data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If the six-digit number $\\overline{201 a b 7}$ is divisible by 11 and 13, then the two-digit number $\\overline{a b}$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 39949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $x^2 + 13x + 30$ can be written as $(x + a)(x + b),$ and the expression $x^2 + 5x - 50$ written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "9 seeds are divided among three pits, labeled A, B, and C, with each pit containing 3 seeds. Each seed has a 0.5 probability of germinating. If at least one seed in a pit germinates, then that pit does not need to be replanted; if no seeds in a pit germinate, then that pit need... | math | {
"style": "rule",
"ground_truth": "0.330"
} | {
"split": "train",
"index": 39951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tree doubled its height every year until it reached a height of 32 feet at the end of 6 years. What was the height of the tree, in feet, at the end of 3 years? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\text{ feet}"
} | {
"split": "train",
"index": 39952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \\(\\frac{10}{3}\\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 39953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive integers can be represented as a difference of two distinct members of the set $\\{1, 2, 3, \\ldots, 14, 15, 16 \\}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is equidistant from the coordinate axes if the vertical distance from the point to the $x$-axis is equal to the horizontal distance from the point to the $y$-axis. The point of intersection of the vertical line $x = a$ with the line with equation $3x + 8y = 24$ is equid... | math | {
"style": "rule",
"ground_truth": "-\\frac{144}{55}"
} | {
"split": "train",
"index": 39955,
"solution": "If $a > 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $a$. If $a < 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $-a$. In each case, there are exactly two points on the vertical line with equatio... |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a > 1"
} | {
"split": "train",
"index": 39956,
"solution": "1. **Understanding the Problem**: We are given an inequality involving the absolute values $|x-4|+|x-3|<a$ and need to determine the range of $a$ for which this inequality holds for some real number $x$.\n\n2. **Geometric Interpretation**: Consider a number line wi... |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = x^2 + px + q \\) and \\( g(x) = x^2 + rx + s \\) be two distinct quadratic polynomials where the \\( x \\)-coordinate of the vertex of \\( f \\) is a root of \\( g \\), and the \\( x \\)-coordinate of the vertex of \\( g \\) is a root of \\( f \\), also both \\( ... | math | {
"style": "rule",
"ground_truth": "-200"
} | {
"split": "train",
"index": 39957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:\n$$\\begin{array}{c@{}c@{}c@{}c@{}c@{}c}\n&&8&6&7&4_h \\\\\n&+&4&3&2&9_h \\\\\n\\cline{2-6}\n&1&3&0&0&3_h.\n\\end{array}$$ Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which number is greater than 0.7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 39959,
"solution": "Each of \\( 0.07, -0.41, 0.35, \\) and \\(-0.9\\) is less than 0.7. The number 0.8 is greater than 0.7."
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=20, B C=10, C A=15$. Let $I$ be the incenter of $A B C$, and let $B I$ meet $A C$ at $E$ and $C I$ meet $A B$ at $F$. Suppose that the circumcircles of $B I F$ and $C I E$ meet at a point $D$ different from $I$. Find the length of the tangent... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{30}"
} | {
"split": "train",
"index": 39960,
"solution": "Solution 1: Let $O=A I \\cap(A E F)$. We claim that $O$ is the circumcenter of $D E F$. Indeed, note that $\\angle E D F=\\angle E C I+\\angle F B I=\\frac{\\angle B+\\angle C}{2}=\\frac{\\angle E O F}{2}$, and $O E=O F$, so the claim is proven. Now note that the c... |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of\n\\[\\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 39961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ have the sums of the first $n$ terms as $S_{n}$ and $T_{n}$, respectively. It is known that $\\frac{{S}_{n}}{{T}_{n}}=\\frac{7n+2}{n+3}$. Find $\\frac{{a}_{7}}{{b}_{7}}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{93}{16}"
} | {
"split": "train",
"index": 39962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan \\left (\\operatorname{arccot} \\frac{4}{7} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 39963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\\triangle AED$ is twice the perimeter of $\\triangle BEC$. Find $AB$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39964,
"solution": "1. **Setup and Diagram**: Points $A, B, C, D$ are collinear with $AB = CD = x$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. We need to find $x$ given that the perimeter of $\\triangle AED$ is twice the perimeter of $\\triangle BEC$.\n\n2. **Prop... |
deepscaler | [
{
"role": "user",
"content": "There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat ... | math | {
"style": "rule",
"ground_truth": "100/3"
} | {
"split": "train",
"index": 39965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an acute triangle with circumcircle $\\omega$ . Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$ , respectively. Lines $BD$ and $CE$ intersect $\\omega$ again at points $P \\neq B$ and $Q \\neq C$ . Suppo... | math | {
"style": "rule",
"ground_truth": "\\sqrt{23}"
} | {
"split": "train",
"index": 39966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation\n\\[g(x) + g(y) = 2m^2\\]holds. What is $g(2187)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 39967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the foci of the ellipse\n\\[\\frac{x^2}{45} + \\frac{y^2}{5} = 9.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{10}"
} | {
"split": "train",
"index": 39968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate \\( \\frac{2}{1} \\times \\frac{2}{3} \\times \\frac{4}{3} \\times \\frac{4}{5} \\times \\frac{6}{5} \\times \\frac{6}{7} \\times \\frac{8}{7} \\). Express the answer in decimal form, accurate to two decimal places. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "1.67"
} | {
"split": "train",
"index": 39969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $\\rm{A}03_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 in order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2563"
} | {
"split": "train",
"index": 39970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.\n\nWhat is the ratio of the areas of the original hexagon to the cut-out shape? Let's think st... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"split": "train",
"index": 39971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Alice is 1.6 meters tall, can reach 50 centimeters above her head, and the ceiling is 3 meters tall, find the minimum height of the stool in centimeters needed for her to reach a ceiling fan switch located 15 centimeters below the ceiling. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 39972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the positive divisors of 18. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 39973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\\left(k_1\\right) + d\\left(k_2\\right)+ \\cdots + d\\left(k_m\\right),$$ where $1 = k_1 < k_2 < \\cdots < k_m = n$ are all divisors of the number $n$. We call an i... | math | {
"style": "rule",
"ground_truth": "1, 3, 18, 36"
} | {
"split": "train",
"index": 39974,
"solution": "\nTo find all almost perfect numbers, we first consider the function \\( f(n) \\). For a given positive integer \\( n \\), we define \\( f(n) \\) as:\n\n\\[\nf(n) = d(k_1) + d(k_2) + \\cdots + d(k_m),\n\\]\n\nwhere \\( 1 = k_1 < k_2 < \\cdots < k_m = n \\) are all ... |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 39975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 39976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of irises to roses in Nikki's garden is 2:5. She currently has 25 roses. She is planning to add 20 more roses and enough irises to keep the same ratio. How many irises will she have in total after this addition? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It fo... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 39978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of the angles of a pentagon are in the ratio of 3:3:3:4:5. What is the number of degrees in the measure of the largest angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"split": "train",
"index": 39979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $[a]$ represent the greatest integer less than or equal to $a$. Determine the largest positive integer solution to the equation $\\left[\\frac{x}{7}\\right]=\\left[\\frac{x}{8}\\right]+1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 39980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 39981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 39982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The s... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 39983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears? Let's think ste... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 39984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "397"
} | {
"split": "train",
"index": 39985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, $m\\angle A = 28^{\\circ}$, $m\\angle B = 74^\\circ$ and $m\\angle C = 26^{\\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x + y$? [asy]\nsize(150);\ndraw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));\ndraw((0,... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 39986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $97 \\times 97$ in your head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9409"
} | {
"split": "train",
"index": 39987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top ve... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 39988,
"solution": "Go to 2020 AMC 10A #19, and connect all of the centers of the faces on the dodecahedron to get the icosahedron. The answer is $\\boxed{810}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{250}{7}"
} | {
"split": "train",
"index": 39989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_464$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $D$ is a point on the side $AB$ of $\\triangle ABC$, and $\\overrightarrow{CD} = \\frac{1}{3}\\overrightarrow{AC} + \\lambda \\cdot \\overrightarrow{BC}$, determine the value of the real number $\\lambda$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 39991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are integers with $4<a<b<22$. If the average (mean) of the numbers $4, a, b, 22$ is 13, how many possible pairs $(a, b)$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39992,
"solution": "Since the average of the four numbers $4, a, b, 22$ is 13, then $\\frac{4+a+b+22}{4}=13$ and so $4+a+b+22=52$ or $a+b=26$. Since $a>4$ and $a$ is an integer, then $a \\geq 5$. Since $a+b=26$ and $a<b$, then $a$ is less than half of 26, or $a<13$. Since $a$ is an in... |
deepscaler | [
{
"role": "user",
"content": "A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\\angle LKN = \\angle ACB$... | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 39993,
"solution": "\nWe are given a triangle \\( ABC \\) with a circle that touches the side \\( AC \\) at its midpoint \\( M \\), passes through the vertex \\( B \\), and intersects \\( AB \\) and \\( BC \\) at \\( K \\) and \\( L \\), respectively. The point \\( N \\) is located on... |
deepscaler | [
{
"role": "user",
"content": "For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for e... | math | {
"style": "rule",
"ground_truth": "4910"
} | {
"split": "train",
"index": 39994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, shown below, $\\cos{C}=\\frac{9\\sqrt{130}}{130}$. Find $AC$.\n\n[asy]\ndraw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));\ndraw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));\nlabel(\"$A$\",(0,0),W);\nlabel(\"$B$\",(7,0),E);\nlabel(\"$C$\... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_{n}\\}$ is an arithmetic sequence, with the sum of its first $n$ terms denoted as $S_{n}$, and $a_{4}=-3$, choose one of the following conditions as known: <br/>$(Ⅰ)$ The arithmetic sequence $\\{a_{n}\\}$'s general formula; <br/>$(Ⅱ)$ The minimum value of $S_{n... | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 39996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On side \\(BC\\) and on the extension of side \\(AB\\) through vertex \\(B\\) of triangle \\(ABC\\), points \\(M\\) and \\(K\\) are located, respectively, such that \\(BM: MC = 4: 5\\) and \\(BK: AB = 1: 5\\). Line \\(KM\\) intersects side \\(AC\\) at point \\(N\\). Find the ra... | math | {
"style": "rule",
"ground_truth": "5/24"
} | {
"split": "train",
"index": 39997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( \\mathrm{S} = \\{1, 2, 3, \\ldots, 10\\} \\). The subset \\( \\mathrm{A} \\) of \\( \\mathrm{S} \\) satisfies \\( \\mathrm{A} \\cap \\{1, 2, 3\\} \\neq \\emptyset \\) and \\( \\mathrm{A} \\cup \\{4, 5, 6\\} \\neq \\mathrm{S} \\). Find the number of such subsets ... | math | {
"style": "rule",
"ground_truth": "888"
} | {
"split": "train",
"index": 39998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7 = 0.$ Given that $P(x)$ is a cubic polynomial such that $P(a) = b + c,$ $P(b) = a + c,$ $P(c) = a + b,$ and $P(a + b + c) = -16,$ find $P(x).$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "2x^3 + 6x^2 + 9x + 11"
} | {
"split": "train",
"index": 39999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the polynomial \\[ x^3 - 3x^2 - 12x + 9, \\] where \\(x\\) is the positive number such that \\(x^2 - 3x - 12 = 0\\) and \\(x \\neq -2\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-23"
} | {
"split": "train",
"index": 40000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $\\Omega$ has radius 13. Circle $\\omega$ has radius 14 and its center $P$ lies on the boundary of circle $\\Omega$. Points $A$ and $B$ lie on $\\Omega$ such that chord $A B$ has length 24 and is tangent to $\\omega$ at point $T$. Find $A T \\cdot B T$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 40001,
"solution": "Let $M$ be the midpoint of chord $A B$; then $A M=B M=12$ and Pythagoras on triangle $A M O$ gives $M O=5$. Note that $\\angle A O M=\\angle A O B / 2=\\angle A P B=\\angle A P T+\\angle T P B$ or $\\tan (\\angle A O M)=\\tan (\\angle A P T+\\angle T P B)$. Applyin... |
deepscaler | [
{
"role": "user",
"content": "Let $x, y, a \\in \\mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\\frac{3}{x} + \\frac{a}{y}$ is $6\\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\\frac{1}{x} + \\frac{2}{y} = 1$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 40002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the sum $\\left[\\log _{2} 1\\right]+\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} 2002\\right]$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17984"
} | {
"split": "train",
"index": 40003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B plan to meet between 8:00 and 9:00 in the morning, and they agreed that the person who arrives first will wait for the other for 10 minutes before leaving on their own. Calculate the probability that they successfully meet. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{36}"
} | {
"split": "train",
"index": 40004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right triangle $ABC$, where $\\angle B = \\angle C$, the length of $AC$ is $8\\sqrt{2}$. Calculate the area of triangle $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 40005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passes through $(2,2,1)$ and $(5,1,-2).$ A point on this line has an $x$-coordinate of 4. Find the $z$-coordinate of the point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 40006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ has $AB = BC = CD$, $m\\angle ABC = 70^\\circ$ and $m\\angle BCD = 170^\\circ$. What is the degree measure of $\\angle BAD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 40007,
"solution": "1. **Assign the unknown and draw necessary diagonals**: Let $\\angle BAD = x$. Draw diagonals $BD$ and $AC$. Let $I$ be the intersection of diagonals $BD$ and $AC$.\n\n2. **Analyze the isosceles triangles**: Since $AB = BC = CD$, triangles $\\triangle ABC$ and $\\t... |
deepscaler | [
{
"role": "user",
"content": "Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number... | math | {
"style": "rule",
"ground_truth": "850"
} | {
"split": "train",
"index": 40008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of positive integers $m, n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1) \\text{ and } (3, 3)"
} | {
"split": "train",
"index": 40009,
"solution": "\nTo solve the problem, we need to find all pairs of positive integers \\( (m, n) \\) such that \\( 9^{|m-n|} + 3^{|m-n|} + 1 \\) is divisible by both \\( m \\) and \\( n \\) simultaneously.\n\nLet's denote \\( d = |m-n| \\). The expression becomes \\( f(d) = 9^d +... |
deepscaler | [
{
"role": "user",
"content": "On a $4 \\times 4 \\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2.1"
} | {
"split": "train",
"index": 40010,
"solution": "1. **Identify the vertices and their relationships**: In a $4 \\times 4 \\times 3$ rectangular parallelepiped, let's assume $D$ is at the origin $(0,0,0)$, and $A$, $B$, $C$ are at $(4,0,0)$, $(0,4,0)$, and $(0,0,3)$ respectively. These vertices are adjacent to $D$... |
deepscaler | [
{
"role": "user",
"content": "A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. ... | math | {
"style": "rule",
"ground_truth": "183"
} | {
"split": "train",
"index": 40011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{\\tan ^{2}\\left(20^{\\circ}\\right)-\\sin ^{2}\\left(20^{\\circ}\\right)}{\\tan ^{2}\\left(20^{\\circ}\\right) \\sin ^{2}\\left(20^{\\circ}\\right)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 40012,
"solution": "If we multiply top and bottom by $\\cos ^{2}\\left(20^{\\circ}\\right)$, the numerator becomes $\\sin ^{2}\\left(20^{\\circ}\\right) \\cdot(1-\\cos ^{2} 20^{\\circ})=\\sin ^{4}\\left(20^{\\circ}\\right)$, while the denominator becomes $\\sin ^{4}\\left(20^{\\circ}\... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$ , consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$ . For $n<100$ , find the largest value of $h_n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 40013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $p$ and $q$ are inversely proportional. If $p=25$ when $q=6$, find the value of $p$ when $q=15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 40014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45486"
} | {
"split": "train",
"index": 40015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse with its focus on the $y$-axis $\\frac{x^2}{m^2} + \\frac{y^2}{4} = 1$ ($m > 0$) and eccentricity $e = \\frac{1}{2}$, where $A$ is the right vertex of the ellipse and $P$ is any point on the ellipse. Find the maximum value of $|PA|$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 40016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A prism has 15 edges. How many faces does the prism have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 40017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Car A and Car B start simultaneously from points $A$ and $B$ respectively, traveling towards each other. The initial speed ratio of car A to car B is 5:4. Shortly after departure, car A has a tire blowout, stops to replace the tire, and then resumes the journey, increasing its ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 40018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the matrix $\\mathbf{A}$ has an inverse and $(\\mathbf{A} - 2 \\mathbf{I})(\\mathbf{A} - 4 \\mathbf{I}) = \\mathbf{0},$ then find\n\\[\\mathbf{A} + 8 \\mathbf{A}^{-1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6 & 0 \\\\ 0 & 6 \\end{pmatrix}"
} | {
"split": "train",
"index": 40019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:\n\\[ x = \\sqrt{29} + \\frac{121}{{\\sqrt{29}+\\frac{121}{{\\sqrt{29}+\\frac{121}{{\\sqrt{29}+\\frac{121}{{\\sqrt{29}+\\frac{121}{x}}}}}}}}}.\\] Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "513"
} | {
"split": "train",
"index": 40020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\left( \\frac{1}{3}\\right)^{ax^2-4x+3}$,\n\n$(1)$ If $a=-1$, find the intervals of monotonicity for $f(x)$;\n\n$(2)$ If $f(x)$ has a maximum value of $3$, find the value of $a$;\n\n$(3)$ If the range of $f(x)$ is $(0,+\\infty)$, find the range of va... | math | {
"style": "rule",
"ground_truth": "\\{0\\}"
} | {
"split": "train",
"index": 40021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.\nA) $\\frac{25}{128}$\nB) $\\frac{25(5 + 2\\sqrt{5})}{128}$\nC) $\\frac{25(5-... | math | {
"style": "rule",
"ground_truth": "\\frac{25(5-2\\sqrt{5})}{128}"
} | {
"split": "train",
"index": 40022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "215"
} | {
"split": "train",
"index": 40023,
"solution": "Let our answer be $n$. Write $n = 42a + b$, where $a, b$ are positive integers and $0 \\leq b < 42$. Then note that $b, b + 42, ... , b + 42(a-1)$ are all primes.\n If $b$ is $0\\mod{5}$, then $b = 5$ because $5$ is the only prime divisible by $5$. We get $n = 215$... |
deepscaler | [
{
"role": "user",
"content": "Find the numerical value of $k$ for which\n\\[\\frac{7}{x + y} = \\frac{k}{x + z} = \\frac{11}{z - y}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 40024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\\mathbb R^3$ . A plane $\\mathcal P$ is tangent to $\\mathcal S$ at the point $(x_0,y_0,z_0)$ , where $x_0$ , $y_0$ , and $z_0$ are all positive. Suppose the intersection of plane $\\ma... | math | {
"style": "rule",
"ground_truth": "40/21"
} | {
"split": "train",
"index": 40025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new sequence, the first term is \\(a_1 = 5000\\) and the second term is \\(a_2 = 5001\\). Furthermore, the values of the remaining terms are designed so that \\(a_n + a_{n+1} + a_{n+2} = 2n\\) for all \\( n \\geq 1 \\). Determine \\(a_{1000}\\). Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "5666"
} | {
"split": "train",
"index": 40026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is inscribed in a circle of radius 1. Find the perimeter of the square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"split": "train",
"index": 40027,
"solution": "The square has diagonal length 2, so side length $\\sqrt{2}$ and perimeter $4 \\sqrt{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? Let'... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 40028,
"solution": "Let $p(n)$ denote the product of the distinct proper divisors of $n$. A number $n$ is nice in one of two instances:\nIt has exactly two distinct prime divisors.\n If we let $n = pq$, where $p,q$ are the prime factors, then its proper divisors are $p$ and $q$, and $... |
deepscaler | [
{
"role": "user",
"content": "Given a \"double-single\" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 40029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the cube root of \\( m \\) is a number in the form \\( n + r \\), where \\( n \\) is a positive integer and \\( r \\) is a positive real number less than \\(\\frac{1}{1000}\\). When \\( m \\) is the smallest positive integer satisfying the above condition, find the v... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 40030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have 10 distinguishable socks in my drawer: 4 white, 4 brown, and 2 blue. In how many ways can I choose a pair of socks, provided that I get two socks of different colors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 40031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{7}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 40032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with an equilateral triangle as shown in diagram a, each side of the triangle is divided into three equal parts, and at the middle segment, new equilateral triangles are constructed outward, as shown in diagram b, forming a \"snowflake hexagon.\" Next, each of the 12 s... | math | {
"style": "rule",
"ground_truth": "40/27"
} | {
"split": "train",
"index": 40033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any integer $n\\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \\ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:\n[list][*] $a_i+b_i+c_i=n$ for all $i=1, \\ldots, N(n... | math | {
"style": "rule",
"ground_truth": "\\left\\lfloor \\frac{2n}{3} \\right\\rfloor + 1"
} | {
"split": "train",
"index": 40034,
"solution": "\nTo determine \\( N(n) \\), the maximum number of triples \\((a_i, b_i, c_i)\\) where each \\( a_i, b_i, c_i \\) are nonnegative integers satisfying the conditions:\n\n1. \\( a_i + b_i + c_i = n \\) for all \\( i = 1, \\ldots, N(n) \\),\n2. If \\( i \\neq j \\) th... |
deepscaler | [
{
"role": "user",
"content": "Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two-digit number is subtracted from the three-digit number, what is the smallest difference? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "269"
} | {
"split": "train",
"index": 40035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a standard deck of 52 cards, what is the probability that the top four cards include only cards from the black suits ($\\spadesuit$ and $\\clubsuit$)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{276}{4998}"
} | {
"split": "train",
"index": 40036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the point M ($4\\sqrt {2}$, $\\frac{\\pi}{4}$) in the polar coordinate system, the polar coordinate equation of the curve C is $\\rho^2 = \\frac{12}{1+2\\sin^{2}\\theta}$. Point N moves on the curve C. Establish a rectangular coordinate system with the pole as the coordin... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 40037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x-y=15$ and $xy=4$, what is the value of $x^2+y^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "233"
} | {
"split": "train",
"index": 40038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\( t(0)-t\\left(\\frac{\\pi}{5}\\right)+t\\left(\\frac{2\\pi}{5}\\right)-t\\left(\\frac{3\\pi}{5}\\right)+\\ldots+t\\left(\\frac{8\\pi}{5}\\right)-t\\left(\\frac{9\\pi}{5}\\right) \\), where \\( t(x) = \\cos 5x + * \\cos 4x + * \\cos 3x + * \\cos 2x + *^2 \\cos x + * \... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 40039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 40040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 40041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangle $A B C D$, let $X$ and $Y$ be points on $A B$ and $B C$, respectively. Suppose the areas of the triangles $\\triangle A X D$, $\\triangle B X Y$, and $\\triangle D Y C$ are 5, 4, and 3, respectively. Find the area of $\\triangle D X Y$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{21}"
} | {
"split": "train",
"index": 40042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Tranquility Town, the streets are all $30$ feet wide and the blocks they enclose are rectangles with lengths of side $500$ feet and $300$ feet. Alice walks around the rectangle on the $500$-foot side of the street, while Bob walks on the opposite side of the street. How many... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 40043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four carpenters were hired by a guest to build a yard. The first carpenter said: \"If only I alone were to build the yard, I would complete it in one year.\" The second carpenter said: \"If only I alone were to build the yard, I would complete it in two years.\" The third carpe... | math | {
"style": "rule",
"ground_truth": "175.2"
} | {
"split": "train",
"index": 40044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 40045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes 24 minutes for Jana to walk one mile. At that rate, how far in miles will she walk in 10 minutes? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.4\\mbox{ miles}"
} | {
"split": "train",
"index": 40046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the share of gold in the currency structure of the National Welfare Fund (NWF) as of December 1, 2022, using one of the following methods:\n\nFirst Method:\n\na) Find the total amount of NWF funds allocated in gold as of December 1, 2022:\n\\[GOLD_{22} = 1388.01 - 41.... | math | {
"style": "rule",
"ground_truth": "8.2"
} | {
"split": "train",
"index": 40047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let be the average value obtained if a teacher is picked at random and the number o... | math | {
"style": "rule",
"ground_truth": "-13.5"
} | {
"split": "train",
"index": 40048,
"solution": "1. **Calculate $t$ (average number of students per teacher):**\n The average number of students per teacher, $t$, is calculated by taking the sum of the number of students in each class and dividing by the number of teachers. Since each teacher teaches one class,... |
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