data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\\frac1{a^2+b}+\\frac1{b^2+c}+\\frac1{c^2+a}$ can be written as $\\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .\n\n*Proposed by M... | math | {
"style": "rule",
"ground_truth": "301"
} | {
"split": "train",
"index": 18349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle ABC$ is similar to $\\triangle DEF$ . What is the number of centimeters in the length of $\\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,0)--(8,-2)--(5,4)--cycle);\nlabel(\"8cm\",(2.5,2),NW);\nlabel(\"5cm\",(6.1,1),NE);\nd... | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 18350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all 3-digit numbers with all digits in the set $\\{1,2,3,4,5,6,7\\}$ (so in particular, all three digits are nonzero). For how many elements $\\overline{a b c}$ of $S$ is it true that at least one of the (not necessarily distinct) 'digit cycles' $\\overlin... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 18351,
"solution": "Since the value of each digit is restricted to $\\{1,2, \\ldots, 7\\}$, there is exactly one digit representative of each residue class modulo 7. Note that $7 \\mid \\overline{a b c}$ if and only if $100 a+10 b+c \\equiv 0(\\bmod 7)$ or equivalently $2 a+3 b+c \\eq... |
deepscaler | [
{
"role": "user",
"content": "Consider the set $\\{2, 7, 12, 17, 22, 27, 32\\}$. Calculate the number of different integers that can be expressed as the sum of three distinct members of this set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$. If $a_1 = -3$, $a_{k+1} = \\frac{3}{2}$, and $S_k = -12$, then calculate the value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $\\frac{1}{5} + \\frac{2}{5} + \\frac{3}{5} + \\dots + \\frac{9}{5} + \\frac{10}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of ... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 18355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\\triangle PQR$. Find the sum of the slope and $y$-intercept of this line. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-20}{3}"
} | {
"split": "train",
"index": 18356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the sum as a common fraction: $.2 + .04 + .006 + .0008 + .00001.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24681}{100000}"
} | {
"split": "train",
"index": 18357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 18358,
"solution": "1. **Define Variables:**\n Let $m$ represent the total amount of money Brianna has. Let $c$ represent the cost of one CD, and let $n$ be the total number of CDs she wants to buy.\n\n2. **Establish Relationships:**\n According to the problem, Brianna uses one fi... |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a-b+c=7$, $b-c+d=8$, $c-d+a=4$, and $d-a+b=1$. Determine the value of $2a+2b+2c+2d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 18359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a set of positive integers \"conspiratorial\" if no three of them are pairwise relatively prime. What is the largest number of elements in any \"conspiratorial\" subset of the integers $1$ to $16$ ? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sequence $\\{a_n\\}$ satisfy that the sum of the first $n$ terms $S_n$ fulfills $S_n + a_1 = 2a_n$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence. Find the value of $a_1 + a_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 18361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=4\\cos x\\cos \\left(x- \\frac {\\pi}{3}\\right)-2$. \n$(I)$ Find the smallest positive period of the function $f(x)$. \n$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $\\left[- \\frac {\\pi}{6}, \\frac {\\pi}{4}\\ri... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 18362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two players, \\(A\\) and \\(B\\), play rock-paper-scissors continuously until player \\(A\\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABCD is an isosceles trapezoid with \\(AB = CD\\). \\(\\angle A\\) is acute, \\(AB\\) is the diameter of a circle, \\(M\\) is the center of the circle, and \\(P\\) is the point of tangency of the circle with the side \\(CD\\). Denote the radius of the circle as \\(x\\). Then \\... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \\sqrt[3]{61{,}629{,}875}$ and $n = \\sqrt[7]{170{,}859{,}375}$ . Given that $m$ and $n$ are both integers, compute $100m+n$ .\n\n*Proposed by Evan Chen* Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "39515"
} | {
"split": "train",
"index": 18365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Caroline can make eleven lassis out of two mangoes. How many lassis can she make out of twelve mangoes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 18366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 15$ and $FQ = 34$. The centroids of $\\triangle{EFQ}$, $\\triangle{FGQ}$, $\\triangle{GHQ}$, and $\\triangle{HEQ}$ are the vertices of a convex quadrilateral. What is the area of that quad... | math | {
"style": "rule",
"ground_truth": "\\frac{1600}{9}"
} | {
"split": "train",
"index": 18367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ represent the smallest integer that satisfies the following conditions:\n$\\frac n2$ is a perfect square.\n$\\frac n3$ is a perfect cube.\n$\\frac n5$ is a perfect fifth.\nHow many divisors does $n$ have that are not multiples of 10?\n Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "242"
} | {
"split": "train",
"index": 18368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a new shape by adding an eighth unit cube to the previously described configuration of seven cubes. Place this new cube adjacent to one of the six outlying cubes from the central cube. What is the ratio of the volume in cubic units to the surface area in square units ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{33}"
} | {
"split": "train",
"index": 18369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{5}{33}$ is expressed in decimal form, what digit is in the 92nd place to the right of the decimal point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ be a chord of a circle $\\omega$, and let $P$ be a point on the chord $\\overline{AB}$. Circle $\\omega_1$ passes through $A$ and $P$ and is internally tangent to $\\omega$. Circle $\\omega_2$ passes through $B$ and $P$ and is internally tangent to $\\omega... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 18371,
"solution": "Connect $AQ,QB$, since $\\angle{AO_1P}=\\angle{AOB}=\\angle{BO_2P}$, so $\\angle{AQP}=\\frac{\\angle{AO_1P}}{2}=\\angle{BQP}=\\frac{\\angle{BO_2P}}{2}, \\angle{AQB}=\\angle{AOB}$ then, so $A,O,Q,B$ are concyclic\n We let $\\angle{AO_1P}=\\angle{AOB}=\\angle{BO_2P}=... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$. The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18372,
"solution": "Notice that for $\\frac{a^3b^3+1}{a^3+b^3}$ to be maximized, $\\frac{ab+1}{a+b}$ has to be maximized. We simplify as above to $2ab + 2 < 3a + 3b$, which is $(a-\\frac{3}{2})(b-\\frac{3}{2}) < \\frac{5}{4}$. To maximize, $a$ has to be as close to $b$ as possible, ma... |
deepscaler | [
{
"role": "user",
"content": "For $\\mathbf{v} = \\begin{pmatrix} 1 \\\\ y \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 9 \\\\ 3 \\end{pmatrix}$,\n\\[\\text{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} -6 \\\\ -2 \\end{pmatrix}.\\]Find $y$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "-23"
} | {
"split": "train",
"index": 18373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=5$, and $|r-s|=6$. What is the sum of all possible values of $|p-s|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ 0 \\\\ -1 \\end{pmatrix}.$ Find the vector $\\mathbf{v}$ that satisfies $\\mathbf{v} \\times \\mathbf{a} = \\mathbf{b} \\times \\mathbf{a}$ and $\\mathbf{v} \\times \... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 3 \\\\ 1 \\\\ -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 18375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "345"
} | {
"split": "train",
"index": 18376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( y = \\tan(2015x) - \\tan(2016x) + \\tan(2017x) \\) has how many zeros in the interval \\([0, \\pi]\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 18377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 18378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAbsent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: taking one pill daily. These pills are effective in $90\\%$ of cases, and in $2\\%$ of cases, there is a side effect - absence of absent-mindedness, if it existed.\n\nAnother doctor pres... | math | {
"style": "rule",
"ground_truth": "0.69"
} | {
"split": "train",
"index": 18379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of cells that need to be marked in a $7 \\times 7$ grid so that in each vertical or horizontal $1 \\times 4$ strip there is at least one marked cell? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A card is secretly removed from a standard deck of 52 cards. Then two cards are dealt at random from the now 51-card deck. What is the probability that both cards drawn are from the same suit and the second card is a face card? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{50}"
} | {
"split": "train",
"index": 18381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man has $10,000 to invest. He invests $4000 at 5% and $3500 at 4%. In order to have a yearly income of $500, he must invest the remainder at: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.4\\%"
} | {
"split": "train",
"index": 18382,
"solution": "1. **Calculate the total amount invested at known rates:**\n The man invests $\\$4000$ at 5% and $\\$3500$ at 4%. We need to calculate the income from these investments.\n \\[\n \\text{Income from } \\$4000 = \\$4000 \\times \\frac{5}{100} = \\$200\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "8.8\\%"
} | {
"split": "train",
"index": 18383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $g$ is defined on the set of integers and satisfies \\[g(n)= \\begin{cases} n-4 & \\mbox{if }n\\ge 1010 \\\\ g(g(n+7)) & \\mbox{if }n<1010. \\end{cases}\\] Find $g(77)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 18384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 18385,
"solution": "Let $c$ be the number of correct answers Olivia had, and $w$ be the number of incorrect answers. Since there are 10 problems in total, we have:\n\\[ c + w = 10 \\]\n\nFor each correct answer, Olivia gains 5 points, and for each incorrect answer, she loses 2 points.... |
deepscaler | [
{
"role": "user",
"content": "Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 18386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semicircle is inscribed in a quarter circle. What fraction of the quarter circle is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The table below lists the air distances in kilometers between several cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than 9000 kilometers? Express your answer as a common fraction.\n\n\\begin{t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 18388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of a set of $60$ numbers is $42$. If three numbers of the set, namely $40$, $50$, and $60$, are discarded, the arithmetic mean of the remaining set of numbers is:\n**A)** 41.3 \n**B)** 41.4 \n**C)** 41.5 \n**D)** 41.6 \n**E)** 41.7 Let's think step by st... | math | {
"style": "rule",
"ground_truth": "41.6"
} | {
"split": "train",
"index": 18389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them? Let'... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 18390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the third quadrant, if $\\sin^{4}{\\theta} + \\cos^{4}{\\theta} = \\frac{5}{9}$, find the value of $\\sin{2\\theta}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 18391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the points \\( 0, z, \\frac{1}{z}, z+\\frac{1}{z} \\) form a parallelogram with an area of \\( \\frac{35}{37} \\). If the real part of \\( z \\) is greater than 0, find the minimum value of \\( \\left| z + \\frac{1}{z} \\right| \\). Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{74}}{37}"
} | {
"split": "train",
"index": 18392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the asymptotes of a hyperbola has equation $y=3x.$ The foci of the hyperbola have the same $x-$coordinate, which is $5.$ Find the equation of the other asymptote of the hyperbola, giving your answer in the form \"$y = mx + b$\". Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "y = -3x + 30"
} | {
"split": "train",
"index": 18393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of real solutions $(x,y,z,w)$ to the system of equations:\n\\begin{align*}\nx &= z+w+zwx, \\\\ \ny &= w+x+wxy, \\\\\nz &= x+y+xyz, \\\\\nw &= y+z+yzw.\n\\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number $x$ is randomly chosen from the interval $[-1, 1]$. What is the probability that the value of $\\cos \\frac{\\pi x}{2}$ lies between $0$ and $\\frac{1}{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 18395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The time right now is 9 o'clock. On a 12-hour clock, what time will it be 2004 hours from now? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a new operation $\\star$ such that for positive integers $a, b, c$, $a \\star b \\star c = \\frac{a \\times b + c}{a + b + c}$. Calculate the value of $4 \\star 8 \\star 2$.\n**A)** $\\frac{34}{14}$ **B)** $\\frac{16}{7}$ **C)** $\\frac{17}{7}$ **D)** $\\frac{32}{14}$ **... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{7}"
} | {
"split": "train",
"index": 18397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has side length $1$ ; circle $\\Gamma$ is centered at $A$ with radius $1$ . Let $M$ be the midpoint of $BC$ , and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\\Gamma$ . Compute $MN$ .\n\n*2018 CCA Math Bonanza Individual R... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 18398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Through the midpoint $D$ of the base of an isosceles triangle, a line is drawn at an angle of $30^{\\circ}$ to this base, on which the angle $\\angle ACB$ intercepts a segment $EF$. It is known that $ED = 6$ and $FD = 4$. Find the height of the triangle drawn to the base. Let's... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly pers... | math | {
"style": "rule",
"ground_truth": "1896"
} | {
"split": "train",
"index": 18400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a store where all items cost an integer number of rubles, there are two special offers:\n1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.\n2) A customer who buys e... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 18401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a set of integers \"spacy\" if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1, 2,\n3, \\dots, 12\\}$, including the empty set, are spacy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 18402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\\qquad$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10829"
} | {
"split": "train",
"index": 18403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of quadrilateral PQRS, made from two similar right-angled triangles PQR and PRS, is given that the length of PQ is 3, the length of QR is 4, and ∠PRQ = ∠PSR. Find the perimeter of PQRS. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 18404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation \\(2008=1111+444+222+99+77+55\\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.\n\ni) Find a similar decomposition for the number 20... | math | {
"style": "rule",
"ground_truth": "1111 + 777 + 66 + 55"
} | {
"split": "train",
"index": 18405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the store \"Everything for School,\" three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the numb... | math | {
"style": "rule",
"ground_truth": "390"
} | {
"split": "train",
"index": 18406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n$$\n\\lim _{h \\rightarrow 0} \\frac{\\sin \\left(\\frac{\\pi}{3}+4 h\\right)-4 \\sin \\left(\\frac{\\pi}{3}+3 h\\right)+6 \\sin \\left(\\frac{\\pi}{3}+2 h\\right)-4 \\sin \\left(\\frac{\\pi}{3}+h\\right)+\\sin \\left(\\frac{\\pi}{3}\\right)}{h^{4}}\n$$ Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 18407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \\textup{lcm}(a,b)>\\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.\n\nFun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortil... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression \\(\\frac{3}{10}+\\frac{3}{100}+\\frac{3}{1000}\\) is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.333"
} | {
"split": "train",
"index": 18409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$ -coordinate, and $B$ and $C$ have the same $x$ -coordinate. Find the greatest value that $N$ can attain con... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"split": "train",
"index": 18410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number \\(2 * 0 * 1 * 6 * 0 * 2 *\\), each of the 6 asterisks needs to be replaced with any of the digits \\(0, 2, 4, 5, 7, 9\\) (digits can be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "5184"
} | {
"split": "train",
"index": 18411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $|x| + x + y = 14$ and $x + |y| - y = 16,$ find $x + y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 18413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Experimental Primary School football team has 42 members, and 17 new members joined. How many members are in the football team now? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 18414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has center $(-10, -4)$ and has radius $13$. Another circle has center $(3, 9)$ and radius $\\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18415,
"solution": "1. **Identify the equations of the circles**: \n The first circle has center $(-10, -4)$ and radius $13$. Its equation is:\n \\[\n (x+10)^2 + (y+4)^2 = 169.\n \\]\n The second circle has center $(3, 9)$ and radius $\\sqrt{65}$. Its equation is:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of eight-digit numbers where the product of the digits equals 3375. The answer must be presented as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 18416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular quadrilateral frustum with lateral edges \\(A A_{1}, B B_{1}, C C_{1}, D D_{1}\\), the side length of the upper base \\(A_{1} B_{1} C_{1} D_{1}\\) is 1, and the side length of the lower base is 7. A plane passing through the edge \\(B_{1} C_{1}\\) perpendicular to ... | math | {
"style": "rule",
"ground_truth": "\\frac{38\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 18417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\) with \\( AB = 8 \\) and \\( AC = 10 \\), the incenter \\( I \\) is reflected across side \\( AB \\) to point \\( X \\) and across side \\( AC \\) to point \\( Y \\). Given that segment \\( XY \\) bisects \\( AI \\), compute \\( BC^2 \\). (The incenter \\... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 18418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of ordered pairs of integers $(m,n)$ for which $mn \\ge 0$ and\n\n$m^3 + n^3 + 99mn = 33^3$\nis equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 18419,
"solution": "1. **Recall the Factorization Identity**: We start by recalling the identity for the sum of cubes:\n \\[\n x^3 + y^3 + z^3 - 3xyz = \\frac{1}{2} \\cdot (x + y + z) \\cdot ((x - y)^2 + (y - z)^2 + (z - x)^2)\n \\]\n This identity holds for any real numbers $... |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c \\in R^{+}$ and $a + b + c = 1$, find the maximum value of $\\sqrt{4a + 1} + \\sqrt{4b + 1} + \\sqrt{4c + 1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 18420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arctan (\\tan 65^{\\circ}-2 \\tan 40^{\\circ})$. (Express your answer in degrees as an angle between $0^{\\circ}$ and $180^{\\circ}$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25^{\\circ}"
} | {
"split": "train",
"index": 18421,
"solution": "First Solution: We have $\\tan 65^{\\circ}-2 \\tan 40^{\\circ}=\\cot 25^{\\circ}-2 \\cot 50^{\\circ}=\\cot 25^{\\circ}-\\frac{\\cot ^{2} 25^{\\circ}-1}{\\cot 25^{\\circ}}=\\frac{1}{\\cot 25^{\\circ}}=\\tan 25^{\\circ}$. Therefore, the answer is $25^{\\circ}$. Secon... |
deepscaler | [
{
"role": "user",
"content": "All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18422,
"solution": "Let the total number of marbles Marcy has be denoted by $n$. According to the problem, the distribution of marbles is as follows:\n- Blue marbles: $\\frac{1}{3}n$\n- Red marbles: $\\frac{1}{4}n$\n- Green marbles: $6$\n\nWe need to find the number of yellow marbles,... |
deepscaler | [
{
"role": "user",
"content": "There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.\n\nAll the students in this class sat at a round table.\n\n- Several students said: \"There is exactly one troublemaker next to me.\"\n- All other stu... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two of the roots of\n\\[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\\]are $-2$ and 3. Find the third root. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 18424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each s... | math | {
"style": "rule",
"ground_truth": "2047"
} | {
"split": "train",
"index": 18425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point on the hyperbola $\\frac{x^{2}}{16} - \\frac{y^{2}}{20} = 1$, and let $F_{1}$ and $F_{2}$ be the left and right foci, respectively. If $|PF_{1}| = 9$, then find $|PF_{2}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sixteen is 64$\\%$ of what number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 18427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \\( A \\) such that, in any such set of numbers, each number does not exceed \\( A \\). Let's think ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest negative value of the expression \\( x - y \\) for all pairs of numbers \\((x, y)\\) satisfying the equation\n\n$$\n(\\sin x + \\sin y)(\\cos x - \\cos y) = \\frac{1}{2} + \\sin(x - y) \\cos(x + y)\n$$ Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 18429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers \\( n \\) are there such that \\( n \\) is a multiple of 4, and the least common multiple of \\( 4! \\) and \\( n \\) equals 4 times the greatest common divisor of \\( 8! \\) and \\( n \\)? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, C, D, E;\nA=(0,20.785);\nB=(0,0);\nC=(9,-5.196);\nD=(13.5,-2.598);\nE=(12,0);\ndraw(A--B--C--D--E--... | math | {
"style": "rule",
"ground_truth": "\\frac{189}{2}\\sqrt{3}"
} | {
"split": "train",
"index": 18431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = \\{1, 2, 3, 4, 5, 6, 7\\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "399"
} | {
"split": "train",
"index": 18432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z, w \\in [0,1] \\). Find the maximum value of \\( S = x^2 y + y^2 z + z^2 w + w^2 x - xy^2 - yz^2 - zw^2 - wx^2 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{27}"
} | {
"split": "train",
"index": 18433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in the blanks with appropriate numbers. \n4 liters 25 milliliters = ___ milliliters \n6.09 cubic decimeters = ___ cubic centimeters \n4.9 cubic decimeters = ___ liters ___ milliliters \n2.03 cubic meters = ___ cubic meters ___ cubic decimeters. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Give an example of a number $x$ for which the equation $\\sin 2017 x - \\operatorname{tg} 2016 x = \\cos 2015 x$ holds. Justify your answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 18435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anna flips an unfair coin 10 times. The coin has a $\\frac{1}{3}$ probability of coming up heads and a $\\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{5120}{19683}"
} | {
"split": "train",
"index": 18436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $(2c-a)\\cos B = b\\cos A$.\n1. Find angle $B$.\n2. If $b=6$ and $c=2a$, find the area of $\\triangle ABC$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 18437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the larger of the two distinct solutions to the equation $$x^2 - 11x - 42 = 0.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\[\n1.047. \\left(\\frac{\\sqrt{561^{2} - 459^{2}}}{4 \\frac{2}{7} \\cdot 0.15 + 4 \\frac{2}{7} : \\frac{20}{3}} + 4 \\sqrt{10}\\right) : \\frac{1}{3} \\sqrt{40}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 18439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_{n}\\}$, $a_{5}a_{8}=6$, $a_{3}+a_{10}=5$, then $\\frac{a_{20}}{a_{13}}=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has sidelengths $AB=1$ , $BC=\\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\\angle EDF = \\angle DFA = 90^{\\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed a... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 18441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer will be called \"sparkly\" if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers $2,3, \\ldots, 2003$ are sparkly? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18442,
"solution": "Suppose $n$ is sparkly; then its smallest divisor other than 1 is some prime $p$. Hence, $n$ has $p$ divisors. However, if the full prime factorization of $n$ is $p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{r}^{e_{r}}$, the number of divisors is $\\left(e_{1}+1\\right)\... |
deepscaler | [
{
"role": "user",
"content": "Given an odd function $f(x)$ defined on $\\mathbb{R}$ that satisfies $f(2-x) - f(x) = 0$, and $f(-1) = 1$, calculate the value of $f(1) + f(2) + f(3) + \\ldots + f(2010)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 18443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the function $f(x) = ax + \\frac{x}{x-1}$ where $x > 1$.\n(1) If $a > 0$, find the minimum value of the function $f(x)$.\n(2) If $a$ is chosen from the set \\{1, 2, 3\\} and $b$ is chosen from the set \\{2, 3, 4, 5\\}, find the probability that $f(x) > b$ always holds t... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 18444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{M}$ if it satisfies $\\mathbf{M} \\mathbf{i} = \\begin{pmatrix} 2 \\\\ 3 \\\\ -8 \\end{pmatrix},$ $\\mathbf{M} \\mathbf{j} = \\begin{pmatrix} 0 \\\\ 5 \\\\ -2 \\end{pmatrix},$ and $\\mathbf{M} \\mathbf{k} = \\begin{pmatrix} 7 \\\\ -1 \\\\ 4 \\end{pmatr... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 & 0 & 7 \\\\ 3 & 5 & -1 \\\\ -8 & -2 & 4 \\end{pmatrix}"
} | {
"split": "train",
"index": 18445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Aeroflot cashier must deliver tickets to five groups of tourists. Three of these groups live in the hotels \"Druzhba,\" \"Russia,\" and \"Minsk.\" The cashier will be given the address of the fourth group by the tourists from \"Russia,\" and the address of the fifth group b... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?\n\nA) 6 \nB) 10 \nC) 12 \nD) 18 \nE) 24 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marty and three other people took a math test. Everyone got a non-negative integer score. The average score was 20. Marty was told the average score and concluded that everyone else scored below average. What was the minimum possible score Marty could have gotten in order to de... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 18448,
"solution": "Suppose for the sake of contradiction Marty obtained a score of 60 or lower. Since the mean is 20, the total score of the 4 test takers must be 80. Then there exists the possibility of 2 students getting 0, and the last student getting a score of 20 or higher. If s... |
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