ as positive integers less than or equal to 15. What is the greatest number of positive factors $b^n$ can have?"
}
] | math | {
"style": "rule",
"ground_truth": "496"
} | {
"index": 34728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of\n\\[\\frac{n}{2} + \\frac{18}{n}\\]is smallest for which positive integer $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where $ a$ , $ b$ , and $ c$ are digits. What is the value of $ 100a \\plus{} 10b \\plus{} c$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "709"
} | {
"index": 34730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $$\\frac {1}{\\tan 20^\\circ} - \\frac {1}{\\cos 10^\\circ}$$ is equal to \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 34731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $(\\sqrt{1+\\sqrt{1+\\sqrt{1}}})^{4}$ is:\n(a) $\\sqrt{2}+\\sqrt{3}$;\n(b) $\\frac{1}{2}(7+3 \\sqrt{5})$;\n(c) $1+2 \\sqrt{3}$;\n(d) 3 ;\n(e) $3+2 \\sqrt{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2 \\sqrt{2}"
} | {
"index": 34732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $1.000 + 0.101 + 0.011 + 0.001$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "1.113"
} | {
"index": 34733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $10^{\\log_{10}7}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34734,
"solution": "1. **Understanding the Expression**: The expression given is $10^{\\log_{10}7}$. This can be interpreted using the definition of logarithms. Recall that if $b^y = x$, then $\\log_b x = y$. \n\n2. **Applying the Definition of Logarithms**: Here, $\\log_{10}7 = x$ implies that $10^x = 7... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $2x+10$ is half of the value of $5x+30$. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 34735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $3x + 15$ is one third of the value of $6x + 45$. After finding $x$, subtract 5 from the result. What is the final value?"
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 34736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $999 + 999$ is"
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 34737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 34738,
"solution": "To solve the expression $[2 - 3(2 - 3)^{-1}]^{-1}$, we start by simplifying the innermost expression first.\n\n1. **Simplify $(2 - 3)^{-1}$:**\n \\[\n (2 - 3)^{-1} = (-1)^{-1} = -1\n \\]\n\n2. **Substitute back into the expression:**\n \\[\n 2 - 3(-1) = 2 + 3 = 5\n \\]\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 34739,
"solution": "To solve the expression $\\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}}$, we will evaluate it from the innermost fraction outward.\n\n1. **Evaluate the innermost fraction:**\n \\[\n 2 - \\frac{1}{2} = 2 - 0.5 = 1.5\n \\]\n So, the expression simplifies to:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\frac{1}{16}a^0+\\left (\\frac{1}{16a} \\right )^0- \\left (64^{-\\frac{1}{2}} \\right )- (-32)^{-\\frac{4}{5}}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34740,
"solution": "We start by simplifying each term in the expression $\\frac{1}{16}a^0+\\left (\\frac{1}{16a} \\right )^0- \\left (64^{-\\frac{1}{2}} \\right )- (-32)^{-\\frac{4}{5}}$.\n\n1. **Simplify $\\frac{1}{16}a^0$:**\n Since any non-zero number raised to the power of 0 is 1, we have $a^0 = 1$... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\frac{2^4 - 2}{2^3 - 1}$ is?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34741,
"solution": "We note that $2^3 = 2 \\times 2 \\times 2 = 8$ and $2^4 = 2^3 \\times 2 = 16$. Therefore, $\\frac{2^4 - 2}{2^3 - 1} = \\frac{16 - 2}{8 - 1} = \\frac{14}{7} = 2$. Alternatively, $\\frac{2^4 - 2}{2^3 - 1} = \\frac{2(2^3 - 1)}{2^3 - 1} = 2$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\frac{x}{2}$ is less than the value of $x^{2}$. The value of $x^{2}$ is less than the value of $x$. Which of the following could be a value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 34742,
"solution": "Since $x^{2}<x$ and $x^{2} \\geq 0$, then $x>0$ and so it cannot be the case that $x$ is negative. Thus, neither (D) nor (E) is the answer. Since $x^{2}<x$, then we cannot have $x>1$. This is because when $x>1$, we have $x^{2}>x$. Thus, (A) is not the answer and so the answer is (B) o... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\left(256\\right)^{.16}\\left(256\\right)^{.09}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34743,
"solution": "1. **Identify the operation and simplify the expression**: We are given the expression $\\left(256\\right)^{0.16}\\left(256\\right)^{0.09}$. Using the property of exponents that states when multiplying like bases you add the exponents, we can simplify this expression:\n \\[\n 256^... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\log_5{3125}$ is between which consecutive integers $c$ and $d$? Find $c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\log_{10}{17}$ is between the consecutive integers $a$ and $b$. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\log_{10}{28471}$ is between the consecutive integers $a$ and $b$. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\log_{10}{579}$ is between the consecutive integers $a$ and $b$. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\sin 210^\\circ$ is equal to $\\frac{-\\sqrt{3}}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 34748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\sqrt{3^{3}+3^{3}+3^{3}}$ is what?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34749,
"solution": "Since $3^{3}=3 \\times 3 \\times 3=3 \\times 9=27$, then $\\sqrt{3^{3}+3^{3}+3^{3}}=\\sqrt{27+27+27}=\\sqrt{81}=9$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\sqrt{73}$ is between two positive, consecutive integers. What is the product of these two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 34750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $\\tan {75}^{{o}}$ is $\\dfrac{\\sqrt{6}+\\sqrt{2}}{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "2+\\sqrt{3}"
} | {
"index": 34751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \\cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \\cos(a x)+4$ have for the same value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "4014"
} | {
"index": 34752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $x + x(x^x)$ when $x = 2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34753,
"solution": "1. **Substitute $x = 2$ into the expression**: We start by substituting $x = 2$ into the expression $x + x(x^x)$.\n \\[\n x + x(x^x) = 2 + 2(2^2)\n \\]\n\n2. **Evaluate the exponentiation**: Next, we calculate $2^2$.\n \\[\n 2^2 = 4\n \\]\n\n3. **Substitute back into the e... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $x$ at the intersection of $y=\\frac{8}{x^2+4}$ and $x+y=2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 34754,
"solution": "1. **Substitute for $y$ in the second equation:** \n Given the equations $y = \\frac{8}{x^2 + 4}$ and $x + y = 2$, we can substitute $y$ from the second equation into the first:\n \\[\n x + y = 2 \\implies y = 2 - x.\n \\]\n Substituting $y = 2 - x$ into $y = \\frac{8}{x^2 ... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $x$ is one-half the value of $y$, and the value of $y$ is one-fifth the value of $z$. If $z$ is 60, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 34756,
"solution": "Recall the identity $\\log_{a^n} b^{m} = \\frac{m}{n}\\log_{a} b$ (which is easily proven using exponents or change of base). Then this problem turns into \\[\\frac{20}{x}\\log_{2} 3 = \\frac{2020}{x+3}\\log_{2} 3\\] Divide $\\log_{2} 3$ from both sides. And we are left with $\\frac{2... |
DeepScaleR | [
{
"role": "user",
"content": "The value of $y$ varies inversely as $\\sqrt x$ and when $x=2$, $y=4$. What is $x$ when $y=1$?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 34757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( 2 \\frac{1}{10} + 3 \\frac{11}{100} \\) can be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "5.21"
} | {
"index": 34758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( 2 \\frac{1}{10} + 3 \\frac{11}{100} \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "5.21"
} | {
"index": 34759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( 333 + 33 + 3 \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "369"
} | {
"index": 34760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( 444 - 44 - 4 \\) is"
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"index": 34761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( \\frac{1}{2} + \\frac{2}{4} + \\frac{4}{8} + \\frac{8}{16} \\) is what?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34762,
"solution": "In the given sum, each of the four fractions is equivalent to \\( \\frac{1}{2} \\). Therefore, the given sum is equal to \\( \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} = 2 \\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\( a \\) is chosen such that the number of roots of the first equation \\( 4^{x} - 4^{-x} = 2 \\cos a x \\) is 2007. How many roots does the second equation \\( 4^{x} + 4^{-x} = 2 \\cos a x + 4 \\) have for the same \\( a \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "4014"
} | {
"index": 34763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\((4 + 44 + 444) \\div 4\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "123"
} | {
"index": 34764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(0.001 + 1.01 + 0.11\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "1.121"
} | {
"index": 34765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(0.9 + 0.99\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "1.89"
} | {
"index": 34766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(1 + 0.01 + 0.0001\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "1.0101"
} | {
"index": 34767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(2 \\frac{1}{10} + 3 \\frac{11}{100} + 4 \\frac{111}{1000}\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "9.321"
} | {
"index": 34768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(987 + 113 - 1000\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 34769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(\\frac{1^{2}-3^{2}+5^{2}-7^{2}+\\cdots+97^{2}-99^{2}}{1-3+5-7+\\cdots+97-99}\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 34770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of \\(\\frac{1}{1+\\frac{1}{1+\\frac{1}{2}}}\\) can be expressed as a simplified fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 34771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the expression \\[(2^{1004}+5^{1005})^2-(2^{1004}-5^{1005})^2\\]is $k\\cdot10^{1004}$ for some positive integer $k$. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the expression \\[(3^{1001}+4^{1002})^2-(3^{1001}-4^{1002})^2\\]is $k\\cdot12^{1001}$ for some positive integer $k$. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the expression $\\frac{\\sin 10^{\\circ}}{1-\\sqrt{3}\\tan 10^{\\circ}}$ can be simplified using trigonometric identities and calculated exactly."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the expression \\(10 - 10.5 \\div [5.2 \\times 14.6 - (9.2 \\times 5.2 + 5.4 \\times 3.7 - 4.6 \\times 1.5)]\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "9.3"
} | {
"index": 34775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the product \\(\\cos \\frac{\\pi}{15} \\cos \\frac{2 \\pi}{15} \\cos \\frac{3 \\pi}{15} \\cdots \\cos \\frac{7 \\pi}{15}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"index": 34776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of the quadratic polynomial $a(x^3 - x^2 + 3x) + b(2x^2 + x) + x^3 - 5$ when $x = 2$ is $-17$. What is the value of this polynomial when $x = -2$?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 34777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The value of x that satisfies the equation \\( x^{x^x} = 2 \\) is calculated."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 34778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of $a$ in the equation: $\\log_{10}(a^2 - 15a) = 2$ are:"
}
] | math | {
"style": "rule",
"ground_truth": "20, -5"
} | {
"index": 34779,
"solution": "1. **Convert the logarithmic equation to exponential form:**\n Given the equation $\\log_{10}(a^2 - 15a) = 2$, we can rewrite it in exponential form:\n \\[\n 10^2 = a^2 - 15a\n \\]\n Simplifying, we get:\n \\[\n 100 = a^2 - 15a\n \\]\n Rearranging the terms, we obtai... |
DeepScaleR | [
{
"role": "user",
"content": "The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 34780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of $f$, $g$, $h$ and $j$ are 5, 6, 7 and 8, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$ and $fj$?"
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 34781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are:"
}
] | math | {
"style": "rule",
"ground_truth": "9 and -7"
} | {
"index": 34782,
"solution": "To find the values of $k$ for which the quadratic equation $2x^2 - kx + x + 8 = 0$ has real and equal roots, we first simplify the equation:\n\\[ 2x^2 - kx + x + 8 = 0 \\]\n\\[ 2x^2 + (1-k)x + 8 = 0 \\]\n\nFor a quadratic equation $ax^2 + bx + c = 0$, the condition for the roots to be... |
DeepScaleR | [
{
"role": "user",
"content": "The values of $x$ and $y$ are always positive, and $x^2$ and $y$ vary inversely. If $y$ is 10 when $x$ is 2, then find $x$ when $y$ is 4000."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"index": 34783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0$, $2x+y+3=0$ may be found by solving:"
}
] | math | {
"style": "rule",
"ground_truth": "$y^{2}+10y-7=0$"
} | {
"index": 34784,
"solution": "1. **Substitute $x$ from the second equation into the first equation:**\n Given the equations:\n \\[ 2x^2 + 6x + 5y + 1 = 0 \\]\n \\[ 2x + y + 3 = 0 \\]\n Solve the second equation for $x$:\n \\[ 2x = -y - 3 \\]\n \\[ x = -\\frac{y+3}{2} \\]\n\n2. **Substitute $x = -\\frac... |
DeepScaleR | [
{
"role": "user",
"content": "The values of a function $f(x)$ are given below:\n\n\\begin{tabular}{|c||c|c|c|c|c|} \\hline $x$ & 3 & 4 & 5 & 6 & 7 \\\\ \\hline $f(x)$ & 10 & 17 & 26 & 37 & 50 \\\\ \\hline \\end{tabular}Evaluate $f^{-1}\\left(f^{-1}(50)\\times f^{-1}(10)+f^{-1}(26)\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of a function $f(x)$ are given in the table below.\n\n\\begin{tabular}{|c||c|c|c|c|c|} \\hline $x$ & 1 & 2 & 3 & 4 & 5 \\\\ \\hline $f(x)$ & 3 & 1 & 5 & 4 & 2 \\\\ \\hline\n\\end{tabular}If $f^{-1}$ exists, then what is $f^{-1}(f^{-1}(f^{-1}(1)))$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The values of a function $f(x)$ are given in the table below. \\begin{tabular}{|r||c|c|c|c|c|c|} \\hline $x$ & 1 & 2 & 3 & 5 & 8 & 13 \\\\ \\hline $f(x)$ & 3 & 13 & 8 & 1 & 0 & 5 \\\\ \\hline \\end{tabular}If $f^{-1}$ exists, what is $f^{-1}\\left(\\frac{f^{-1}(5) +f^{-1}(13)}... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The variables $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 24, their difference is 6. What is $b$ when $a$ equals 5?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The variables \\(a, b, c, d, e\\), and \\(f\\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that\n\n\\[\n\\begin{aligned}\n& a + b = c, \\\\\n& b + c = d, \\\\\n& c + e = f,\n\\end{aligned}\n\\]\n\nwhat is the value of \\(a + c + f\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 34789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The variables \\(a, b, c, d, e\\), and \\(f\\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that\n\\[\n\\begin{aligned}\n& a + b = c, \\\\\n& b + c = d, \\\\\n& c + e = f,\n\\end{aligned}\n\\]\nDetermine the value of \\(a + c + f\\)."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 34790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting vector."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 \\sqrt{2} \\\\ -\\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\end{pmatrix}"
} | {
"index": 34791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vector $\\vec{a} =(-1,3)$, $\\vec{b} =(3,-4)$, then the projection of vector $\\vec{a}$ in the direction of vector $\\vec{b}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 34792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},$ $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ -3 \\\\ 1 \\end{pmatrix},$ and $\\mathbf{c} = \\begin{pmatrix} 4 \\\\ 1 \\\\ -5 \\end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ su... | math | {
"style": "rule",
"ground_truth": "\\left( 2, -\\frac{13}{7}, -\\frac{4}{7} \\right)"
} | {
"index": 34793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vectors $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ 1 \\\\ -2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\\ 2 \\\\ -1 \\end{pmatrix}.$ There exist scalars $p,$ $q,$ and $r$ such that\n\\[\\begin{pmatrix} 4 \\\\ 1 \\\\ -4 \\end{pmatrix} = p \\mathbf{a} + q \\mathbf{... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{6}"
} | {
"index": 34794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vectors $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ satisfy $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{c}\\| = 2,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{c}) + \\mathbf{b} = \\mathbf{0}.\\]If $\\theta$ is the angle between $\\mathbf{a}$... | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"index": 34795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertex of a parabola is \\( O \\) and its focus is \\( F \\). When a point \\( P \\) moves along the parabola, find the maximum value of the ratio \\( \\left|\\frac{P O}{P F}\\right| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 34796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34797,
"solution": "1. **Identify the Vertex Formula**: The vertex form of a parabola given by $y = ax^2 + bx + c$ is $(h, k)$ where $h = -\\frac{b}{2a}$ and $k = c - \\frac{b^2}{4a}$. For the given parabola $y = x^2 - 8x + c$, we have $a = 1$ and $b = -8$.\n\n2. **Calculate the x-coordinate of the Verte... |
DeepScaleR | [
{
"role": "user",
"content": "The vertex of the parabola $y^2 = 4x$ is $O$, and the coordinates of point $A$ are $(5, 0)$. A line $l$ with an inclination angle of $\\frac{\\pi}{4}$ intersects the line segment $OA$ (but does not pass through points $O$ and $A$) and intersects the parabola at points $M$ and $... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"index": 34798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertex of the parabola described by the equation $y=-3x^2-30x-81$ is $(m,n)$. What is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 34799,
"solution": ""
} |
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