id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-60725bcbf0ac62404f27f22e9e93ca58f9f11acddabb8ef4fa12d4c10cbb6558 | Which set of five numbers, such that their sum equals their product, has the largest sum? | [
"10"
] | [
"Which set of five numbers, such that their sum equals their product, has the largest sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-96c006abcf77d11b6241f16a08871a0db64dbca165f34cac6dc13d8214a03a9b | Which sets $S$ of positive integers are such that the product of any number of elements from $S$, allowing repetitions, has the form $a^b$ for integers $a$ and $b$ with $a \geq 1$ and $b \geq 2$? | [
"S \\text{ is a set of perfect } d\\text{th powers for some integer } d \\geq 2"
] | [
"Which sets $S$ of positive integers are such that the product of any number of elements from $S$, allowing repetitions, has the form $a^b$ for integers $a$ and $b$ with $a \\geq 1$ and $b \\geq 2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-429f427b49c22f6c2a2768c13dc7113747bbbc60ed90ed15bb66d30337f003b6 | Which solutions to the equation $x^{y} = y^{x}$ are integral and which are rational? | [
"Integral solutions: \\((t, t)\\) for integer \\( t \\) and \\((2, 4)\\) and \\((4, 2)\\). Rational solutions: \\((t, t)\\) for rational \\( t \\) and \\((2, 4)\\) and \\((4, 2)\\)."
] | [
"Which solutions to the equation $x^{y} = y^{x}$ are integral and which are rational?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-865ef7718b56d959204d8743dc25424607d2778855dd7bc06147b23e2eab66eb | Which string is more likely to appear first when a machine generates 0's and 1's with equal probability (1/2), starting at time \( t = 0 \): 011 or 110? | [
"011"
] | [
"Which string is more likely to appear first when a machine generates 0's and 1's with equal probability (1/2), starting at time \\( t = 0 \\): 011 or 110?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-2f44d2cb1fd159d71ea3d08254a1ae18172169b33cb673aad98a5def7f2ef4d4 | Which subsets of complex numbers are closed under addition (+), subtraction (-), multiplication (×), and division (÷) but do not contain the number 1? | [
"\\{0\\}"
] | [
"Which subsets of complex numbers are closed under addition (+), subtraction (-), multiplication (×), and division (÷) but do not contain the number 1?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-064b51b3b123ad2121998f71793d9976721deb82876b87ccf6b914638111a83e | Which term in the equation \( y = -\frac{1}{20}x^2 + 3x + 5 \) determines the height at which the ball was thrown? Does changing this term change the coordinates of the maximum height of the ball? Explain. | [
"The constant term \\(5\\) determines the height at which the ball was thrown. Changing this term changes the coordinates of the maximum height of the ball.",
"The term that determines the height at which the ball was thrown is the constant term 5. Changing this term does not change the coordinates of the maximum... | [
"Which term in the equation \\( y = -\\frac{1}{20}x^2 + 3x + 5 \\) determines the height at which the ball was thrown? Does changing this term change the coordinates of the maximum height of the ball? Explain."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 4,
"tir": 24
} | 62 | 0 | 0 |
omr-ws-afda86d50eb77ba71bb175ca6229b295ec53d0afd271ad905c10c4bf823a9121 | Which term in the expansion of $(x+x^{-2})^{12}$ does not contain $x$? | [
"$k = 4$"
] | [
"Which term in the expansion of $(x+x^{-2})^{12}$ does not contain $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c071b9796f443307f6dd67887e87f399204c549f14945645e4184ae17d12e22 | Which term in the sequence \(1! \times 2! \times 3! \times \cdots \times 2010!\) can be removed to make the product a perfect square? | [
"1005!"
] | [
"Which term in the sequence \\(1! \\times 2! \\times 3! \\times \\cdots \\times 2010!\\) can be removed to make the product a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2a08b4f9a831b18db00f0ce56cfe7d2f4457fdfe9f72a43ca59d2298a449c228 | Which terms must be removed from the sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \frac{1}{12}$ if the remaining sum is to equal $1$? | [
"$\\frac{1}{8}, \\frac{1}{10}$"
] | [
"Which terms must be removed from the sum $\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{6} + \\frac{1}{8} + \\frac{1}{10} + \\frac{1}{12}$ if the remaining sum is to equal $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 28
} | 38 | 0 | 0 |
omr-ws-cf88172a50735021dd2c9596f2227eb13e381e039c506e3a44cdcd93d2b2cd65 | Which terms must be removed from the sum \[ \frac12 + \frac14 + \frac16 + \frac18 + \frac1{10} + \frac1{12} \] so that the sum of the remaining terms is equal to $1$? | [
"\\(\\frac18 \\text{ and } \\frac1{10}\\)"
] | [
"Which terms must be removed from the sum\n\\[ \\frac12 + \\frac14 + \\frac16 + \\frac18 + \\frac1{10} + \\frac1{12} \\]\nso that the sum of the remaining terms is equal to $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-291cc82c49a4d63447ac04b2fb4a33a43a63ffafe17b09c8e85e2fe4d3cf54ae | Which three-digit number has the greatest number of different factors? | [
"840"
] | [
"Which three-digit number has the greatest number of different factors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03760b300479486b435ab500ec9ccd9e61c2984ee660ab5f9cda419d2adbe15b | Which triangles can be divided into 3 triangles, all having equal circumradii? | [
"\\text{Equilateral triangles}"
] | [
"Which triangles can be divided into 3 triangles, all having equal circumradii?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-93a88ab6590576e825e128459f50862af183c665066c1e199041290dba5b9e49 | Which triangles can be divided into three triangles such that each of the three resulting triangles has circumcircles with equal radii? | [
"\\text{Equilateral triangles}"
] | [
"Which triangles can be divided into three triangles such that each of the three resulting triangles has circumcircles with equal radii?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-9097e7969309645ea7f30150720c456548f9249630bb9eb55f49b52b1cf35259 | Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b,$ and $c$ are the sides of a triangle? | [
"a = b \\text{ or } c^2 = a^2 + b^2"
] | [
"Which triangles satisfy the equation $\\frac{c^2-a^2}{b}+\\frac{b^2-c^2}{a}=b-a$ when $a, b,$ and $c$ are the sides of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-339f7ec40b71751258f5fe8398694236bcc72c9a6933a399707c095826ef7f1b | Which triangles with integer sides are more: with a perimeter of 2008 or with a perimeter of 2005? | [
"2008"
] | [
"Which triangles with integer sides are more: with a perimeter of 2008 or with a perimeter of 2005?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-ff859283fa54c582d976782ad78f81360c84aa572e0ad75211a63f68819997f1 | Which triangular numbers \( u(n) = \frac{n(n+1)}{2} \) are perfect squares? | [
"\\( x_n = \\left( \\frac{(1+\\sqrt{2})^n + (1-\\sqrt{2})^n}{2} \\right)^2 \\) for odd \\( n \\) and \\( x_n = \\left( \\frac{(1+\\sqrt{2})^n - (1-\\sqrt{2})^n}{2} \\right)^2 \\) for even \\( n \\)"
] | [
"Which triangular numbers \\( u(n) = \\frac{n(n+1)}{2} \\) are perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-a23bde9a88d531801c6bade6fb33b64d4ae045ce5538ab81d27022889b97aea6 | Which trigonometric ratios cannot be equal to $\pi/2$? | [
"\\(\\tan\\) and \\(\\sec\\)"
] | [
"Which trigonometric ratios cannot be equal to $\\pi/2$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28e29ba1ebeef23137ddc493b003385627bc7a27b54410f1306a180e7515c60e | Which trigonometric ratios cannot be equal to $\pi/4$? | [
"\\csc \\theta \\text{ and } \\sec \\theta"
] | [
"Which trigonometric ratios cannot be equal to $\\pi/4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-487a13c359bde39cb6ef9505e4be2b753cf48aa2ec64e2674c504060ca07a070 | Which triplets \(a, b, c \in \mathbb{Z}\) satisfy the Diophantine equation \[ c^2 + 1 = (a^2 - 1)(b^2 - 1)? \] | [
"\\( (0, 0, 0) \\)"
] | [
"Which triplets \\(a, b, c \\in \\mathbb{Z}\\) satisfy the Diophantine equation\n\\[ c^2 + 1 = (a^2 - 1)(b^2 - 1)? \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-a7fa0c2ea4f81a624f58bd9b177deb2abb267d41c344de42e26fc0a4bd4744a7 | Which two prime numbers have a sum and a difference that are both perfect numbers? Calculate the product of these two prime numbers. | [
"187"
] | [
"Which two prime numbers have a sum and a difference that are both perfect numbers? Calculate the product of these two prime numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-149c8d11db0661bb873ef240e4205c52e1b8e51b136bc90af07c28289250e6dd | Which two transformations transform the graph of $y=f(x)$ to that of $y=f(2x+1)$? Does the order of these transformations matter? | [
"Horizontal contraction by a factor of 2 followed by a horizontal translation of 1 unit to the left. Order matters."
] | [
"Which two transformations transform the graph of $y=f(x)$ to that of $y=f(2x+1)$? Does the order of these transformations matter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 3,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-5c5c920b1b64d401fec502be385573dd73312e3a1c1988853708423eb571ec1c | Which two-digit numbers have the same last two digits as their cubes? | [
"24, 25, 49, 51, 76, 99"
] | [
"Which two-digit numbers have the same last two digits as their cubes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98e8c357df567bac0c89c78602d14869909240a8041a71171ef048fd2437803e | Which type of conic section is described by the equation \[(x+5)^2 = (4y-3)^2 - 140?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | [
"(\\text{H})"
] | [
"Which type of conic section is described by the equation \\[(x+5)^2 = (4y-3)^2 - 140?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fbf17449fd57e83daa52587515d4b523b4fdd9da26c7d5bc845bf09fda54791b | Which type of conic section is described by the equation \[\sqrt{x^2 + (y-1)^2} + \sqrt{(x-5)^2 + (y+3)^2} = 10?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | [
"\\text{(E)}"
] | [
"Which type of conic section is described by the equation \\[\\sqrt{x^2 + (y-1)^2} + \\sqrt{(x-5)^2 + (y+3)^2} = 10?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c413699dc40ae706fe5be169ceb2a304d6e1606c7b5204b05add22497c2bb0b2 | Which type of conic section is described by the equation \[|y+5| = \sqrt{(x-2)^2 + y^2}?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | [
"\\text{(P)}"
] | [
"Which type of conic section is described by the equation \\[|y+5| = \\sqrt{(x-2)^2 + y^2}?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-207d68d02d8f8d208de364d8315442075c1ca888888cc2aae7a975d01c222a15 | Which type of six-digit numbers are there more of: those that can be represented as the product of two three-digit numbers or those that cannot? | [
"\\text{Those that cannot}"
] | [
"Which type of six-digit numbers are there more of: those that can be represented as the product of two three-digit numbers or those that cannot?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-861d240a0c50c8a187e02049128c0d6f7643d59b42cf45bc169f99b57b529116 | Which value of $n$ satisfies $n^{29} \equiv 7 \pmod{65}$? | [
"37"
] | [
"Which value of $n$ satisfies $n^{29} \\equiv 7 \\pmod{65}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 21
} | 38 | 0 | 0 |
omr-ws-a15d2c62bd2a912b8dd339919c2487fc66dd031f8381ab9ed84f43642e66111a | Which value of $x$ satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$? | [
"45"
] | [
"Which value of $x$ satisfies the equation $\\sqrt[3]{6+\\sqrt{x}} + \\sqrt[3]{6-\\sqrt{x}} = \\sqrt[3]{3}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-68aec4ed6045e3650cce38b8672253ce5ccb3ec718b2d62c86d456f01a99527d | Which well-known ring is the quotient ring \(\frac{\mathbb{Z}_5[X,Y]}{(X^2 + Y + 1, Y - X)}\) isomorphic to? | [
"\\(\\mathbb{F}_{25}\\)"
] | [
"Which well-known ring is the quotient ring \\(\\frac{\\mathbb{Z}_5[X,Y]}{(X^2 + Y + 1, Y - X)}\\) isomorphic to?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-70d028aa841497b2eb7fbde505f8547f0c4df3a4a4f7b7a29ea96a6d3b4681c2 | Which year in the 18th century, when added to the product of its digits, equals 2018? | [
"1766"
] | [
"Which year in the 18th century, when added to the product of its digits, equals 2018?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-78fd0a068a316989695cf7f5cbf6373e8e5fc60a2a8fa70bd8f5e52b02131171 | Which year in the third millennium (2001 to 3000) has more factors than the number 2016? | [
"2520"
] | [
"Which year in the third millennium (2001 to 3000) has more factors than the number 2016?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-26543ed5972e79270434697d7635de6e69d80c94128c20757f24e3eb890259ed | Which year in the third millennium (2001-3000) has more factors than the number 2016? | [
"2520"
] | [
"Which year in the third millennium (2001-3000) has more factors than the number 2016?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-a958a7a324ca89d3428dea72b0ff40a6f7e378dcc6d32ea8966259036bd599d7 | While adding all positive integers from 1 to 100, we omitted \( k \) odd numbers and the sum became 3775. Find how many possible values of \( k \) exist. | [
"11"
] | [
"While adding all positive integers from 1 to 100, we omitted \\( k \\) odd numbers and the sum became 3775. Find how many possible values of \\( k \\) exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 15
} | 67 | 0 | 0 |
omr-ws-fd5178ab67c9eb289e6ebf1305e032a2433e1fa023c43e4a8ab4ca88739c9aef | While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m\pi}{n-\pi}... | [
"900"
] | [
"While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\\frac{m\\pi}... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-beca1964325cbce74e4dad41ad355d207f425e427b5e184dba27d49fcf7458c4 | While shopping at the local mall, Ann walked slowly (at a constant speed) down a descending escalator, taking 10 steps to reach the last step. She then ran up the same (descending) escalator with a speed five times greater than the speed with which she had walked down, taking 25 steps to reach the top. Find the number ... | [
"20"
] | [
"While shopping at the local mall, Ann walked slowly (at a constant speed) down a descending escalator, taking 10 steps to reach the last step. She then ran up the same (descending) escalator with a speed five times greater than the speed with which she had walked down, taking 25 steps to reach the top. Find the nu... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c79cdcfccb4df1a64cf02ea2fef94e085587a93fb928f804db549c1788789c6c | While sleeping in Mama Bear’s bed, Goldilocks dreams about different colored sheep. She sees two blue sheep, two red sheep, and one green sheep. How many ways are there to arrange the sheep in a line if no two sheep of the same color can be standing next to each other? Note that sheep of the same color are indistinguis... | [
"10"
] | [
"While sleeping in Mama Bear’s bed, Goldilocks dreams about different colored sheep. She sees two blue sheep, two red sheep, and one green sheep. How many ways are there to arrange the sheep in a line if no two sheep of the same color can be standing next to each other? Note that sheep of the same color are indisti... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d361a1e272a61eae1251881242775aeafdcf4009955fd38d0bd0524510917e76 | While standing in line to buy concert tickets, Kit moved 60 feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window? | [
"105"
] | [
"While standing in line to buy concert tickets, Kit moved 60 feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-620fc14375a569f64259f283ba2008c6580c4b73924064fd00848e96337ffe36 | While staying at the Denver Marriott Tech Center for the 2008 Lockheed Martin MATHCOUNTS Competition, Darryl enters an elevator that travels through 20 floors. He enters the elevator on the second floor. Darryl tosses a fair, six-sided die. If the number appearing is prime, Darryl moves down one floor, otherwise, he go... | [
"\\dfrac{3}{16}"
] | [
"While staying at the Denver Marriott Tech Center for the 2008 Lockheed Martin MATHCOUNTS Competition, Darryl enters an elevator that travels through 20 floors. He enters the elevator on the second floor. Darryl tosses a fair, six-sided die. If the number appearing is prime, Darryl moves down one floor, otherwise, ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1b54cde09a0ca696f75488106e6fbecab3461ce00c0c12a40ff610d1a9efee79 | While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is 3.5 degrees. After driving 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain. | [
"1.3"
] | [
"While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is 3.5 degrees. After driving 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-21564aa6dc5943c68c9e46a5f3b360edb2c18969386896bacf692a8f93df75e4 | While waiting for the bishop to arrive at St Stephen's Anglican Church to lead a service, every person present greeted every other person with a hand shake. Arriving late, the bishop shook hands with only some of the people as he made his way in. Altogether 1933 handshakes had taken place. How many people shook hands w... | [
"42"
] | [
"While waiting for the bishop to arrive at St Stephen's Anglican Church to lead a service, every person present greeted every other person with a hand shake. Arriving late, the bishop shook hands with only some of the people as he made his way in. Altogether 1933 handshakes had taken place. How many people shook ha... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e58ede9e13ced2c28a888584cdab6e50cde20bb9551dec2aa40fff040858dc4a | While walking down a descending escalator at a constant speed, Ann took 10 steps to reach the last step. When running up the same descending escalator at a speed five times greater, she took 25 steps to reach the top. Find the number of steps showing on the escalator at any one moment. | [
"20"
] | [
"While walking down a descending escalator at a constant speed, Ann took 10 steps to reach the last step. When running up the same descending escalator at a speed five times greater, she took 25 steps to reach the top. Find the number of steps showing on the escalator at any one moment."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-39ed170566680a7011ada2a358e3d6fe65eb0638a174ab2c8928daea7f99d11c | While walking on a plane surface, a traveler first headed 18 miles north, then 11 miles west, then 6 miles south and finally 6 miles east. How many miles from the starting point was the traveler after these four legs of the journey? | [
"13"
] | [
"While walking on a plane surface, a traveler first headed 18 miles north, then 11 miles west, then 6 miles south and finally 6 miles east. How many miles from the starting point was the traveler after these four legs of the journey?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-325b95fa6409ab5b98ba812d2c7e3780617dd8674628e0200227d4d80dfcca47 | While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show? | [
"10"
] | [
"While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b034d98668d408a65910f384f1ba563b9da66f23a706481b8e808cb13bafd75 | While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next... | [
"090",
"90"
] | [
"While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 16,
"tir": 12
} | 53 | 0 | 0 |
omr-ws-5fe617e1cd032e45f4df9e3761a2f46304ebcf29fa67f929d56198f35a55c378 | Why does the identity $\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n} = \frac{\sqrt{n}}{2^{n-1}}$ not hold for $n=1$? | [
"\\text{The identity does hold for } n=1"
] | [
"Why does the identity $\\prod_{k=1}^{n-1} \\cos\\frac{k\\pi}{2n} = \\frac{\\sqrt{n}}{2^{n-1}}$ not hold for $n=1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-8cef8fb6c2715a5f8ffac2ab30ea02e28c161bc9ad2260e14f170546204c7814 | Why does the inequality \(E \leq 3V - 6\) not hold for the graph \(K_{2}\), which consists of two vertices and one edge? | [
"E \\leq 3V - 6 \\text{ requires at least 3 vertices, but } K_2 \\text{ has only 2.}"
] | [
"Why does the inequality \\(E \\leq 3V - 6\\) not hold for the graph \\(K_{2}\\), which consists of two vertices and one edge?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-e069b40c25862e42d44a0ad5086dc0e82df721576dbbddace00a99a4929e982b | Wildan and William toss a coin alternately, with Wildan going first. The probability of the coin landing on tails is $\frac{1}{3}$. Wildan wins if his toss results in tails, while William wins if his toss results in heads. Find the probability that Wildan wins the game. Write your answer in its simplest form. | [
"\\(\\frac{3}{7}\\)"
] | [
"Wildan and William toss a coin alternately, with Wildan going first. The probability of the coin landing on tails is $\\frac{1}{3}$. Wildan wins if his toss results in tails, while William wins if his toss results in heads. Find the probability that Wildan wins the game. Write your answer in its simplest form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-e7e296e6f22e223ae5983ec63df9cfe991e6a2020d936e8db893d03cb9f0abb9 | Will is given a list of 10 positive integers, which together have exactly 9 distinct values. By adding 9 of the numbers, he can make any integral sum from 86 to 94, inclusive. Find the integer that occurs twice. | [
"10"
] | [
"Will is given a list of 10 positive integers, which together have exactly 9 distinct values. By adding 9 of the numbers, he can make any integral sum from 86 to 94, inclusive. Find the integer that occurs twice."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-fc0f9e52e5070c4fef3484d9a3e1741e9940813c8a1ff49bfeb7927761bd19bb | Will is given a list of 10 positive integers, which together have exactly 9 distinct values. Will finds that by adding 9 of the numbers, he can make any integral sum from 86 to 94, inclusive. Find the integer that occurs twice. | [
"10"
] | [
"Will is given a list of 10 positive integers, which together have exactly 9 distinct values. Will finds that by adding 9 of the numbers, he can make any integral sum from 86 to 94, inclusive. Find the integer that occurs twice."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f57e4e8287ff5ebd2f6711e34ee23353af3ce84d82a43f5de20b16fe42780778 | Will it suffice to buy 4000 pieces of glazed tile of size 10 cm by 10 cm to face the walls of the above-water portion of a swimming pool having the shape of a rhombus of area 450 square meters with sides 0.5 meter above the water level? | [
"No"
] | [
"Will it suffice to buy 4000 pieces of glazed tile of size 10 cm by 10 cm to face the walls of the above-water portion of a swimming pool having the shape of a rhombus of area 450 square meters with sides 0.5 meter above the water level?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e94a8799b652025a6c0c810aac985cc63998b5fee4cb24e82f82409491dc643 | Will the number $1/1996$ decrease or increase and by how many times if in the decimal notation of this number the first non-zero digit after the decimal point is crossed out? | [
"50"
] | [
"Will the number $1/1996$ decrease or increase and by how many times if in the decimal notation of this number the first non-zero digit after the decimal point is crossed out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-f10063ff5fd4f549650c544a09dfa869e98a859da342dcc12d7b4728e796fbda | Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the n... | [
"52"
] | [
"Will wants to insert some × symbols between the following numbers: $$1\\,\\,\\,2\\,\\,\\,3\\,\\,\\,4\\,\\,\\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\\times$ symbols: $$1 \\times 23 \\times 4 \\times 6 = 552.$$ Will discovers that he can obtain the number $276$. Wha... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-469d8f81ca3e3392fdfbd743112971a19a4726f64dfeba812bdb6982702de0d0 | William Hu's phone number is $(555)-555-5555$. Find the last three digits of $555^{555^{5555}}$. | [
"875"
] | [
"William Hu's phone number is $(555)-555-5555$. Find the last three digits of $555^{555^{5555}}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73116a3cb184be7d93cf15f34d8f5296077a22a4e73f97a8be21d269cce8b4e0 | William draws a triangle $\vartriangle ABC$ with $AB = \sqrt{3}$, $BC = 1$, and $AC = 2$. Let the angle bisector of $\angle ABC$ meet $AC$ at point $D$. He folds $\vartriangle ABD$ over $BD$, and the new location of point $A$ is $A'$. After folding $\vartriangle A'CD$ over $CD$, what area of the resulting figure is cov... | [
"\\(\\frac{2\\sqrt{3} - 3}{4}\\)"
] | [
"William draws a triangle $\\vartriangle ABC$ with $AB = \\sqrt{3}$, $BC = 1$, and $AC = 2$. Let the angle bisector of $\\angle ABC$ meet $AC$ at point $D$. He folds $\\vartriangle ABD$ over $BD$, and the new location of point $A$ is $A'$. After folding $\\vartriangle A'CD$ over $CD$, what area of the resulting fig... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-9bacdaba3c19286f22d51c3f8e0f556c6f5d415b12b098de39260a072f424c66 | William drew a circle with \( n \ge 3 \) cells around the circumference and wrote a positive number in each cell. He then erased the numbers and wrote in each cell the geometric mean of the numbers previously written in the two neighboring cells. Let \( a_i \) be the original number in the \( i \)-th cell and \( b_i \)... | [
"n"
] | [
"William drew a circle with \\( n \\ge 3 \\) cells around the circumference and wrote a positive number in each cell. He then erased the numbers and wrote in each cell the geometric mean of the numbers previously written in the two neighboring cells. Let \\( a_i \\) be the original number in the \\( i \\)-th cell a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a9da9230acc3906702187d60668a804be4df533e6b5193d3eaee2905a0f9ba8 | William has a lucky charm consisting of a vertical wire $AB$ of length $20$ cm with midpoint $M$. At the bottom, there is a right square pyramid with side length $10\sqrt{2}$ cm and height $AM$. On the top, there is a regular tetrahedron with height $MB$, where $M$ is a vertex belonging to both the pyramid and the tetr... | [
"72"
] | [
"William has a lucky charm consisting of a vertical wire $AB$ of length $20$ cm with midpoint $M$. At the bottom, there is a right square pyramid with side length $10\\sqrt{2}$ cm and height $AM$. On the top, there is a regular tetrahedron with height $MB$, where $M$ is a vertex belonging to both the pyramid and th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e84c19e5cfa856d9405d2aa40908094cf9838518157f154f9000e7822bfeb9c6 | William has some number of apples and some number of crates. If he packs 70 apples in each crate, he will have 50 apples left. If he packs 90 apples in each crate, he will be 10 apples short. How many apples and how many crates does William have? | [
"260 apples, 3 crates"
] | [
"William has some number of apples and some number of crates. If he packs 70 apples in each crate, he will have 50 apples left. If he packs 90 apples in each crate, he will be 10 apples short. How many apples and how many crates does William have?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-688bb21f593ae6642683160128c0dd1c338f916b59700b6bcd0c35722fb1f4da | William is a bacteria farmer. He would like to give his fiance 2021 bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those 2021 bacteria. The infinite plane petri dish starts off empt... | [
"8"
] | [
"William is a bacteria farmer. He would like to give his fiance 2021 bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those 2021 bacteria. The infinite plane petri dish starts off... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-7893b6f8d23964386ae4648920d799349bcdbd222a666ebcd47ab07d64ba6ef9 | William is taking the 25-question, multiple choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last four questions. What is the probability that he will get at least one of these final four questions right? | [
"\\frac{369}{625}"
] | [
"William is taking the 25-question, multiple choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last four questions. What is the probability that he will get at least one of these final four questions right?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f039f772e5fccaba976739b470acb383b6ff3a1caad9b7f2dc6b340850c56fa3 | William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William if $m$ divides his number. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee determining William's nu... | [
"15"
] | [
"William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William if $m$ divides his number. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee determining William... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e68e4a6caba695aadd2080638945f8c7c5b8622f6a7564a3a43b5de328211c0 | William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder whe... | [
"173"
] | [
"William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainde... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-36d349c72a1b727cd38eb42918b81c89b525444022d11234e8d73d99a99379f3 | William's Pachirisu, named Tony Ho, has the moves Thunder (which does 110 damage, but only hits 70% of the time) and Rain Dance (which does no damage, but makes it so any future Thunders must hit). If Tony Ho randomly picks a move 2 turns in a row, what is the expected damage done, to the nearest integer? | [
"77"
] | [
"William's Pachirisu, named Tony Ho, has the moves Thunder (which does 110 damage, but only hits 70% of the time) and Rain Dance (which does no damage, but makes it so any future Thunders must hit). If Tony Ho randomly picks a move 2 turns in a row, what is the expected damage done, to the nearest integer?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64102a6a3ec16048f5761ad4a760a3811eb0c03ecb748bd02930ce05702f6e09 | Willy walks towards a street light at a rate of 5 feet per minute. The light is 20 feet high and Willy's height is 6 feet. Find the rate of change of Willy's shadow's length when Willy is 24 feet away from the light. | [
"-\\dfrac{15}{7}"
] | [
"Willy walks towards a street light at a rate of 5 feet per minute. The light is 20 feet high and Willy's height is 6 feet. Find the rate of change of Willy's shadow's length when Willy is 24 feet away from the light."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a07782d8ad13bfc60cbdb004948fd104032f220ad3218408a4bf5cb3f8e44a90 | Win has two coins. He will do the following procedure over and over again as long as he still has coins: toss all the coins he has at the same time; every coin that appears with a number side will be given to Albert. Determine the probability that Win will repeat this procedure more than three times. | [
"\\dfrac{15}{64}"
] | [
"Win has two coins. He will do the following procedure over and over again as long as he still has coins: toss all the coins he has at the same time; every coin that appears with a number side will be given to Albert. Determine the probability that Win will repeat this procedure more than three times."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d57278ab9a0edf1767c8b4123305cc3c485f9644345bd9974e6f5c9fd8f4cf74 | Winnie The Pooh has a pouch containing 100 candies numbered 1 through 100, with 50 white and 50 yellow candies. At each move, he draws a candy and places it on a tray. He can then choose two candies of the same color from the tray and eat them, earning points equal to the absolute difference of their numbers. After 100... | [
"2500"
] | [
"Winnie The Pooh has a pouch containing 100 candies numbered 1 through 100, with 50 white and 50 yellow candies. At each move, he draws a candy and places it on a tray. He can then choose two candies of the same color from the tray and eat them, earning points equal to the absolute difference of their numbers. Afte... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-461b95a1e223177652f303576e06e380ae6a6be3ae844b8cc10b6ce70384ae1a | Winnie The Pooh has a pouch containing 100 candies numbered 1 through 100, with 50 white and 50 yellow candies. At each move, he randomly draws a candy and places it on a tray. He can then choose two candies of the same color from the tray and eat them, earning points equal to the absolute difference of their numbers. ... | [
"1250"
] | [
"Winnie The Pooh has a pouch containing 100 candies numbered 1 through 100, with 50 white and 50 yellow candies. At each move, he randomly draws a candy and places it on a tray. He can then choose two candies of the same color from the tray and eat them, earning points equal to the absolute difference of their numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 8,
"tir": 1
} | 32 | 0 | 0 |
omr-ws-41b411f0e7d5d2a2072572ce59bc88d1ba23dc570e5e16e4aa8304e7465f218d | Winnie and Piglet left their houses at the same time to visit one another. Due to the fog, they didn't notice each other. Piglet reached Winnie's place in 4 minutes and Winnie's place 1 minute after they "met." How long was each of them on their way? | [
"3"
] | [
"Winnie and Piglet left their houses at the same time to visit one another. Due to the fog, they didn't notice each other. Piglet reached Winnie's place in 4 minutes and Winnie's place 1 minute after they \"met.\" How long was each of them on their way?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-ca5768f83c2c1fbeb2d045f32725699b491929458af3cb570f8aa7341f6dac41 | Winnie has 17 red balloons, 33 white balloons, 65 green balloons, and 83 chartreuse balloons. With no regard to color, Winnie gives each of her 8 nearest and dearest friends some balloons. Each friend gets the same number of balloons and she hands out as many balloons as possible. How many balloons does Winnie end up k... | [
"6"
] | [
"Winnie has 17 red balloons, 33 white balloons, 65 green balloons, and 83 chartreuse balloons. With no regard to color, Winnie gives each of her 8 nearest and dearest friends some balloons. Each friend gets the same number of balloons and she hands out as many balloons as possible. How many balloons does Winnie ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e029c5ee1d792659e9aa7f2c4daeaeb78dec76d5c3ac057d478a18c963d4d200 | Winnie has 45 cherry lollipops, 116 wintergreen lollipops, 4 grape lollipops, and 229 shrimp cocktail lollipops. With no regard to flavor, Winnie gives each of her 11 nearest and dearest friends some lollipops. Each friend gets the same number of lollipops and she hands out as many lollipops as possible. How many lolli... | [
"9"
] | [
"Winnie has 45 cherry lollipops, 116 wintergreen lollipops, 4 grape lollipops, and 229 shrimp cocktail lollipops. With no regard to flavor, Winnie gives each of her 11 nearest and dearest friends some lollipops. Each friend gets the same number of lollipops and she hands out as many lollipops as possible. How ma... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8256cc3989dc0f3bed4e23a883705203c9ce619302d60887fcad5b170b167651 | Winnie the Pooh has $2n+1$ jars of honey. No matter which jar he excludes, he is always able to separate the remaining $2n$ jars into $2$ groups of $n$ jars such that both groups have the same total amount of honey. Given that the total amount of honey in all jars is $T$, find the value of $T$ if the amount of honey in... | [
"\\dfrac{(2n + 1)S}{n}"
] | [
"Winnie the Pooh has $2n+1$ jars of honey. No matter which jar he excludes, he is always able to separate the remaining $2n$ jars into $2$ groups of $n$ jars such that both groups have the same total amount of honey. Given that the total amount of honey in all jars is $T$, find the value of $T$ if the amount of hon... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86e5b39bc0f985d47ebb03c335e7dc9e23774472a82ba676f7d911dbdd965f58 | Winnie the Pooh has \(2n+1\) jars of honey. No matter which jar he excludes, he is always able to separate the remaining \(2n\) jars into 2 groups of \(n\) jars such that both groups have the same total amount of honey. If the total amount of honey in all jars is \(T\), find the value of \(T\) given that each jar conta... | [
"(2n + 1)k"
] | [
"Winnie the Pooh has \\(2n+1\\) jars of honey. No matter which jar he excludes, he is always able to separate the remaining \\(2n\\) jars into 2 groups of \\(n\\) jars such that both groups have the same total amount of honey. If the total amount of honey in all jars is \\(T\\), find the value of \\(T\\) given that... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2472709fe123f1e63c15788ddb4c9e5961fd2fdaec00e2ce0eca053aa16a0b7 | Winnie the Pooh has found a beehive! He knows that there are many bees, over $9000$! (Ten thousand to be exact.) He wants to get the honey, but doesn't know if it's worth the risk. He wants to know if he has a guaranteed $90\%$ chance of staying alive. The probability that a bee stings Winnie is $0.1\%$. Write an expre... | [
"1 - (0.999)^{10000}"
] | [
"Winnie the Pooh has found a beehive! He knows that there are many bees, over $9000$! (Ten thousand to be exact.) He wants to get the honey, but doesn't know if it's worth the risk. He wants to know if he has a guaranteed $90\\%$ chance of staying alive. The probability that a bee stings Winnie is $0.1\\%$. Write a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cfd261212c92ae3d91f655144b8cc8ba9af32035380e65b51699b74434478a5a | Winnie-the-Pooh has 27 honey pots containing a total of 17 kilograms of honey, with each pot containing a different amount of honey. Every day, Winnie selects 7 pots and eats exactly $x$ kilograms of honey from each of the selected pots, where $x$ is a real number. Determine the minimum number of days required for Winn... | [
"54"
] | [
"Winnie-the-Pooh has 27 honey pots containing a total of 17 kilograms of honey, with each pot containing a different amount of honey. Every day, Winnie selects 7 pots and eats exactly $x$ kilograms of honey from each of the selected pots, where $x$ is a real number. Determine the minimum number of days required for... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5e429397219f9ecd8554a51f1250840eb4782f5cd7b1b1b520ba50afdbae0f0f | Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each of 25 donkeys standing in a row, such that successive donkeys receive balloons of different colours, and at least one balloon of each colour is given. How many ways can Winnie-the-Pooh achieve this? | [
"\\(\\sum_{m = 0}^7 (-1)^{m-1}\\binom {7} {m} m(m-1)^{24}\\)"
] | [
"Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each of 25 donkeys standing in a row, such that successive donkeys receive balloons of different colours, and at least one balloon of each colour is given. How many ways can Winnie-the-Pooh achieve this?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-8f0fda5b3129879ebb45c3063e24e9a6751bf228a0e5ec04756e0bc647c26251 | Winona can purchase a pencil and two pens for 250 cents, or two pencils and three pens for 425 cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens. | [
"950"
] | [
"Winona can purchase a pencil and two pens for 250 cents, or two pencils and three pens for 425 cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-acfc669e8e050a16a21849a90414c3edba7edf27ae6f81d43ffc201edd6b1a28 | Winry and Riza are each writing the integers $1, 2, \ldots, N$ on a chalkboard. Winry erases all occurrences of the digit 1, and Riza erases all occurrences of the digit 3. They notice that at $N = 13$, the sum of the digits written on the chalkboard on Winry’s list is the same as the sum of the digits written on the c... | [
"172"
] | [
"Winry and Riza are each writing the integers $1, 2, \\ldots, N$ on a chalkboard. Winry erases all occurrences of the digit 1, and Riza erases all occurrences of the digit 3. They notice that at $N = 13$, the sum of the digits written on the chalkboard on Winry’s list is the same as the sum of the digits written on... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-9e6e8eac40b25bf64b98e3a17e3522dce05f9a2ac61ac36dfd04c963e93c4099 | Winston is in the bottom left hand corner of a 4x4 grid, and Julia is in the top right hand corner of the same grid. Each minute, Winston randomly moves 1 step up or to the right along the edges, and Julia simultaneously randomly moves 1 step down or to the left along the edges. After four minutes, find the probability... | [
"\\dfrac{35}{128}"
] | [
"Winston is in the bottom left hand corner of a 4x4 grid, and Julia is in the top right hand corner of the same grid. Each minute, Winston randomly moves 1 step up or to the right along the edges, and Julia simultaneously randomly moves 1 step down or to the left along the edges. After four minutes, find the probab... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35368eac46f86cb95b054693661c742ff3de0655b6100d86a5362dbea11f1772 | Winter is 90 days long. 10 days have precipitation, and 70 are below 0 degrees. When it's below 0 and precipitates, it's snow, but if it's above 0, it is rain. Snow is on the ground the day it snows, and for as long as the temperature stays below 0. What is the expected value of the number of days with snow on the grou... | [
"\\dfrac{100}{3}"
] | [
"Winter is 90 days long. 10 days have precipitation, and 70 are below 0 degrees. When it's below 0 and precipitates, it's snow, but if it's above 0, it is rain. Snow is on the ground the day it snows, and for as long as the temperature stays below 0. What is the expected value of the number of days with snow on the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-94f2799a5d5de12499de8574414178e73868318884c327aa3c13253d5f15f0ea | With $(2n-1)^3$ same small cubes, we build a giant cube. A small cube is considered free if at least three of its faces are not shared with any other small cube. We begin by removing a free cube, which may cause other cubes to become free. What is the minimum number of moves required to remove the small cube which is i... | [
"\\( n^3 \\)"
] | [
"With $(2n-1)^3$ same small cubes, we build a giant cube. A small cube is considered free if at least three of its faces are not shared with any other small cube. We begin by removing a free cube, which may cause other cubes to become free. What is the minimum number of moves required to remove the small cube which... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-140acc1469fa6f18b71d7ab3585437fe031b99acab5ebb3991fb3a69927b65e1 | With $k$ different colours, we want to colour the numbers $2, 3, 4, \ldots, 1024$ such that no number has the same colour as any of its multiples. Find the least possible value of $k$. | [
"10"
] | [
"With $k$ different colours, we want to colour the numbers $2, 3, 4, \\ldots, 1024$ such that no number has the same colour as any of its multiples. Find the least possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-282ea79d36e73816e7c29120aea2d19549a0a2b33b9da153144aa055b521f482 | With 100 planes passing through its center, at most how many parts can a spherical region be divided into? | [
"166751"
] | [
"With 100 planes passing through its center, at most how many parts can a spherical region be divided into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-b6170d7be72a681b3a4a6839e84ce07308a886286caac6c329a6d1559b985a95 | With 8 species of fungus in the lab, Caelestor wants to throw at least 2 of them into the pot and record their reactions. However, he knows that mixing the Beta and Omega samples together could be hazardous. How many possible safe combinations of reactants can Caelestor create (this includes the one mentioned in the pr... | [
"183"
] | [
"With 8 species of fungus in the lab, Caelestor wants to throw at least 2 of them into the pot and record their reactions. However, he knows that mixing the Beta and Omega samples together could be hazardous. How many possible safe combinations of reactants can Caelestor create (this includes the one mentioned in t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-e07d6b7a58b0e1a1cc62fd72e57a0f962bac5daefb385999220c70f41734b623 | With \( \delta \) determined as in Problem 1, define the sequence \( (a_n) \) by \( a_1 = \frac{\delta}{2} \) and \( a_{n+1} = f(a_n) \) for all \( n \ge 1 \). Study the convergence of the series \( \sum_{n=1}^{\infty} a_n^r \) for \( r \in \mathbb{R} \). | [
"r > 0"
] | [
"With \\( \\delta \\) determined as in Problem 1, define the sequence \\( (a_n) \\) by \\( a_1 = \\frac{\\delta}{2} \\) and \\( a_{n+1} = f(a_n) \\) for all \\( n \\ge 1 \\). Study the convergence of the series \\( \\sum_{n=1}^{\\infty} a_n^r \\) for \\( r \\in \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-9811412d38f3f7819a103333f15ddfed2aad8d238c9594c5a0d3ae4c527edc5a | With \( p \in \mathbb{N} \), what is the smallest integer \( n \) such that there exists a set with \( n \) elements containing \( p \) subsets \( (E_i) \) where each subset \( E_i \) has 3 elements and any two distinct subsets \( E_i \) and \( E_j \) have at most one element in common? | [
"\\left\\lceil \\frac{1 + \\sqrt{1 + 24p}}{2} \\right\\rceil"
] | [
"With \\( p \\in \\mathbb{N} \\), what is the smallest integer \\( n \\) such that there exists a set with \\( n \\) elements containing \\( p \\) subsets \\( (E_i) \\) where each subset \\( E_i \\) has 3 elements and any two distinct subsets \\( E_i \\) and \\( E_j \\) have at most one element in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2caa83be0888dc1c5ab1b059399b0d376ad31d3594f76d80a76870a082cd4bd5 | With \(1 \le x, y, z \le 3\), find the maximum value of \(P = 7x + y + z + 9y^2 + 2z^3 - 3xz - 26xyz\). Solve without using derivatives. | [
"13"
] | [
"With \\(1 \\le x, y, z \\le 3\\), find the maximum value of \\(P = 7x + y + z + 9y^2 + 2z^3 - 3xz - 26xyz\\). Solve without using derivatives."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-b34ed0be2bd04cd22c84e6d23abceef11feec91c219769cd1834711d84e66e88 | With \(a\) and \(b\) as in Problem 1, determine the possible values of \(\frac{a^2 + b^2 - 1}{ab}\). | [
"2"
] | [
"With \\(a\\) and \\(b\\) as in Problem 1, determine the possible values of \\(\\frac{a^2 + b^2 - 1}{ab}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-457729927add6cd5b1bf5bc49190aa61557ef6ef2465721c7dc890e279625b2d | With a biased coin where the probability of getting heads is \( \frac{1}{5} \), a player tosses the coin successively until they get two consecutive heads. What is the expected number of such tosses? | [
"30"
] | [
"With a biased coin where the probability of getting heads is \\( \\frac{1}{5} \\), a player tosses the coin successively until they get two consecutive heads. What is the expected number of such tosses?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b8b60bbc7b776d84925d8c6e8c8cf9ea683cffc8ddbe35ab277071b68499561 | With a fair coin, what is the probability of getting two consecutive tails (CC) and the probability of getting one head followed by one tail (CK)? | [
"\\dfrac{1}{4}"
] | [
"With a fair coin, what is the probability of getting two consecutive tails (CC) and the probability of getting one head followed by one tail (CK)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88017c1ff6d454d82255c983a67e633d25d350039dd68c35d91831ca82ed126b | With a positive integer \( n \), denote \( f(n) \) as the largest integer such that \( 2^{f(n)} \) divides \( n+1 \). A pair \((n, p)\) has property \( T \) if \( 2^{f(n)} > p \). Find all triples \((n, p, q)\) such that all pairs \((n, p)\), \((p, q)\), and \((n + p + q, n)\) have property \( T \). | [
"(2^{k}(2^{s} - 1) - 1, 2^{k} - 1, 1)"
] | [
"With a positive integer \\( n \\), denote \\( f(n) \\) as the largest integer such that \\( 2^{f(n)} \\) divides \\( n+1 \\). A pair \\((n, p)\\) has property \\( T \\) if \\( 2^{f(n)} > p \\). Find all triples \\((n, p, q)\\) such that all pairs \\((n, p)\\), \\((p, q)\\), and \\((n + p + q, n)\\) have property \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-dafbc9ac8abef1610f5006412feb10948759dce7c16deed1f7f827aeb70aa7fb | With a rigged coin where the probability of obtaining heads is 0.2, a player tosses the coin successively until he obtains two consecutive heads. What is the expected number of trials? | [
"30"
] | [
"With a rigged coin where the probability of obtaining heads is 0.2, a player tosses the coin successively until he obtains two consecutive heads. What is the expected number of trials?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-8c8f5c8c9f29e6e82ea413282524f02e30b9407faa5629f21d02b8ad0f9eb780 | With a roller shear, rectangular sheets of $1420$ mm wide should be made, namely with a width of $500$ mm and a total length of $1000$ m as well as a width of $300$ mm and a total length of $1800$ m can be cut. So far it has been based on the attached drawing cut, in which the gray area represents the waste, which is q... | [
"3.6\\%"
] | [
"With a roller shear, rectangular sheets of $1420$ mm wide should be made, namely with a width of $500$ mm and a total length of $1000$ m as well as a width of $300$ mm and a total length of $1800$ m can be cut. So far it has been based on the attached drawing cut, in which the gray area represents the waste, which... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-21f6be025275fcbcddb88621ae15d1994304e1d27dc7430954fe32dd965bba8d | With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | [
"91"
] | [
"With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 101,
"genselect": 5,
"tir": 24
} | 130 | 0 | 0 |
omr-ws-03fc66ef03bb12ee5bafc5ddf8a7bbdaa79abb3f1d9e9df1e0af715395fbb4cb | With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | [
"091",
"91"
] | [
"With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.",
"With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.\n"... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 6
} | 43 | 0 | 0 |
omr-ws-150ad821360f0e3277781d6e6a723560419f7ec7d8b3cee1de2678a6b31739ec | With center at the vertices and at the midpoints of the sides of an equilateral triangle with side length 4, arcs of a circle are drawn to form a figure. What is the shaded area? | [
"\\( 4\\pi - 6\\sqrt{3} \\)"
] | [
"With center at the vertices and at the midpoints of the sides of an equilateral triangle with side length 4, arcs of a circle are drawn to form a figure. What is the shaded area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-8a795b8d162deb2eb806e0f10ae3ba1a1c4f10eb1f3b348f325754f73a33bac8 | With constant acceleration the body covered the distance of 4 m from the point $A$ during the first second, and during the first three seconds the distance between the body and the point $A$ increased to 16 m. Find the relationship between the distance covered by the body and the time elapsed if it is known that at $t ... | [
"$S(t)=\\frac{2}{3}t^2+\\frac{10}{3}t$. %"
] | [
"With constant acceleration the body covered the distance of 4 m from the point $A$ during the first second, and during the first three seconds the distance between the body and the point $A$ increased to 16 m. Find the relationship between the distance covered by the body and the time elapsed if it is known that a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36e336f7ce85148667bd14770e997f16cae5891afe6d0844ee23d1b19e713157 | With five squares with sides of $27$ cm, we form a sequence of figures. In the $4$th figure, what is the area of the gray square? In the $5$th figure, what is the area of the gray square? | [
"64 cm^2 and 28 \\frac{4}{9} cm^2"
] | [
"With five squares with sides of $27$ cm, we form a sequence of figures. In the $4$th figure, what is the area of the gray square? In the $5$th figure, what is the area of the gray square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e932a461b7e6ffb3f2eb8c50b2d4ab84f5f63cc69eaf0e2694d3cc6e5a3b1c25 | With four women and six men, how many ways are there to select 2 teams each of 5 people such that there is at least one woman on each team? | [
"120"
] | [
"With four women and six men, how many ways are there to select 2 teams each of 5 people such that there is at least one woman on each team?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc4eafe804a9fda18caeb8db77f4d7d592096b6be1d02bf6de1147bb229ae3bc | With how many zeros can the decimal writing of $1^n + 2^n + 3^n + 4^n$ end, where $n$ is a positive integer? | [
"2"
] | [
"With how many zeros can the decimal writing of $1^n + 2^n + 3^n + 4^n$ end, where $n$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-c7663e3cb40a064b8603f1d8535e26890ea5a408775298e7bf41e8eea3611293 | With only a 5-gallon bucket and a 3-gallon bucket, how can you isolate one gallon of a 50-50 water-alcohol solution from an infinite supply of pure water and 5 gallons of pure alcohol? | [
"1 gallon of 50-50 solution"
] | [
"With only a 5-gallon bucket and a 3-gallon bucket, how can you isolate one gallon of a 50-50 water-alcohol solution from an infinite supply of pure water and 5 gallons of pure alcohol?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-143bd8a062d64cd4347b42a3d93a346d5b18e677ec338c079a3ce2b12ee349d7 | With the numbers from [1 - N] in increasing order, how many sub-lists have the sum N? Example: If N = 5, {1, 2, 3, 4, 5} The number of sublists that have a sum of 5 is 2: {2,3} and {5}. | [
"\\text{Number of odd divisors of } N"
] | [
"With the numbers from [1 - N] in increasing order, how many sub-lists have the sum N?\n\nExample:\nIf N = 5,\n{1, 2, 3, 4, 5}\nThe number of sublists that have a sum of 5 is 2:\n{2,3} and {5}."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.