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-dd756abe618186579639e4e98ac62084a97c1303b73e24eda519c0730b28e19c | An infinite sequence \(a_n\) is defined by \(a_1 = 2\), \(a_2 = 3\), and \(a_{k+2} = \frac{a_{k+1}}{a_k}\) for \(k \ge 1\). Find \(a_{2010}\). | [
"\\(\\frac{1}{3}\\)"
] | [
"An infinite sequence \\(a_n\\) is defined by \\(a_1 = 2\\), \\(a_2 = 3\\), and \\(a_{k+2} = \\frac{a_{k+1}}{a_k}\\) for \\(k \\ge 1\\). Find \\(a_{2010}\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6adef7eee4597be5f305eb615b9402e76602b919be0edc5df1cb97c1979107a | An infinite sequence of integers, \(a_0, a_1, a_2, \dots\), with \(a_0 > 0\), has the property that for \(n \ge 0\), \(a_{n+1} = a_n - b_n\), where \(b_n\) is the number having the same sign as \(a_n\) but with the digits written in reverse order. For example, if \(a_0 = 1210\), then \(a_1 = 1089\) and \(a_2 = -8712\),... | [
"1012"
] | [
"An infinite sequence of integers, \\(a_0, a_1, a_2, \\dots\\), with \\(a_0 > 0\\), has the property that for \\(n \\ge 0\\), \\(a_{n+1} = a_n - b_n\\), where \\(b_n\\) is the number having the same sign as \\(a_n\\) but with the digits written in reverse order. For example, if \\(a_0 = 1210\\), then \\(a_1 = 1089\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a8328fca4abc8ced50a4a38c6fd212cf2d1d54aaad14ca321965e23b37ae4a8c | An infinite sequence of positive integers $(a_n)$ is defined by the following rules: $a_1=1$, for any positive integer $n\geq 2$, $a_n$ is the smallest positive integer different from $a_1,a_2,...,a_{n-1}$ so that $\sum_{i=1}^na_i$ is divisible by $n$. A positive integer $t$ is called "bad" if $a_t\leq t$. In the real ... | [
"no_valid_answer_found"
] | [
"An infinite sequence of positive integers $(a_n)$ is defined by the following rules: $a_1=1$, for any positive integer $n\\geq 2$, $a_n$ is the smallest positive integer different from $a_1,a_2,...,a_{n-1}$ so that $\\sum_{i=1}^na_i$ is divisible by $n$. A positive integer $t$ is called \"bad\" if $a_t\\leq t$. In... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed97fe5f62eb637ae8fe675470c6f12bf2f2d1e96c8647f5eb508672536327f3 | An infinite sequence of positive integers \(a_1, a_2, a_3, \ldots\) has the property that \(a_1a_2a_3 \cdots a_n = a_{n+1}\) for all positive integers \(n\). Given \(a_1 = k\), the 2014th term of the sequence can be expressed as \(k^m\) for some real number \(m\). Furthermore, \(m\) can be expressed as \(x^y\) for real... | [
"2014"
] | [
"An infinite sequence of positive integers \\(a_1, a_2, a_3, \\ldots\\) has the property that \\(a_1a_2a_3 \\cdots a_n = a_{n+1}\\) for all positive integers \\(n\\). Given \\(a_1 = k\\), the 2014th term of the sequence can be expressed as \\(k^m\\) for some real number \\(m\\). Furthermore, \\(m\\) can be expresse... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0055e72893eb3217d2dcdaf25dccc7905ce02fbb7b2f1869f0008f1225a8e1fb | An infinite sequence of positive real numbers \( x_0, x_1, x_2, \ldots \) is called \( \text{vasco} \) if it satisfies the following properties: (a) \( x_0 = 1 \) and \( x_1 = 3 \); (b) \( x_0 + x_1 + \cdots + x_{n-1} \ge 3x_n - x_{n+1} \) for every \( n \ge 1 \). Find the greatest real number \( M \) such that, for ev... | [
"2"
] | [
"An infinite sequence of positive real numbers \\( x_0, x_1, x_2, \\ldots \\) is called \\( \\text{vasco} \\) if it satisfies the following properties:\n(a) \\( x_0 = 1 \\) and \\( x_1 = 3 \\);\n(b) \\( x_0 + x_1 + \\cdots + x_{n-1} \\ge 3x_n - x_{n+1} \\) for every \\( n \\ge 1 \\).\nFind the greatest real number ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-e74068ff5c7d5f58ac7dbb7772de1554dea25b095f0f61db1993db4c6743986a | An infinite sequence of positive real numbers is defined by $a_0 = 1$ and $a_{n+2} = 6a_n - a_{n+1}$ for $n = 0, 1, 2, \cdots$. Find the possible value(s) of $a_{2007}$. | [
"2^{2007}"
] | [
"An infinite sequence of positive real numbers is defined by $a_0 = 1$ and $a_{n+2} = 6a_n - a_{n+1}$ for $n = 0, 1, 2, \\cdots$. Find the possible value(s) of $a_{2007}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ebf57f4097fcd93445a5371bb2723e5a68e5669c1bffb309b901c46ef88dc08 | An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2 \times 3$ rectangle, there are exactly two red squares. How many red squares are there in an arbitrary $9 \times 11$ rectangle? (The sides of all considered rectangles go along the grid lines.) | [
"33"
] | [
"An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2 \\times 3$ rectangle, there are exactly two red squares. How many red squares are there in an arbitrary $9 \\times 11$ rectangle? (The sides of all considered rectangles go along the grid lines.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-04d250a2d1037b0820a81f595e064ea44aebb701d31ed1251c144c95bc1557e2 | An infinite square grid is colored using $n^2$ colors such that all unit squares of any $n \times n$ square are of different colors. It is known that any infinite row contains all the colors. Determine the minimum number of colors that must appear in some infinite column. | [
"n"
] | [
"An infinite square grid is colored using $n^2$ colors such that all unit squares of any $n \\times n$ square are of different colors. It is known that any infinite row contains all the colors. Determine the minimum number of colors that must appear in some infinite column."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-7112f892ffc7658e566dce2ba09e9aec58b804663b59962eb2e226d1b83900c1 | An infinite square grid is colored with $n^2$ colors such that all unit squares in any $n \times n$ square are of different colors. It is known that any infinite row contains all the colors. If there exists an infinite column containing exactly $k$ colors, find the maximum possible value of $k$. | [
"n^2"
] | [
"An infinite square grid is colored with $n^2$ colors such that all unit squares in any $n \\times n$ square are of different colors. It is known that any infinite row contains all the colors. If there exists an infinite column containing exactly $k$ colors, find the maximum possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-dd5645c4f943ca3d7d389e7b8f07774b4db358dba0bc32cb70a8072ac9e457e7 | An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1 - p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads... | [
"\\dfrac{7}{9}"
] | [
"An infinitely large bag of coins is such that for every $0.5 < p \\le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1 - p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8427b5dd8d962022d5a038d03ac14ed717821b744a94f9c32b484c34d2d8a5e7 | An innkeeper has a 1-liter bottle full of wine and a half-liter empty bottle. The innkeeper pours some wine into the half-liter bottle, fills the rest of the half-liter bottle with water, and then pours the mixture back into the 1-liter bottle. What is the maximum percentage of water that can replace the wine in the 1-... | [
"50%"
] | [
"An innkeeper has a 1-liter bottle full of wine and a half-liter empty bottle. The innkeeper pours some wine into the half-liter bottle, fills the rest of the half-liter bottle with water, and then pours the mixture back into the 1-liter bottle. What is the maximum percentage of water that can replace the wine in t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df04395179b5be346b5cbc3f2c2fe03b3cf72dfdab2a2d33d431c37883819856 | An inscribed $n$-gon ($n > 3$) is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles that can be obtained? (An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle.) | [
"\\( \\lfloor \\frac{n}{2} \\rfloor \\)"
] | [
"An inscribed $n$-gon ($n > 3$) is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles that can be obtained? (An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 3,
"tir": 6
} | 33 | 0 | 0 |
omr-ws-61db35e97717251c6fa520d65503733aeec869214c0b345ef456463aa625dc1e | An inscribed and circumscribed $n$-gon is divided by a line into two inscribed and circumscribed polygons with different numbers of sides. Find $n$. | [
"6"
] | [
"An inscribed and circumscribed $n$-gon is divided by a line into two inscribed and circumscribed polygons with different numbers of sides. Find $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16fede582b18fced243f697acf59c182b6e14bfab5ee7ffdaad69c1ffdb1e128 | An inscribed and circumscribed \(n\)-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find \(n\). | [
"5"
] | [
"An inscribed and circumscribed \\(n\\)-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-7758d859240ed7a2c386e50542b6e702f59f2d6ffd168776116864311959c6b2 | An inscribed quadrilateral $ABCD$ is given. Lines $AD$ and $BC$ intersect at point $E$, lines $AC$ and $BD$ intersect at point $F$. Point $M$ is the midpoint of $CD$. On the circumscribed circle of triangle $ABM$, a point $N$ distinct from $M$ is taken such that $\frac{AN}{NB}=\frac{AM}{MB}$. Calculate the value of $\f... | [
"2"
] | [
"An inscribed quadrilateral $ABCD$ is given. Lines $AD$ and $BC$ intersect at point $E$, lines $AC$ and $BD$ intersect at point $F$. Point $M$ is the midpoint of $CD$. On the circumscribed circle of triangle $ABM$, a point $N$ distinct from $M$ is taken such that $\\frac{AN}{NB}=\\frac{AM}{MB}$. Calculate the value... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-a9787a850318991a4584a3d656eb72422d5f648fe67cc7ed46ffde60c22dd712 | An inscribed quadrilateral $ABCD$ is given. We consider the variable circular arc $\Omega$ passing through points $A$ and $D$ and lying in the half-plane with respect to $AD$ that does not contain quadrilateral $ABCD$. The lines through $B$ and $C$ touch $\Omega$ at points $X$ and $Y$, respectively, with ray $BA$ lying... | [
"no_valid_answer_found"
] | [
"An inscribed quadrilateral $ABCD$ is given. We consider the variable circular arc $\\Omega$ passing through points $A$ and $D$ and lying in the half-plane with respect to $AD$ that does not contain quadrilateral $ABCD$. The lines through $B$ and $C$ touch $\\Omega$ at points $X$ and $Y$, respectively, with ray $BA... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff4e262ae80ee9e935334a94e92bc156589f725312c620c905ee5b8768a195f6 | An inscribed quadrilateral \(ABCD\) has \(AD\) as a diameter. Given \(AB = 3\), \(BC = 4\), and \(CD = 5\), find the value of \(d\) where \(DA = d\). | [
"2\\sqrt{13}"
] | [
"An inscribed quadrilateral \\(ABCD\\) has \\(AD\\) as a diameter. Given \\(AB = 3\\), \\(BC = 4\\), and \\(CD = 5\\), find the value of \\(d\\) where \\(DA = d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-160ea70a31f6f68bc6b008de9146093ae493d26380b930517a1a4544b2a674a8 | An inscribed quadrilateral \(ABCD\) is given. Let \(E\) be the intersection point of its diagonals, and \(K\), \(L\), \(M\), \(N\) the midpoints of the sides \(AB\), \(BC\), \(CD\), \(DA\) respectively. If the radius of the circle circumscribed around triangle \(KLE\) is \(r\), find the value of \(r^2\) given that the ... | [
"\\dfrac{25}{2}"
] | [
"An inscribed quadrilateral \\(ABCD\\) is given. Let \\(E\\) be the intersection point of its diagonals, and \\(K\\), \\(L\\), \\(M\\), \\(N\\) the midpoints of the sides \\(AB\\), \\(BC\\), \\(CD\\), \\(DA\\) respectively. If the radius of the circle circumscribed around triangle \\(KLE\\) is \\(r\\), find the val... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-f90cfa12170b6f00dea020dc986ede9aba7249bcf2c0e8be34ce4188709bbc04 | An inscribed quadrilateral \(ABCD\) is given. Let \(H\) be the projection of point \(A\) on \(BC\), and let \(A'\) be the point symmetric to point \(A\) with respect to \(BC\). The point \(P\) is chosen on the side \(CD\), and the point \(K\) is chosen on the circumscribed circle of the triangle \(ABH\) so that \(HK \p... | [
"2"
] | [
"An inscribed quadrilateral \\(ABCD\\) is given. Let \\(H\\) be the projection of point \\(A\\) on \\(BC\\), and let \\(A'\\) be the point symmetric to point \\(A\\) with respect to \\(BC\\). The point \\(P\\) is chosen on the side \\(CD\\), and the point \\(K\\) is chosen on the circumscribed circle of the triangl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9dec6dec86dda965c895bd8f2051cb6e94c6116de7f5dd43bfd134fd98230826 | An inscribed quadrilateral \(ABCD\) is given. Let \(P\), \(Q\), \(R\), and \(S\) be the centers of the circles inscribed in triangles \(ABD\), \(ABC\), \(BCD\), and \(ACD\) respectively. If the lengths of the sides of the rectangle formed by \(P\), \(Q\), \(R\), and \(S\) are \(a\) and \(b\), calculate the value of \(a... | [
"no_valid_answer_found"
] | [
"An inscribed quadrilateral \\(ABCD\\) is given. Let \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) be the centers of the circles inscribed in triangles \\(ABD\\), \\(ABC\\), \\(BCD\\), and \\(ACD\\) respectively. If the lengths of the sides of the rectangle formed by \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) are \\(a\\) and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c73bd9775eb43de835b7b8cd3bd815aceef7cfa3df7993a53826c862c4c71cb | An inscribed sphere of a regular $n$-gonal prism touches all the faces of the prism. Another sphere is circumscribed about the prism. Find the ratio of the volume of the latter to that of the former. | [
"\\left(1+\\frac{1}{\\cos^2 \\frac{\\pi }{n}}\\right)^\\frac{3}{2}"
] | [
"An inscribed sphere of a regular $n$-gonal prism touches all the faces of the prism. Another sphere is circumscribed about the prism. Find the ratio of the volume of the latter to that of the former."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e8876f3a62d3440585a5cb03697c13f7d62912ec83e65f3f494aaffcf4d318a | An insect has just told me that she has laid $154_6$ eggs. In base 10, how many eggs did she lay? | [
"70"
] | [
"An insect has just told me that she has laid $154_6$ eggs. In base 10, how many eggs did she lay?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c03f6b3d4df1bc1aa0514161c43be8b2014924666f244a4dbae24c31e77f372 | An insect is situated on a vertex of a regular tetrahedron. It can move to any of the three adjacent vertices in each move and can retrace its path. In how many ways can the insect move such that after 6 steps it returns to its starting position? | [
"183"
] | [
"An insect is situated on a vertex of a regular tetrahedron. It can move to any of the three adjacent vertices in each move and can retrace its path. In how many ways can the insect move such that after 6 steps it returns to its starting position?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-5df91d7204f790c8d1f808042b1539f58d64a3025f92317e1d05c089797b87df | An insect lives on the surface of a regular tetrahedron with edges of length $1$. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common ... | [
"1"
] | [
"An insect lives on the surface of a regular tetrahedron with edges of length $1$. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no co... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b62841b679db9e5886043831024ab40a6310c3f1c55104fd3bb3b16cdd03fcf3 | An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | [
"1"
] | [
"An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no comm... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7a564ee7771b68da344793535038c705806b23dc47ac7ba1c09682fa6c27e6b4 | An insect starts at the southernmost point of a horizontal disk with a diameter of 3 inches, which rotates once every 15 seconds. The insect crawls at a speed of 1 inch per second, always facing due north. Determine where the insect reaches the edge of the disk again. | [
"36 degrees east of the northernmost point.",
"\\( \\frac{\\pi}{5} \\) radians from the northernmost point."
] | [
"An insect starts at the southernmost point of a horizontal disk with a diameter of 3 inches, which rotates once every 15 seconds. The insect crawls at a speed of 1 inch per second, always facing due north. Determine where the insect reaches the edge of the disk again."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 13
} | 31 | 0 | 0 |
omr-ws-55952eb2391c8f33e1054dca71fd1a1d6a9729188340472ecb2556b3206be14a | An insect walks along the diagonal of a rectangular floor with dimensions 321 units by 123 units, tiled in a checkerboard fashion with square tiles of side length 1 unit. How many 4-corner intersection points does the insect cross on its walk between diagonally opposite corners? | [
"3"
] | [
"An insect walks along the diagonal of a rectangular floor with dimensions 321 units by 123 units, tiled in a checkerboard fashion with square tiles of side length 1 unit. How many 4-corner intersection points does the insect cross on its walk between diagonally opposite corners?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d480f5d5e73aec24d2e7bb36539e5a3b15cd1c22f4a337e1d45b151989a3fb7 | An inspecting officer starts at the rear of a one-kilometer-long column of men marching in a straight line and moves forward at a constant speed until he reaches the front. He then turns around and travels at the same speed until he reaches the last man in the rear. By this time, the column has moved one kilometer forw... | [
"1 + \\sqrt{2}"
] | [
"An inspecting officer starts at the rear of a one-kilometer-long column of men marching in a straight line and moves forward at a constant speed until he reaches the front. He then turns around and travels at the same speed until he reaches the last man in the rear. By this time, the column has moved one kilometer... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fea537406c223a47af6317c2ba68fdc4983394b226bc72ca41eca66f98805498 | An inspector in a factory has a 95% chance of correctly identifying a defective part and a 2% chance of incorrectly classifying a good part as defective. 1% of all items are defective. If an item is classified as good, what is the probability that it is actually good? | [
"0.99948"
] | [
"An inspector in a factory has a 95% chance of correctly identifying a defective part and a 2% chance of incorrectly classifying a good part as defective. 1% of all items are defective. If an item is classified as good, what is the probability that it is actually good?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-5db59472495927a6f52f4f2436171baa6cf7ad38e4b8417555d67e5cdc0119cb | An insurance company designates 10% of its customers as high risk and 90% as low risk. The number of claims made by a customer in a calendar year is Poisson distributed with mean 'theta' and is independent of the number of claims made by a customer in any previous calendar year. For high-risk customers, theta = 0.6, wh... | [
"\\(\\frac{0.036 \\cdot e^{-0.6} + 0.009 \\cdot e^{-0.1}}{0.06 \\cdot e^{-0.6} + 0.09 \\cdot e^{-0.1}}\\)"
] | [
"An insurance company designates 10% of its customers as high risk and 90% as low risk. The number of claims made by a customer in a calendar year is Poisson distributed with mean 'theta' and is independent of the number of claims made by a customer in any previous calendar year. For high-risk customers, theta = 0.... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0adba80505a52e794fb190dcafb03c89fdad299ff06869d634e8f76be15fe48 | An insurance policy sells for \$800. Based on past data, an average of 1 in 100 policyholders will file a \$15,000 claim, and an average of 1 in 250 policyholders will file an \$80,000 claim. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected profit or l... | [
"3300000"
] | [
"An insurance policy sells for \\$800. Based on past data, an average of 1 in 100 policyholders will file a \\$15,000 claim, and an average of 1 in 250 policyholders will file an \\$80,000 claim. Find the expected value (to the company) per policy sold. If the company sells 10,000 policies, what is the expected pro... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1774b37c806c3bbfcb64271b2526b428710d9981d7028e95ed94d5aee7be202 | An integer $A$ is called a "lucky number of Euler" if the expression $x^2 - x + A$ is prime for all $0 \leq x < A$. Given that $\mathbb{Q}(\sqrt{1-4A})$ is a unique factorisation domain, find the sum of all "lucky numbers of Euler" less than 100. | [
"79"
] | [
"An integer $A$ is called a \"lucky number of Euler\" if the expression $x^2 - x + A$ is prime for all $0 \\leq x < A$. Given that $\\mathbb{Q}(\\sqrt{1-4A})$ is a unique factorisation domain, find the sum of all \"lucky numbers of Euler\" less than 100."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e080ccb36e25fa373ba19d443da7a0af0d60f48f9ef1a19f83de10b023b13da | An integer $N$ is worth $1$ point for each adjacent pair of digits that form a perfect square of a positive integer. Compute the smallest positive integer that is worth $5$ points. | [
"1641649"
] | [
"An integer $N$ is worth $1$ point for each adjacent pair of digits that form a perfect square of a positive integer. Compute the smallest positive integer that is worth $5$ points."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-e428137154978a00ba9f8b64f618555824037ddb2eaf7723c50f35aff2a4e213 | An integer $a$ has just two digits. When the digits are reversed, the resulting number $b$ is $p\, \%$ larger than $a$. Given that $p$ is an odd integer, find the largest value of $p$. | [
"75%"
] | [
"An integer $a$ has just two digits. When the digits are reversed, the resulting number $b$ is $p\\, \\%$ larger than $a$. Given that $p$ is an odd integer, find the largest value of $p$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-3767672e6b60d9a3f2fd08724cf1ad30fcbaf5047d38be093465a106dd16f7de | An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. Let $S$ be the sum of all friendly integers in the set $\{ 1,2,\ldots ,2012\}$. Find the remainder when $S$ is divided by $1000$. | [
"733"
] | [
"An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. Let $S$ be the sum of all friendly integers in the set $\\{ 1,2,\\ldots ,2012\\}$. Find the remainder when $S$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d081ad6ecbbd631b1d8d4379eb7aef37f5548ee94ec067a6f8c04d0afcd68e47 | An integer $k > 1$ is called clean if it can be expressed as the sum of two positive integers, each of which is not divisible by $3$. Given that $k$ is clean and not divisible by $3$, find the remainder when $k + 1$ is divided by $6$. | [
"2"
] | [
"An integer $k > 1$ is called clean if it can be expressed as the sum of two positive integers, each of which is not divisible by $3$. Given that $k$ is clean and not divisible by $3$, find the remainder when $k + 1$ is divided by $6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-25db248243dfb8483f28619c346fa202098e7771bbdac85d3c4da2ec16d1c7af | An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$. | [
"\\( n = 3 \\)"
] | [
"An integer $n \\ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9020ccd32b4b13b724b9ad19b798ec2853b768ac2797ac1f041baa5344364af7 | An integer $n \ge 1$ is called conservative if the smallest prime divisor of $(n!)^n + 1$ is at most $n + 2015$. Let $S$ be the sum of the first 10 conservative integers. Find the remainder when $S$ is divided by $1000$. | [
"74"
] | [
"An integer $n \\ge 1$ is called conservative if the smallest prime divisor of $(n!)^n + 1$ is at most $n + 2015$. Let $S$ be the sum of the first 10 conservative integers. Find the remainder when $S$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf2eb5d45afae9e43cba73e05346605f248dee921c849aa95c4c87839e25f49b | An integer $n$ has been randomly chosen from the interval $[1, 2002]$ with uniform probability. Your objective is to guess $n$ in an odd number of guesses. After each incorrect guess, you are informed whether $n$ is higher or lower, and you must guess an integer on your next turn among the numbers that are still feasib... | [
"11"
] | [
"An integer $n$ has been randomly chosen from the interval $[1, 2002]$ with uniform probability. Your objective is to guess $n$ in an odd number of guesses. After each incorrect guess, you are informed whether $n$ is higher or lower, and you must guess an integer on your next turn among the numbers that are still f... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6659468db2284829e18a1af9b73c4c1f3ae9b4d1ce9747b5810fc02038babb6f | An integer $n$ is called bad if there exist distinct primes $p$ and $q$ such that $pq$ divides $n^p + n^q$. Find the smallest bad integer greater than $1000$. | [
"1001"
] | [
"An integer $n$ is called bad if there exist distinct primes $p$ and $q$ such that $pq$ divides $n^p + n^q$. Find the smallest bad integer greater than $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a064d4168386f4ca06c1834b5304592efea4456849cdea55b0196759d9889945 | An integer $n$ is called fatal if it cannot be expressed as the sum of two positive perfect squares and a positive perfect cube. Determine the sum of all integers greater than 16 and less than 1000 that are not multiples of 6 and are fatal. | [
"1060"
] | [
"An integer $n$ is called fatal if it cannot be expressed as the sum of two positive perfect squares and a positive perfect cube. Determine the sum of all integers greater than 16 and less than 1000 that are not multiples of 6 and are fatal."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-3c2eee0565c64519d6aa033e9d1df6a4726bfdde9599752b15c5ab1c5072d6a3 | An integer $n$ is called naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$, and $90$ are naoish, whereas $9900$, $2009$, and $9$ are not. Express the number $2020$ as a sum of naoish integers: \[ 2020 = n_{1} + n_{2} + \ldots + n_{k} \] What is... | [
"8"
] | [
"An integer $n$ is called naoish if $n \\geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$, and $90$ are naoish, whereas $9900$, $2009$, and $9$ are not. Express the number $2020$ as a sum of naoish integers:\n \\[\n 2020 = n_{1} + n_{2} + \\ldots ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8cefa4e76c0a5674d31b5d4867b98da034af890802659c08aa9bd2c0658faee2 | An integer $n$ is chosen at random from the set $\{1, 2, ..., 2014\}$ and an integer $k$ is chosen at random from the set $\{0, 1, 2\}$. Then a deck of $3n$ cards labeled $\{1, 1, 1, 2, 2, 2, ..., n, n, n\}$ is produced, and a card is randomly selected and removed from this deck. At this point $k$ cards labeled $0$ are... | [
"\\dfrac{2}{6043}"
] | [
"An integer $n$ is chosen at random from the set $\\{1, 2, ..., 2014\\}$ and an integer $k$ is chosen at random from the set $\\{0, 1, 2\\}$. Then a deck of $3n$ cards labeled $\\{1, 1, 1, 2, 2, 2, ..., n, n, n\\}$ is produced, and a card is randomly selected and removed from this deck. At this point $k$ cards labe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cff80c94d3a433ad106504156c197d891e487d47ed6aa10d28be5b3319b92345 | An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | [
"40"
] | [
"An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eea3b8edc74a82895c8317b428b09b0c370c3dd01a253b803663e657653881e | An integer \( N = \overline{a_n a_{n-1} \ldots a_2 a_1} \) is called a "Googly" number if \( \overline{a_n a_1} \) divides \( N \). Find all such "Googly" numbers for \( n = 2, 3, 4 \). | [
"854"
] | [
"An integer \\( N = \\overline{a_n a_{n-1} \\ldots a_2 a_1} \\) is called a \"Googly\" number if \\( \\overline{a_n a_1} \\) divides \\( N \\). Find all such \"Googly\" numbers for \\( n = 2, 3, 4 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a42cbc26ba187983f0449b5ba1dc8b3c5c9e28664667984f30b5dc3562a4a33e | An integer \( N \) is reduced to \( \frac{1}{9} \) its value when one digit is deleted, and the resulting number \( M \) is again divisible by 9. If \( M \) is divided by 9, the resulting integer \( K \) has one fewer digit than \( M \). Find the sum of all possible values of \( K \) if \( N \) is a 5-digit number. | [
"3500"
] | [
"An integer \\( N \\) is reduced to \\( \\frac{1}{9} \\) its value when one digit is deleted, and the resulting number \\( M \\) is again divisible by 9. If \\( M \\) is divided by 9, the resulting integer \\( K \\) has one fewer digit than \\( M \\). Find the sum of all possible values of \\( K \\) if \\( N \\) is... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6238eee26d8cc2fd85861cf4efb1c454742f98c911cd86e5ebd3ed04e5cb4289 | An integer \( k \) is called "developed" if the tens digit of \( k^2 + k \) is equal to 0. How many positive integers \( x \leq 1000 \) are developed? | [
"180"
] | [
"An integer \\( k \\) is called \"developed\" if the tens digit of \\( k^2 + k \\) is equal to 0. How many positive integers \\( x \\leq 1000 \\) are developed?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 23
} | 33 | 0 | 0 |
omr-ws-23c46d36695c005771ca759a88a3ee5414b9e0f7dc4a5b94f9529f570fe84c54 | An integer \( k \) is called cultivated if the tens digit of \( k^2 + k \) is equal to 0. How many integers \( 0 < x \leq 1000 \) are cultivated? | [
"180"
] | [
"An integer \\( k \\) is called cultivated if the tens digit of \\( k^2 + k \\) is equal to 0. How many integers \\( 0 < x \\leq 1000 \\) are cultivated?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 22
} | 34 | 0 | 0 |
omr-ws-80006d9ccd416c97e099de192110ced3fc4d6486bccce2ae71e387e0fe65a38b | An integer \( m \ge 1 \) is a Bunafezzu number if at least one of \( 5m^2 - 4 \) and \( 5m^2 + 4 \) is a perfect square. Find all Bunafezzu numbers. | [
"Fibonacci numbers"
] | [
"An integer \\( m \\ge 1 \\) is a Bunafezzu number if at least one of \\( 5m^2 - 4 \\) and \\( 5m^2 + 4 \\) is a perfect square. Find all Bunafezzu numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-b8388426743af2bb4658435f0ceb9b624fdecc377d8da6ad720c9be7d9d46495 | An integer \( n > 1 \) is called [i]good[/i] if for any natural numbers \( 1 \le b_1, b_2, \ldots, b_{n-1} \le n-1 \) and any \( i \in \{0, 1, \ldots, n-1\} \), there exists a subset \( I \) of \( \{1, \ldots, n-1\} \) such that \( \sum_{k \in I} b_k \equiv i \pmod{n} \). Find all good numbers. | [
"Prime numbers"
] | [
"An integer \\( n > 1 \\) is called [i]good[/i] if for any natural numbers \\( 1 \\le b_1, b_2, \\ldots, b_{n-1} \\le n-1 \\) and any \\( i \\in \\{0, 1, \\ldots, n-1\\} \\), there exists a subset \\( I \\) of \\( \\{1, \\ldots, n-1\\} \\) such that \\( \\sum_{k \\in I} b_k \\equiv i \\pmod{n} \\). Find all good nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4a00c17ddd1c416cf187faf532572057794a5bbbf814e83420ffe05dd6e2e723 | An integer \( n \) has 302481 digits in base-9261. What is the difference between the largest and smallest number of digits that \( n \) could have in base 21? | [
"2"
] | [
"An integer \\( n \\) has 302481 digits in base-9261. What is the difference between the largest and smallest number of digits that \\( n \\) could have in base 21?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c019333c19eb905a839dbffd2c1fe7e5cae760bb42d0e968abf073dfe59bdaed | An integer \( n \) is called "good" if it can be written as \( n = a_1 + a_2 + \cdots + a_k \) where \( a_1, a_2, \ldots, a_k \) are positive integers (not necessarily distinct) satisfying \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1 \). Given that the integers 33 through 73 are good, find the smallest... | [
"33"
] | [
"An integer \\( n \\) is called \"good\" if it can be written as \\( n = a_1 + a_2 + \\cdots + a_k \\) where \\( a_1, a_2, \\ldots, a_k \\) are positive integers (not necessarily distinct) satisfying \\( \\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_k} = 1 \\). Given that the integers 33 through 73 are go... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-b38a0f2970b83fb7148bc832ff242be2d1f72e8a6f87a1db6368d6800a56b5c7 | An integer \( n \) is called "good" if it can be written as \( n = a_1 + a_2 + \cdots + a_k \), where \( a_1, a_2, \ldots, a_k \) are positive integers satisfying \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1 \). Given that the integers 33 through 73 are good, find the smallest integer \( m \geq 33 \) s... | [
"78"
] | [
"An integer \\( n \\) is called \"good\" if it can be written as \\( n = a_1 + a_2 + \\cdots + a_k \\), where \\( a_1, a_2, \\ldots, a_k \\) are positive integers satisfying \\( \\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_k} = 1 \\). Given that the integers 33 through 73 are good, find the smallest inte... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-ba3f48522a2b1a17db0a6fbab7c0af5b89596949bd65229566772e26b4583904 | An integer \( n \) is called a good number if and only if \( |n| \) is not the square of another integer. Find all integers \( m \) such that they can be written in infinitely many ways as the sum of three different good numbers, and the product of these three numbers is the square of an odd number. | [
"m \\equiv 3 \\pmod{4}"
] | [
"An integer \\( n \\) is called a good number if and only if \\( |n| \\) is not the square of another integer. Find all integers \\( m \\) such that they can be written in infinitely many ways as the sum of three different good numbers, and the product of these three numbers is the square of an odd number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-1478fc53d89ce722954ae4428dbcd0bf106e5a082d695f13d97a684a486f4501 | An integer \( n \) is called apocalyptic if the sum of 6 different positive divisors of \( n \) equals 3528. Find the smallest positive apocalyptic number. | [
"1440"
] | [
"An integer \\( n \\) is called apocalyptic if the sum of 6 different positive divisors of \\( n \\) equals 3528. Find the smallest positive apocalyptic number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-f7764e37fcd2a707deafb97fdf04aa1fd352ed170037166b3f177a081e7e8ff3 | An integer \( n \) is called good if \( |n| \) is not a square of an integer. Find all integers \( m \) such that \( m \) can be represented in infinitely many ways as a sum of three distinct good numbers, the product of which is the square of an odd integer. | [
"\\text{All odd integers}"
] | [
"An integer \\( n \\) is called good if \\( |n| \\) is not a square of an integer. Find all integers \\( m \\) such that \\( m \\) can be represented in infinitely many ways as a sum of three distinct good numbers, the product of which is the square of an odd integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-bf3bae951724518fc40f0a6e99e2d05a116f25cfd007fa2d114dc8b03ec3e45d | An integer \( n \) is called silesian if there exist positive integers \( a, b, \) and \( c \) such that \[ n = \frac{a^2 + b^2 + c^2}{ab + bc + ca}. \] Find the smallest positive integer \( k \) that is not silesian. | [
"3"
] | [
"An integer \\( n \\) is called silesian if there exist positive integers \\( a, b, \\) and \\( c \\) such that \n\\[ n = \\frac{a^2 + b^2 + c^2}{ab + bc + ca}. \\]\nFind the smallest positive integer \\( k \\) that is not silesian."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-368561961b2a657eb06523119f96dc4ec107482ecefc91fe768676c06ae0e2db | An integer \( n \) is said to be good if \( |n| \) is not the square of an integer. Determine all integers \( m \) with the following property: \( m \) can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. | [
"\\( m \\equiv 3 \\mod 4 \\)"
] | [
"An integer \\( n \\) is said to be good if \\( |n| \\) is not the square of an integer. Determine all integers \\( m \\) with the following property: \\( m \\) can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-cfebc2a818f77e9446c5cf82a7526d0e9643d1eb21b5c9d256b3f627cf3d68d7 | An integer \( n \ge 1 \) is called "good" if any positive integer divisible by each of the nine numbers \( n + 1, n + 2, \ldots, n + 9 \) is also divisible by \( n + 10 \). How many good integers \( n \ge 1 \) are there? | [
"38"
] | [
"An integer \\( n \\ge 1 \\) is called \"good\" if any positive integer divisible by each of the nine numbers \\( n + 1, n + 2, \\ldots, n + 9 \\) is also divisible by \\( n + 10 \\). How many good integers \\( n \\ge 1 \\) are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-82187fb5a665668c9a5776dc8077f7327b7a467988a58246fc0aef40df629699 | An integer \( n \geq 2 \) is said to be "tuanis" if the sum of its smallest prime divisor and its largest prime divisor (which can be the same) is odd. Calculate the sum of all "tuanis" numbers that are less than or equal to \( 2023 \). | [
"1021086"
] | [
"An integer \\( n \\geq 2 \\) is said to be \"tuanis\" if the sum of its smallest prime divisor and its largest prime divisor (which can be the same) is odd. Calculate the sum of all \"tuanis\" numbers that are less than or equal to \\( 2023 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-9e4b9de3b4bb5aca2e4b89fea46125b794e285c5b06bfbf00cc58844f5702e22 | An integer \( n \geq 3 \) is called *fabulous* if there exists an integer \( a \) with \( 2 \leq a \leq n - 1 \) such that \( a^n - a \) is divisible by \( n \). Find all the *fabulous* integers. | [
"All integers \\( n \\geq 3 \\) that are not powers of 2."
] | [
"An integer \\( n \\geq 3 \\) is called *fabulous* if there exists an integer \\( a \\) with \\( 2 \\leq a \\leq n - 1 \\) such that \\( a^n - a \\) is divisible by \\( n \\). Find all the *fabulous* integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 6,
"tir": 17
} | 58 | 0 | 0 |
omr-ws-89f01c6b83c573ae066860fa5aa911dd2878ca2312eca2687cbcf859e2f0ee62 | An integer \( n \geq 3 \) is called *poli-pythagorean* if there exist \( n \) distinct positive integers that can be placed at the vertices of a regular \( n \)-gon such that the sum of the squares of the numbers at any two consecutive vertices is a perfect square. Determine all poli-pythagorean integers. | [
"\\( n \\geq 3 \\)"
] | [
"An integer \\( n \\geq 3 \\) is called *poli-pythagorean* if there exist \\( n \\) distinct positive integers that can be placed at the vertices of a regular \\( n \\)-gon such that the sum of the squares of the numbers at any two consecutive vertices is a perfect square. Determine all poli-pythagorean integers."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-67337210732747d3bc87e8b0bbd7193740561ae767c764638f153d1d12e7dac7 | An integer \( n \geq 3 \) is called special if it does not divide \( (n-1)! \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n-1} \right) \). Find all special numbers \( n \) such that \( 10 \leq n \leq 100 \). | [
"94"
] | [
"An integer \\( n \\geq 3 \\) is called special if it does not divide \\( (n-1)! \\left( 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n-1} \\right) \\). Find all special numbers \\( n \\) such that \\( 10 \\leq n \\leq 100 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e316aa49d71b1c3eb992594433e1db966f119084bd4303b50301443081fed522 | An integer \( n \geq 3 \) is called special if it does not divide \( \left( n-1 \right)! \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n-1} \right) \). Find all special numbers \( n \) such that \( 10 \leq n \leq 100 \). | [
"\\( 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94 \\)"
] | [
"An integer \\( n \\geq 3 \\) is called special if it does not divide \\( \\left( n-1 \\right)! \\left( 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n-1} \\right) \\). Find all special numbers \\( n \\) such that \\( 10 \\leq n \\leq 100 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-4766090eac08d90b53e180a0d96246624990590b1e19eece7aab75d3411b3380 | An integer \( n \geq 3 \) is given. We call an \( n \)-tuple of real numbers \((x_1, x_2, \dots, x_n)\) [i]Shiny[/i] if for each permutation \( y_1, y_2, \dots, y_n \) of these numbers, we have \[ \sum_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1. \] Find the largest constant \( K = ... | [
"\\(\\frac{1-n}{2}\\)"
] | [
"An integer \\( n \\geq 3 \\) is given. We call an \\( n \\)-tuple of real numbers \\((x_1, x_2, \\dots, x_n)\\) [i]Shiny[/i] if for each permutation \\( y_1, y_2, \\dots, y_n \\) of these numbers, we have\n\\[\n\\sum_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \\cdots + y_{n-1}y_n \\geq -1.\n\\]\nFind the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-50ef0dc82e7b0f91087bffd76f7431c1e5cffe9deb725052509d1c09883417bb | An integer \(N \geq 2\) is given. A collection of \(N(N + 1)\) soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove \(N(N - 1)\) players from this row, leaving a new row of \(2N\) players. Define \(f(N)\) as the number of distinct ways Sir Alex can achieve this such that the f... | [
"48"
] | [
"An integer \\(N \\geq 2\\) is given. A collection of \\(N(N + 1)\\) soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove \\(N(N - 1)\\) players from this row, leaving a new row of \\(2N\\) players. Define \\(f(N)\\) as the number of distinct ways Sir Alex can achieve this... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-83e055163db577e4134ad2b9f2445f06e8bd71679e96a31bb959f31a4ec1543b | An integer \(k\) is given. Determine all functions \(f: \mathbb{N} \to \mathbb{N}\) satisfying \[ f(n) + f(m) \mid n^2 + k f(n) m + f(m)^2 \] for every natural numbers \(m, n\). | [
"f(n) = n"
] | [
"An integer \\(k\\) is given. Determine all functions \\(f: \\mathbb{N} \\to \\mathbb{N}\\) satisfying\n\\[ f(n) + f(m) \\mid n^2 + k f(n) m + f(m)^2 \\]\nfor every natural numbers \\(m, n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-c253a44277268a1ae612afd0db234fd43250ef6b73682939efe3bf622da0a093 | An integer \(n \geq 3\) is given. We call an \(n\)-tuple of real numbers \((x_1, x_2, \ldots, x_n)\) Shiny if for each permutation \(y_1, y_2, \ldots, y_n\) of these numbers, we have \[ \sum_{i=1}^{n-1} y_i y_{i+1} \geq -1. \] Find the largest constant \(K = K(n)\) such that \[ \sum_{2 \leq i < j \leq n} x_i x_j \geq K... | [
"-1"
] | [
"An integer \\(n \\geq 3\\) is given. We call an \\(n\\)-tuple of real numbers \\((x_1, x_2, \\ldots, x_n)\\) Shiny if for each permutation \\(y_1, y_2, \\ldots, y_n\\) of these numbers, we have\n\\[\n\\sum_{i=1}^{n-1} y_i y_{i+1} \\geq -1.\n\\]\nFind the largest constant \\(K = K(n)\\) such that\n\\[\n\\sum_{2 \\l... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-8283c4c50b9925da27538db4bb75bb0575d049ed1b5181a6390fc58355b2bb4b | An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? | [
"615"
] | [
"An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e194c528095a0a0abb639966c6e0ec4f1b9f67f6994d879569bb9375215d1fc5 | An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? | [
"\\dfrac{56}{225}"
] | [
"An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53c25f39d20d4465ee9918d7f183adb4f87a54646bda77370e8c758e6dc72ca5 | An integer between 2 and 99 inclusive is randomly selected, and that number of pennies is placed on a table. Todd and Wes take turns removing 1 to 9 pennies each turn. The player who takes the last penny wins. Assuming both players are intelligent and try to win, what is the probability that Wes wins if Todd goes first... | [
"\\(\\frac{9}{98}\\)"
] | [
"An integer between 2 and 99 inclusive is randomly selected, and that number of pennies is placed on a table. Todd and Wes take turns removing 1 to 9 pennies each turn. The player who takes the last penny wins. Assuming both players are intelligent and try to win, what is the probability that Wes wins if Todd goes ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d2ca50036ff4334d0ebcbe9db7753d270b5f016ed92c489f7669d64c41c84b07 | An integer greater than 0 is written on each face of a cube. Each vertex is assigned the product of the numbers on the three faces that intersect at that vertex. The sum of all the numbers assigned to the vertices is 1001. Compute the sum of the numbers written on the faces of the cube. | [
"31"
] | [
"An integer greater than 0 is written on each face of a cube. Each vertex is assigned the product of the numbers on the three faces that intersect at that vertex. The sum of all the numbers assigned to the vertices is 1001. Compute the sum of the numbers written on the faces of the cube."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-3366034470fb63511dc1a51cd58f62d0509a8b5bbd1238f9f39f1d5aeade8603 | An integer greater than 10 is said to be $good$ if its digits can be divided into two groups such that the sum of the digits of one group is equal to the sum of the digits of the other group. Find the smallest natural number $n$ such that $n$ is good and $n + 1$ is also good. | [
"549"
] | [
"An integer greater than 10 is said to be $good$ if its digits can be divided into two groups such that the sum of the digits of one group is equal to the sum of the digits of the other group. Find the smallest natural number $n$ such that $n$ is good and $n + 1$ is also good."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 22
} | 31 | 0 | 0 |
omr-ws-16aaa0178451ae4f5aafb178626a52f6d823105d442d381491602a20dbc19c6f | An integer greater than 9 and less than 100 is randomly chosen. What is the probability that its digits are different? | [
"\\frac{9}{10}"
] | [
"An integer greater than 9 and less than 100 is randomly chosen. What is the probability that its digits are different?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-760bb51bff70da67a9b06df3286e0221a522479f316b4d20f6aea5c5d5e25e01 | An integer has 2018 digits and is divisible by 7. The first digit is \( d \), while all the other digits are 2. What is the value of \( d \)? | [
"4"
] | [
"An integer has 2018 digits and is divisible by 7. The first digit is \\( d \\), while all the other digits are 2. What is the value of \\( d \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-df09472d725d6acd0641cc205f70656b26887f9c419204ba18d28e9d83eaa4b0 | An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppos... | [
"126"
] | [
"An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. S... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-288e28a45585f9a8a1f40ef68c79e14254590ebc4d38623d6ad01cec460626cb | An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two verti... | [
"6033"
] | [
"An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd8921e593bcca2c7e916f3e92452cc5c5397592b2a2f61388a9d14a4fc355d1 | An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer \( m \) from each of the integers at two neighboring vertices and adding \( 2m \) to the opposite vertex, which is not adjacent to either of the first t... | [
"0"
] | [
"An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer \\( m \\) from each of the integers at two neighboring vertices and adding \\( 2m \\) to the opposite vertex, which is not adjacent to either of th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-f9a0184983a5b54c614ae1dec5bb856d43f8bcfb3379f563d667063c80f93202 | An integer is called a double integer if it can be expressed as $2n$ for some integer $n$. Find the smallest positive integer $k$ such that there exist at least $10$ distinct double integers between $k^2$ and $(k+1)^2$ inclusive. | [
"9"
] | [
"An integer is called a double integer if it can be expressed as $2n$ for some integer $n$. Find the smallest positive integer $k$ such that there exist at least $10$ distinct double integers between $k^2$ and $(k+1)^2$ inclusive."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26114dfec7d5aa78d286b4e0d95b2de66b1c5e0f8e4f1b9601a009206f11ef12 | An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there? | [
"640"
] | [
"An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-6708f974c4e9ae2b4a56f1d6b7b290fe3df3e794e1ce31b2dec192491775c5cf | An integer is called snakelike if its decimal representation \( a_1a_2a_3 \ldots a_k \) satisfies \( a_i < a_{i+1} \) if \( i \) is odd and \( a_i > a_{i+1} \) if \( i \) is even. How many snakelike integers between 1000 and 9999 have 4 distinct digits? | [
"882"
] | [
"An integer is called snakelike if its decimal representation \\( a_1a_2a_3 \\ldots a_k \\) satisfies \\( a_i < a_{i+1} \\) if \\( i \\) is odd and \\( a_i > a_{i+1} \\) if \\( i \\) is even. How many snakelike integers between 1000 and 9999 have 4 distinct digits?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-39fba454fe7d849f3d971a18635d8eddb2f394034a72b6ce107c40f9b219856a | An integer is called super odd if it is of the form $2^n(2m+1)$ for non-negative integers $n$ and $m$. Find the smallest even positive integer that cannot be expressed as the sum of two super odd numbers. | [
"2"
] | [
"An integer is called super odd if it is of the form $2^n(2m+1)$ for non-negative integers $n$ and $m$. Find the smallest even positive integer that cannot be expressed as the sum of two super odd numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-3b717c2ccfc1422bc400f148c8e18444f7847c0d88da4fd900769e7adcab68cb | An integer is two-separable if it can be represented as the product of two positive integers that differ by two. An integer is nine-separable if it can be represented as the product of two positive integers that differ by nine. What is the sum of all positive integers that are both two-separable and nine-separable? | [
"360"
] | [
"An integer is two-separable if it can be represented as the product of two positive integers that differ by two. An integer is nine-separable if it can be represented as the product of two positive integers that differ by nine. What is the sum of all positive integers that are both two-separable and nine-separable... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32f37746754679d1f41f1a2119bbfdebe6f62b55f5d9b37e7e7d1cfa715f2b1f | An integer is written in each square of a $100 \times 100$ checkerboard. An allowed operation consists of choosing the four squares of an $L$-tetromino (or any rotation of it) and adding $1$ to the numbers written in each square. The goal is to obtain, by means of allowed operations, a board with the least possible amo... | [
"1"
] | [
"An integer is written in each square of a $100 \\times 100$ checkerboard. An allowed operation consists of choosing the four squares of an $L$-tetromino (or any rotation of it) and adding $1$ to the numbers written in each square. The goal is to obtain, by means of allowed operations, a board with the least possib... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-524d3dd40de3661049c0cf3d35754ac363229b5358abd203edd748927aa0f64a | An integer is written on each square of a $2m \times 2n$ board. The permitted operation is to choose three squares forming an L-trimino and to add 1 to the number in each of the three squares. Determine the necessary and sufficient condition, depending on $m$, $n$, and the starting numbers, so that all numbers on the b... | [
"The sum of the numbers on the board must be divisible by \\(4mn\\)."
] | [
"An integer is written on each square of a $2m \\times 2n$ board. The permitted operation is to choose three squares forming an L-trimino and to add 1 to the number in each of the three squares. Determine the necessary and sufficient condition, depending on $m$, $n$, and the starting numbers, so that all numbers on... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c53a0994f387874d3b6ff38b9ee96cead994cdd83668905e0244d7c14a01bd3 | An integer lattice point in the Cartesian plane is a point \((x,y)\) where \(x\) and \(y\) are both integers. Suppose nine integer lattice points are chosen such that no three of them lie on the same line. Out of all 36 possible line segments between pairs of those nine points, some line segments may contain integer la... | [
"6"
] | [
"An integer lattice point in the Cartesian plane is a point \\((x,y)\\) where \\(x\\) and \\(y\\) are both integers. Suppose nine integer lattice points are chosen such that no three of them lie on the same line. Out of all 36 possible line segments between pairs of those nine points, some line segments may contain... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-5a00553024b8909640a3a2981889ecf5e03e84ee74dcb8ddecddcfd5175b008d | An integer number \( m \geq 1 \) is called "mexica" if it is of the form \( n^{d(n)} \), where \( n \) is a positive integer and \( d(n) \) is the number of positive divisors of \( n \). Find all mexica numbers less than 2019. | [
"\\( 1, 4, 9, 64, 25, 1296, 49, 729, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849 \\)"
] | [
"An integer number \\( m \\geq 1 \\) is called \"mexica\" if it is of the form \\( n^{d(n)} \\), where \\( n \\) is a positive integer and \\( d(n) \\) is the number of positive divisors of \\( n \\). Find all mexica numbers less than 2019."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 29
} | 30 | 0 | 0 |
omr-ws-f22d8595454e90ec09cfcfe8ca318b771e60181d5aea22b00364c5500dc89a7a | An integer sequence $(a_1, a_2, \ldots)$ is said to be white if for all $n > 2008$, $a_n$ is equal to the total number of indices $i$, $1 \leq i \leq n-1$, for which $a_i + i \geq n$. An integer $L$ is an important element of the sequence $(a_1, a_2, \ldots)$ if $a_j = L$ for infinitely many different indices $j$. What... | [
"2"
] | [
"An integer sequence $(a_1, a_2, \\ldots)$ is said to be white if for all $n > 2008$, $a_n$ is equal to the total number of indices $i$, $1 \\leq i \\leq n-1$, for which $a_i + i \\geq n$. An integer $L$ is an important element of the sequence $(a_1, a_2, \\ldots)$ if $a_j = L$ for infinitely many different indices... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-fb2dfc6494c8e26b6cd17c2995a71dd55f3648eef8c97e755354cf31ae4c0c1e | An integer sequence $\{a_n\}_{n \geq 1}$ is defined by $\sum_{d \mid n} a_d = 2^n$. Find the value of $\frac{a_{12}}{12}$. | [
"335"
] | [
"An integer sequence $\\{a_n\\}_{n \\geq 1}$ is defined by $\\sum_{d \\mid n} a_d = 2^n$. Find the value of $\\frac{a_{12}}{12}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99f55cd26f23fa68e34948d4da6a5886ffeafbb6e468586bbef09a152007e0d4 | An integer sequence \((a_1, a_2, a_3, \ldots)\) is defined by the starting values \(a_1 = 1\) and \(a_2 = 2\) and by the recurrence relation \(a_n = 3 \cdot a_{n-1} + 5 \cdot a_{n-2}\) for all \(n \geq 3\). Determine whether there exists an integer \(j > 1\) such that \(a_j\) divides the product \(a_{j+1} \cdot a_{j+2}... | [
"\\text{No}"
] | [
"An integer sequence \\((a_1, a_2, a_3, \\ldots)\\) is defined by the starting values \\(a_1 = 1\\) and \\(a_2 = 2\\) and by the recurrence relation \\(a_n = 3 \\cdot a_{n-1} + 5 \\cdot a_{n-2}\\) for all \\(n \\geq 3\\). Determine whether there exists an integer \\(j > 1\\) such that \\(a_j\\) divides the product ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05c0fff527094f6513470b8741ad7d4882b72379abc98060006161fd32e35ccf | An integer set \( T \) is called orensan if there exist integers \( a < b < c \), where \( a \) and \( c \) are elements of \( T \), but \( b \) is not an element of \( T \). Count the number of subsets \( T \) of \(\{1, 2, \ldots, 2019\}\) which are orensan. | [
"\\(2^{2019} - 2020 - \\binom{2019}{2}\\)"
] | [
"An integer set \\( T \\) is called orensan if there exist integers \\( a < b < c \\), where \\( a \\) and \\( c \\) are elements of \\( T \\), but \\( b \\) is not an element of \\( T \\). Count the number of subsets \\( T \\) of \\(\\{1, 2, \\ldots, 2019\\}\\) which are orensan."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 2,
"tir": 17
} | 57 | 0 | 0 |
omr-ws-99a265fa7dc48c53dd3d1494860e86f7ba2c7ab412c21b65684f45fb37b38c0d | An integer-valued function $f$ is called tenuous if $f(x) + f(y) > y^2$ for all positive integers $x$ and $y.$ Let $g$ be a tenuous function such that $g(1) + g(2) + \dots + g(20)$ is as small as possible. Compute the minimum possible value for $g(14).$ | [
"136"
] | [
"An integer-valued function $f$ is called tenuous if $f(x) + f(y) > y^2$ for all positive integers $x$ and $y.$ Let $g$ be a tenuous function such that $g(1) + g(2) + \\dots + g(20)$ is as small as possible. Compute the minimum possible value for $g(14).$"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-179a308b2423ef2cbb4624250cd8ed459d56decb02fd3cf1ceb95434c53a1246 | An integer-valued function \( f \) is called tenuous if \( f(x) + f(y) > x^2 \) for all positive integers \( x \) and \( y \). Let \( g \) be a tenuous function such that \( g(1) + g(2) + \cdots + g(20) \) is as small as possible. Compute the minimum possible value for \( g(14) \). | [
"136",
"195"
] | [
"An integer-valued function \\( f \\) is called tenuous if \\( f(x) + f(y) > x^2 \\) for all positive integers \\( x \\) and \\( y \\). Let \\( g \\) be a tenuous function such that \\( g(1) + g(2) + \\cdots + g(20) \\) is as small as possible. Compute the minimum possible value for \\( g(14) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 9,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-94551d8b8cd81ea3a0b67f61d58f178d24401e90b2d91ab8a196af7d68b3170a | An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, 3, ..., 9 such that the first digit of the code is non-zero. The code can potentially create confusion when read upside down (e.g., 91 appears as 16). How many codes are there for which no such confusion can arise? | [
"71"
] | [
"An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, 3, ..., 9 such that the first digit of the code is non-zero. The code can potentially create confusion when read upside down (e.g., 91 appears as 16). How many codes are there for which no such confusion can arise?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-4cde8743673a66b4d1e12ef80b424003db03235f79b7e757b699295932a309df | An interior point \( P \) of a square \( ABCD \) is such that \( PA = a \), \( PB = b \), and \( PC = b + c \), where \( a, b, \) and \( c \) satisfy the relationship \( a^2 = b^2 + c^2 \). If \( \angle BPC = 90^\circ \), calculate the value of \( \frac{a^2 + b^2}{c^2} \). | [
"3"
] | [
"An interior point \\( P \\) of a square \\( ABCD \\) is such that \\( PA = a \\), \\( PB = b \\), and \\( PC = b + c \\), where \\( a, b, \\) and \\( c \\) satisfy the relationship \\( a^2 = b^2 + c^2 \\). If \\( \\angle BPC = 90^\\circ \\), calculate the value of \\( \\frac{a^2 + b^2}{c^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8cdce7bddd129218bf0c44043baa3965a3d245da5f5b4f2d32647e18bb834470 | An interior point \( P \) of an equilateral triangle \( ABC \) is taken such that \( \angle APC = 150^\circ \). If \( BP = 1 \), find the maximum possible value of \( AP + CP \). | [
"\\sqrt{2}"
] | [
"An interior point \\( P \\) of an equilateral triangle \\( ABC \\) is taken such that \\( \\angle APC = 150^\\circ \\). If \\( BP = 1 \\), find the maximum possible value of \\( AP + CP \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-ab494e839bdec0d63102ce27b3ce6044f83cf02daca6a2e9afaf83233a577113 | An international conference has 250 participants. For every pair of people A and B, there is a language they both speak, a language A speaks and B doesn't, and a language B speaks and A doesn't. What is the smallest number of languages that must be spoken in the conference? | [
"9"
] | [
"An international conference has 250 participants. For every pair of people A and B, there is a language they both speak, a language A speaks and B doesn't, and a language B speaks and A doesn't. What is the smallest number of languages that must be spoken in the conference?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2743ac69a89db720c008dbfb608b0438f7d2995f23b15ce479000964d7488f49 | An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm? | [
"10"
] | [
"An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-40420f992a6b84523122e432dbe3bc5bfd2c7ba8ef276eeae2a1783ec81c1b05 | An internet service provider has two connection lines for its customers. Eighty percent of customers are connected through Line I, which has a Gamma(3, 1/2) connection time (in minutes), while the remaining twenty percent are connected through Line II, which has a U(20, 50) connection time (in seconds). Find the probab... | [
"0.84486"
] | [
"An internet service provider has two connection lines for its customers. Eighty percent of customers are connected through Line I, which has a Gamma(3, 1/2) connection time (in minutes), while the remaining twenty percent are connected through Line II, which has a U(20, 50) connection time (in seconds). Find the p... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c62756f0aab5cce1cd90abb7fc7a9537db3f0c2fc272959b2c1cb257ecd8ef0a | An interstellar hotel has 100 rooms with capacities of 101, 102, ..., 200 people. These rooms are occupied by $n$ people in total. The owner wants to ensure that he can always choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding $B$'s capacity, to provide a VIP guest with a personal room. ... | [
"8824"
] | [
"An interstellar hotel has 100 rooms with capacities of 101, 102, ..., 200 people. These rooms are occupied by $n$ people in total. The owner wants to ensure that he can always choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding $B$'s capacity, to provide a VIP guest with a personal r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
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