data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9444"
} | {
"split": "train",
"index": 20749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\sin (\\arcsin (0.4)+\\arcsin (0.5)) \\cdot \\sin (\\arcsin (0.5)-\\arcsin (0.4))$ where for $x \\in[-1,1]$, $\\arcsin (x)$ denotes the unique real number $y \\in[-\\pi, \\pi]$ such that $\\sin (y)=x$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "0.09 \\text{ OR } \\frac{9}{100}"
} | {
"split": "train",
"index": 20750,
"solution": "Use the difference of squares identity 1 to get $0.5^{2}-0.4^{2}=0.3^{2}=0.09=\\frac{9}{100}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $L$ be the intersection point of the diagonals $CE$ and $DF$ of a regular hexagon $ABCDEF$ with side length 2. Point $K$ is defined such that $\\overrightarrow{LK} = \\overrightarrow{AC} - 3 \\overrightarrow{BC}$. Determine whether point $K$ lies inside, on the boundary, or... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 20751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum $1 + 3 + 5 + \\cdots + 15 + 17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 20752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have that $3 \\cdot f(x) + 4 \\cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 20753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using five nines (9), arithmetic operations, and exponentiation, create the numbers from 1 to 13. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 20754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\\{1, 2, 3, 4, 5, 6\\}$. Find $P(7)$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "51840"
} | {
"split": "train",
"index": 20755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 90 children in a chess club. During a session, they were divided into 30 groups of 3 people each, and in each group, everyone played one game with everyone else. No other games were played. A total of 30 \"boy vs. boy\" games and 14 \"girl vs. girl\" games were played... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 20756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:\n\n| Number of Trees Planted | 5 | 6 | 7 |\n|--------------------------|-----|-----... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 20757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the operation defined as \\(a \\odot b \\odot c = a \\times b \\times c + (a \\times b + b \\times c + c \\times a) - (a + b + c)\\), calculate \\(1 \\odot 43 \\odot 47\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4041"
} | {
"split": "train",
"index": 20758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyrami... | math | {
"style": "rule",
"ground_truth": "1280"
} | {
"split": "train",
"index": 20759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \\frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{16}"
} | {
"split": "train",
"index": 20760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\\bigtriangleup ACE$. Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 20761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with its right focus at $F(c, 0)$. A circle centered at the origin $O$ with radius $c$ intersects the hyperbola in the first quadrant at point $A$. The tangent to the circle at point $A$ ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 20762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "April has four different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 20763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of eleven numbers is 32. If three numbers $x$, $y$, and $z$ are added to this list, the mean of the fourteen-member list becomes 45. What is the mean of $x$, $y$, and $z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{278}{3}"
} | {
"split": "train",
"index": 20764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 15 married couples among 30 people. Calculate the total number of handshakes that occurred among these people. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "301"
} | {
"split": "train",
"index": 20765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right $ \\triangle ABC$ has $ AB \\equal{} 3$ , $ BC \\equal{} 4$ , and $ AC \\equal{} 5$ . Square $ XYZW$ is inscribed in $ \\triangle ABC$ with $ X$ and $ Y$ on $ \\overline{AC}$ , $ W$ on $ \\overline{AB}$ , and $ Z$ on $ \\overline{BC}$ . What is the sid... | math | {
"style": "rule",
"ground_truth": "\\frac {60}{37}"
} | {
"split": "train",
"index": 20766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer $n$ quixotic if the value of $\\operatorname{lcm}(1,2,3, \\ldots, n) \\cdot\\left(\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\ldots+\\frac{1}{n}\\right)$ is divisible by 45 . Compute the tenth smallest quixotic integer. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "573"
} | {
"split": "train",
"index": 20767,
"solution": "Let $L=\\operatorname{lcm}(1,2,3, \\ldots, n)$, and let $E=L\\left(1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}\\right)$ denote the expression. In order for $n$ to be quixotic, we need $E \\equiv 0(\\bmod 5)$ and $E \\equiv 0(\\bmod 9)$. We consider these two c... |
deepscaler | [
{
"role": "user",
"content": "$A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 20768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers and $\\theta$ an angle such that $\\theta \\neq \\frac{\\pi}{2} n$ for any integer $n$. Suppose $$\\frac{\\sin \\theta}{x}=\\frac{\\cos \\theta}{y}$$ and $$\\frac{\\cos ^{4} \\theta}{x^{4}}+\\frac{\\sin ^{4} \\theta}{y^{4}}=\\frac{97 \\s... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 20769,
"solution": "From the first relation, there exists a real number $k$ such that $x=k \\sin \\theta$ and $y=k \\cos \\theta$. Then we have $$\\frac{\\cos ^{4} \\theta}{\\sin ^{4} \\theta}+\\frac{\\sin ^{4} \\theta}{\\cos ^{4} \\theta}=\\frac{194 \\sin \\theta \\cos \\theta}{\\sin... |
deepscaler | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 11 \\pmod{53}$, find $36^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 20770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n,$ $-90 \\le n \\le 90,$ such that $\\sin n^\\circ = \\cos 456^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 20771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Rosa's Rose Shop, a bouquet containing a dozen roses costs $\\$20$. If the price of a bouquet is directly proportional to the number of roses it contains, how many dollars will a bouquet of 39 roses cost? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 20772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all positive integers \\( N < 1000 \\) for which \\( N + 2^{2015} \\) is divisible by 257. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2058"
} | {
"split": "train",
"index": 20773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?\n\n[asy]\ndraw((0,0)--(3... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 20774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathbf{u}, \\mathbf{v},\\) and \\(\\mathbf{w}\\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:\n\\[(\\mathbf{u} \\times \\mathbf{v}) \\times \\mathbf{w} = \\frac{1}{2} \\|\\mathbf{v}\\| \\|\\mathbf{w}\\| \\mathbf{u}.\\]\nLet \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 20775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $555_6$, $55_6$ and $5_6$ in base $6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1103_6"
} | {
"split": "train",
"index": 20776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(AB = 2\\) and \\(AD = 1\\), point \\(P\\) is a moving point on side \\(DC\\) (including points \\(D\\) and \\(C\\)), and point \\(Q\\) is a moving point on the extension line of \\(CB\\) (including point \\(B\\)), such that \\(|\\overrightarrow{DP}| ... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 20777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has sides of lengths 40 units, 50 units, and 70 units. An altitude is dropped from the vertex opposite the side of length 70 units. Calculate the length of this altitude.\nA) $\\frac{40\\sqrt{7}}{7}$ units\nB) $\\frac{80\\sqrt{7}}{7}$ units\nC) $\\frac{120\\sqrt{7}}{... | math | {
"style": "rule",
"ground_truth": "\\frac{80\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 20778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{x}$, $g(x)=-x^{2}+2x+b(b\\in\\mathbb{R})$, denote $h(x)=f(x)- \\frac {1}{f(x)}$ \n(I) Determine the parity of $h(x)$ and write down the monotonic interval of $h(x)$, no proof required; \n(II) For any $x\\in[1,2]$, there exist $x_{1}$, $x_{2}\\in[1,... | math | {
"style": "rule",
"ground_truth": "e^{2}-1"
} | {
"split": "train",
"index": 20779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**The first term of a sequence is $2089$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $2089^{\\text{th}}$ term of the sequence?** Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 20780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p(x) = x^4 + ax^3 + bx^2 + cx + d \\), where \\( a, b, c, \\) and \\( d \\) are constants. Given \\( p(1) = 1993 \\), \\( p(2) = 3986 \\), \\( p(3) = 5979 \\), find \\( \\frac{1}{4} [p(11) + p(-7)] \\). Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "5233"
} | {
"split": "train",
"index": 20781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.\n(1) Find t... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $b$ that satisfies the equation $161_{b}+134_{b}=315_{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 20783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times must a die be rolled so that the probability of the inequality\n\n\\[ \\left| \\frac{m}{n} - \\frac{1}{6} \\right| \\leq 0.01 \\]\n\nis at least as great as the probability of the opposite inequality, where \\( m \\) is the number of times a specific face appears... | math | {
"style": "rule",
"ground_truth": "632"
} | {
"split": "train",
"index": 20784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of \\( m \\). If \\( m \\) is replaced by \\( m+10 \\), the new sequence has a mean of 24 and a median of \\( m+10 \\). If \\( m \\) is replaced by \\( m-8 \\), the new sequence has a m... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of 1200 people polled, $30\\%$ do not like radio, and $10\\%$ of the people who do not like radio also do not like music. How many people polled do not like both radio and music? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $DEF$ with $\\angle D = 90^\\circ$, we have $DE = 8$ and $DF = 15$. Find $\\cos F$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{17}"
} | {
"split": "train",
"index": 20787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \\cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 20788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "$20"
} | {
"split": "train",
"index": 20789,
"solution": "1. **Identify the total number of days and daily cost:**\n Al needs to take one green pill and one pink pill each day for two weeks. Since there are 14 days in two weeks, we first calculate the total daily cost of the pills:\n \\[\n \\text{Daily cost} = \\fra... |
deepscaler | [
{
"role": "user",
"content": "The quadratic $x^2-3x+9=x+41$ has two solutions. What is the positive difference between these solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 60 empty boxes $B_1,\\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.\nIn the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each sub... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 20791,
"solution": "To solve this problem, we need to find the smallest integer \\( n \\) such that Alice can always prevent Bob from winning regardless of how the game progresses. The setup is as follows:\n\n1. Alice and Bob are playing a game with 60 boxes, \\( B_1, B_2, \\ldots, B_... |
deepscaler | [
{
"role": "user",
"content": "The value of \\( 2 \\frac{1}{10} + 3 \\frac{11}{100} \\) can be calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.21"
} | {
"split": "train",
"index": 20792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Along an alley, 75 trees consisting of maples and larches were planted in a single row. It is known that there are no two maples with exactly 5 trees between them. What is the maximum number of maples that could have been planted along the alley? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 20793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary and James each sit in a row of 7 chairs. They choose their seats at random. What is the probability that they don't sit next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 20794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What integer value will satisfy the equation $$ 14^2 \\times 35^2 = 10^2 \\times (M - 10)^2 \\ ? $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 20795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $2b\\cos A+a=2c$, $c=8$, $\\sin A=\\frac{{3\\sqrt{3}}}{{14}}$. Find:\n\n$(Ⅰ)$ $\\angle B$;\n\n$(Ⅱ)$ the area of $\\triangle ABC$. 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": 20796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic,\\[x^2-sx+p,\\]with roots $r_1$ and $r_2$. He notices that\\[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\\cdots=r_1^{2007... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Archimedes Academy, there are three teachers, Mrs. Algebra, Mr. Calculus, and Ms. Statistics, teaching students preparing for the AMC 8 contest. Mrs. Algebra has 13 students, Mr. Calculus has 10 students, and Ms. Statistics has 12 students. If 3 students are enrolled in both... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 20798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using systematic sampling to select a sample of size 20 from 180 students, the students are randomly numbered from 1 to 180. They are then divided into 20 groups in order of their number (group 1: numbers 1-9, group 2: numbers 10-18, ..., group 20: numbers 172-180). If the numb... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 20799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\\overline{A E}$... | math | {
"style": "rule",
"ground_truth": "\\frac{1024}{45}"
} | {
"split": "train",
"index": 20800,
"solution": "Using Heron's formula we arrive at $[A B C]=\\frac{3 \\sqrt{15}}{4}$. Now invoking the relation $[A B C]=\\frac{a b c}{4 R}$ where $R$ is the circumradius of $A B C$, we compute $R^{2}=\\left(\\frac{2 \\cdot 3}{[A B C]^{2}}\\right)=$ $\\frac{64}{15}$. Now observe t... |
deepscaler | [
{
"role": "user",
"content": "\nEvaluate the limit as \\( n \\) approaches infinity:\n\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{(1+2n)^{3} - 8n^{5}}{(1+2n)^{2} + 4n^{2}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\\omega$ . $X$ , $Y$ , and $Z$ are points on $\\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A... | math | {
"style": "rule",
"ground_truth": "7500"
} | {
"split": "train",
"index": 20802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper whose length is $\\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The pape... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 20803,
"solution": "1. **Assumptions and Setup**: Assume the width of the paper is $w$ and the length is $\\sqrt{3}w$. The area of the paper, $A$, is given by:\n \\[\n A = w \\times \\sqrt{3}w = \\sqrt{3}w^2.\n \\]\n Without loss of generality, let $w = 1$. Then the area $A = ... |
deepscaler | [
{
"role": "user",
"content": "Determine the volume of the region enclosed by \\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 6.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 20804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 20805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many divisors of 63 are also divisors of 72? (Recall that divisors may be positive or negative.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Based on the definition of the derivative, find $f^{\\prime}(0)$:\n\n$$\nf(x)=\\left\\{\\begin{array}{c}\n\\frac{2^{\\operatorname{tg} x}-2^{\\sin x}}{x^{2}}, x \\neq 0 \\\\\n0, x=0\n\\end{array}\\right.\n$$ Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\ln \\sqrt{2}"
} | {
"split": "train",
"index": 20807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a \"cluster\" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computer... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 20808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 20809,
"solution": "To determine the possible values of $x$, the cost of each ticket, we need to consider the total amounts spent by the 9th and 10th graders and find a common ticket price that divides both totals.\n\n1. **Identify the total amounts spent by each grade:**\n - 9th gr... |
deepscaler | [
{
"role": "user",
"content": "The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is ... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 20810,
"solution": "Let the terms in the sequence be defined as \\[a_1, a_2, ..., a_6.\\]\n Since this is an arithmetic sequence, we have $a_1+a_6=a_2+a_5=a_3+a_4.$ So, \\[\\sum_{i=1}^6 a_i=3(a_1+a_6)=990.\\] Hence, $(a_1+a_6)=330.$ And, since we are given that $a_6=2a_1,$ we get $3a_... |
deepscaler | [
{
"role": "user",
"content": "There exists a real number $k$ such that the equation\n\\[\\begin{pmatrix} 3 \\\\ 5 \\end{pmatrix} + t \\begin{pmatrix} 4 \\\\ -7 \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ -2 \\end{pmatrix} + s \\begin{pmatrix} -1 \\\\ k \\end{pmatrix}\\]does not have any solutions in $t$ and $s... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 20811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number results when 50 is increased by $120\\%$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 20813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 20814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the degree measure of an angle whose measure is double the measure of its complement? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 20815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a group of 6 students, 4 are to be selected to participate in competitions for four subjects: mathematics, physics, chemistry, and biology. If two students, A and B, cannot participate in the biology competition, determine the number of different selection plans. Let's thi... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 20816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\\left\\{\\begin{array}{l}{x=\\frac{4}{1+{t}^{2}}},\\\\{y=\\frac{4t}{1+{t}^{2}}}\\end{array}\\right.$ $(t\\in R)$.<br/>$(Ⅰ)$ Find the rectangular coordinate equation of curve $C$;<br/>$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{3}"
} | {
"split": "train",
"index": 20817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum \\[\\frac{a}{3b} + \\frac{b}{5c} + \\frac{c}{6a},\\] where \\( a, b, \\) and \\( c \\) are positive real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{90}}"
} | {
"split": "train",
"index": 20818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $y=x, y=2 x$, and $y=3 x$ are the three medians of a triangle with perimeter 1. Find the length of the longest side of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{\\sqrt{58}}{2+\\sqrt{34}+\\sqrt{58}}}"
} | {
"split": "train",
"index": 20819,
"solution": "The three medians of a triangle contain its vertices, so the three vertices of the triangle are $(a, a),(b, 2 b)$ and $(c, 3 c)$ for some $a, b$, and $c$. Then, the midpoint of $(a, a)$ and $(b, 2 b)$, which is $\\left(\\frac{a+b}{2}, \\frac{a+2 b}{2}\\right)$, mus... |
deepscaler | [
{
"role": "user",
"content": "What is the largest value of $n$ less than 100,000 for which the expression $10(n-3)^5 - n^2 + 20n - 30$ is a multiple of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99999"
} | {
"split": "train",
"index": 20820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n(1) $(x-2)^2=25$; \n(2) $x^2+4x+3=0$; \n(3) $2x^2+4x-1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-2-\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 20821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) In $\\triangle ABC$, if $2\\lg \\tan B=\\lg \\tan A+\\lg \\tan C$, then the range of values for $B$ is ______. \n(2) Find the maximum value of the function $y=7-4\\sin x\\cos x+4\\cos ^{2}x-4\\cos ^{4}x$ ______. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive three-digit palindromes is 436,995. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1332"
} | {
"split": "train",
"index": 20823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 20824,
"solution": "1. **Identify the given values and their implications:**\n - The mean of the five test scores is $90$. This implies that the sum of all five scores is $90 \\times 5 = 450$.\n - The median of the scores is $91$, which is the middle value in the ordered list of s... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real t such that\n\\[ x_1 + x_3 = 2tx_2, \\]\n\\[ x_2 + x_4 = 2tx_3, \\]\n\\[ x_3 + x_5 = 2tx_4 \\]\nhas a solution \\( x_1, x_2, x_3, x_4, x_5 \\) in non-negative reals, not all zero. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"split": "train",
"index": 20825,
"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"
} | {
"split": "train",
"index": 20826,
"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 a... |
deepscaler | [
{
"role": "user",
"content": "A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 20827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ,$ $XY = 4,$ $XZ = 5,$ $YZ = 7,$ and $W$ lies on $\\overline{YZ}$ such that $\\overline{XW}$ bisects $\\angle YXZ.$ Find $\\cos \\angle YXW.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{2}{5}}"
} | {
"split": "train",
"index": 20828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the right focus of the ellipse $\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\\frac{\\sqrt{6}}{3},-1)$, ca... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 20829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\) with \\(X\\) as the midpoint of the diagonal \\(AC\\), it turns out that \\(CD \\parallel BX\\). Find \\(AD\\) if it is known that \\(BX = 3\\), \\(BC = 7\\), and \\(CD = 6\\). Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 20830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$, $a_{n+1}a_{n-1}=a_{n}^{2}+2007$.\n Find the greatest integer that does not exceed $\\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}.$ Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 20831,
"solution": "We are given that\n$a_{n+1}a_{n-1}= a_{n}^{2}+2007$, $a_{n-1}^{2}+2007 = a_{n}a_{n-2}$.\n Add these two equations to get\n$a_{n-1}(a_{n-1}+a_{n+1}) = a_{n}(a_{n}+a_{n-2})$\n$\\frac{a_{n+1}+a_{n-1}}{a_{n}}= \\frac{a_{n}+a_{n-2}}{a_{n-1}}$.\n This is an invariant. De... |
deepscaler | [
{
"role": "user",
"content": "If $13^{3n}=\\left(\\frac{1}{13}\\right)^{n-24}$, find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Start with a three-digit positive integer $A$ . Obtain $B$ by interchanging the two leftmost digits of $A$ . Obtain $C$ by doubling $B$ . Obtain $D$ by subtracting $500$ from $C$ . Given that $A + B + C + D = 2014$ , fi\fnd $A$ . Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "344"
} | {
"split": "train",
"index": 20833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan{\\alpha}$ and $\\tan{\\beta}$ are the roots of $x^2 - px + q = 0$, and $\\cot{\\alpha}$ and $\\cot{\\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{p}{q^2}"
} | {
"split": "train",
"index": 20834,
"solution": "1. **Identify the roots of the quadratic equations:** Given that $\\tan{\\alpha}$ and $\\tan{\\beta}$ are the roots of $x^2 - px + q = 0$, and $\\cot{\\alpha}$ and $\\cot{\\beta}$ are the roots of $x^2 - rx + s = 0$.\n\n2. **Apply Vieta's formulas:** By Vieta's for... |
deepscaler | [
{
"role": "user",
"content": "A regular tetrahedron is formed by joining the centers of four non-adjacent faces of a cube. Determine the ratio of the volume of the tetrahedron to the volume of the cube. 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": 20835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "These two spinners are divided into thirds and quarters, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.\n\n[asy]\n\nsize(5cm,5cm);\n... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 20836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, if $\\cos B = \\frac{{\\sqrt{2}}}{2}$, then the minimum value of $(\\tan ^{2}A-3)\\sin 2C$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2} - 6"
} | {
"split": "train",
"index": 20837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 20839,
"solution": "1. **Calculate the total number of dots on three dice**: Each die has faces numbered from 1 to 6. The sum of the numbers on one die is calculated as follows:\n \\[\n 1 + 2 + 3 + 4 + 5 + 6 = 21\n \\]\n Since there are three dice, the total number of dots on ... |
deepscaler | [
{
"role": "user",
"content": "There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 20840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone has for its base a circle having the same radius as a given sphere.\nThe volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "2/1"
} | {
"split": "train",
"index": 20841,
"solution": "1. **Identify the given information:**\n - The base of the cone and the sphere have the same radius, denoted as $r$.\n - The volume of the cone is half that of the sphere.\n\n2. **Write down the formulas for the volumes:**\n - Volume of the sphere, $V_{\\text... |
deepscaler | [
{
"role": "user",
"content": "Let $\\Gamma$ be the region formed by the points $(x, y)$ that satisfy\n$$\n\\left\\{\\begin{array}{l}\nx \\geqslant 0, \\\\\ny \\geqslant 0, \\\\\nx+y+[x]+[y] \\leqslant 5\n\\end{array}\\right.\n$$\nwhere $[x]$ represents the greatest integer less than or equal to the real num... | math | {
"style": "rule",
"ground_truth": "9/2"
} | {
"split": "train",
"index": 20842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive solution to\n\\[\\tan 2x + \\tan 3x = \\sec 3x\\]in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{14}"
} | {
"split": "train",
"index": 20843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular polygons $I C A O, V E N T I$, and $A L B E D O$ lie on a plane. Given that $I N=1$, compute the number of possible values of $O N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20844,
"solution": "First, place $A L B E D O$. We then note that $I C A O$ has two orientations, both of which have $I$ on $E O$. Next, we note that for any given orientation of $I C A O$, the two orientations of $V E N T I$ have $N$ symmetric to line $E I$. Thus, for any given orien... |
deepscaler | [
{
"role": "user",
"content": "Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?\n$\\textbf{(A)}\\ \\sqrt{15} \\qquad \\textbf{(B)}\\ \\sqrt{21} \\qquad \\textbf{(C)... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 20845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To actively create a \"demonstration school for the prevention and control of myopia in children and adolescents in the city\" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semester. $20$ student answer sheets we... | math | {
"style": "rule",
"ground_truth": "1755"
} | {
"split": "train",
"index": 20846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere has a volume of \\( 288\\pi \\) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \\( \\pi \\... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 20847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha \\neq 1$ be a complex number such that the distance from $\\alpha^2$ to 1 is twice the distance from $\\alpha$ to 1, while the distance from $\\alpha^4$ to 1 is four times the distance from $\\alpha$ to 1. Enter all possible values of $\\alpha,$ separated by comma... | math | {
"style": "rule",
"ground_truth": "i \\sqrt{3}, -i \\sqrt{3}"
} | {
"split": "train",
"index": 20848,
"solution": ""
} |
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