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-b70021a3f7fae9406616707bafdee760fc8b20bd09759f78137b51b05e345d6f | An antique scale shows the correct weight for objects lighter than $1000$g but shows a random number above $1000$g for objects heavier or equal to $1000$g. Given five weights $A$, $B$, $C$, $D$, and $E$ each under $1000$g, the scale shows the following weights when they are weighed in pairs: $B+D=1200$, $C+E = 2100$, $... | [
"D"
] | [
"An antique scale shows the correct weight for objects lighter than $1000$g but shows a random number above $1000$g for objects heavier or equal to $1000$g. Given five weights $A$, $B$, $C$, $D$, and $E$ each under $1000$g, the scale shows the following weights when they are weighed in pairs: $B+D=1200$, $C+E = 210... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d046e663625e82f43147123bce260b3850902fa350593735ca6a0c910f7c30de | An antique shop bought two items for 225 roubles and then sold them and made a profit of 40\%. What did the shop pay for each item, if the first of them yielded a profit of 25\% and the second, a profit of 50\%? | [
"Some first item was bought for 90 roubles, the second, for 135 roubles"
] | [
"An antique shop bought two items for 225 roubles and then sold them and made a profit of 40\\%. What did the shop pay for each item, if the first of them yielded a profit of 25\\% and the second, a profit of 50\\%?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22a8a01e43fd35b8fdea979467dc2891f72fcfbae6dcf157a0b1c3b6422a9421 | An antivirus program must perform three operations (find, quarantine, and destroy) on each of 2020 viruses. The operations for each virus must be performed in order, but the antivirus can switch between viruses at any point. In how many different ways can the antivirus complete all operations for all 2020 viruses? | [
"\\(\\frac{(3 \\times 2020)!}{6^{2020}}\\)"
] | [
"An antivirus program must perform three operations (find, quarantine, and destroy) on each of 2020 viruses. The operations for each virus must be performed in order, but the antivirus can switch between viruses at any point. In how many different ways can the antivirus complete all operations for all 2020 viruses?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-db763a35cafea1b9c802ae95b48f21484fe2ec2e9f501421d8fbb85c2bf90842 | An aquarium has the shape of a rectangular parallelepiped with width $\ell$, length $L$, and height $h = 30$ cm, where $\ell < h < L$. When the aquarium is rotated around one of the two longest edges of its base until the base makes an angle of $45^\circ$ with the ground, it loses a third of its water. When rotated aro... | [
"1800"
] | [
"An aquarium has the shape of a rectangular parallelepiped with width $\\ell$, length $L$, and height $h = 30$ cm, where $\\ell < h < L$. When the aquarium is rotated around one of the two longest edges of its base until the base makes an angle of $45^\\circ$ with the ground, it loses a third of its water. When rot... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d80663d696865dee59330054a9378cd0ac342e881a44ca040ab6073ebc3c471e | An aquarium on a level table has rectangular faces and is 10 inches wide and 8 inches high. When it was tilted, the water in it covered an 8" × 10" end but only three-fourths of the rectangular area. What was the depth of the water when the bottom was again made level? | [
"3\""
] | [
"An aquarium on a level table has rectangular faces and is 10 inches wide and 8 inches high. When it was tilted, the water in it covered an 8\" × 10\" end but only three-fourths of the rectangular area. What was the depth of the water when the bottom was again made level?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-906435ff4e35e0d135133fa511d16cdf169f6e6f9daf93c17e1a0ee7edf8bae3 | An aquarium on a level table has rectangular faces and is 10 inches wide and 8 inches high. When it was tilted, the water in it covered an \(8 \times 10\) end but only \(\frac{3}{4}\) of the rectangular room. The depth of the water when the bottom was again made level was. | [
"6"
] | [
"An aquarium on a level table has rectangular faces and is 10 inches wide and 8 inches high. When it was tilted, the water in it covered an \\(8 \\times 10\\) end but only \\(\\frac{3}{4}\\) of the rectangular room. The depth of the water when the bottom was again made level was."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-30b916f20b9089c240b0edb58b794b29f27e52753e98e6c5be9edb3a8c4f8569 | An arbelos consists of three semicircular arcs. A circle is placed inside the arbelos so that it is tangent to all three semicircles. Suppose that the radii of the two smaller semicircles are \( a \) and \( b \), and that the radius of the circle is \( r \). Assuming that \( a > b > r \) and that \( a \), \( b \), and ... | [
"\\( \\frac{a}{b} = 2 \\)"
] | [
"An arbelos consists of three semicircular arcs. A circle is placed inside the arbelos so that it is tangent to all three semicircles. Suppose that the radii of the two smaller semicircles are \\( a \\) and \\( b \\), and that the radius of the circle is \\( r \\). Assuming that \\( a > b > r \\) and that \\( a \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5631e65dddd2eb58cd0fc25928f6800a3dc41d5af8345a0a23c71e28024376b8 | An arbitrary circle can intersect the graph \( y = \sin x \) in at most how many points? | [
"more than 16 points"
] | [
"An arbitrary circle can intersect the graph \\( y = \\sin x \\) in at most how many points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af5ca4c799e8c99b3a2d7635c0329782496476921b506ffc762d26a555d85cfd | An arbitrary circle can intersect the graph of $y = \cos x$ in at most how many points? | [
"more than 9 points"
] | [
"An arbitrary circle can intersect the graph of $y = \\cos x$ in at most how many points?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-6030a853f06e3fb88d02c7bd395bcd37544b4268a21f8c934eb24f10b1b382e9 | An arbitrary line $\ell$ intersects sides $BC$, $CA$, and $AB$ at points $A'$, $B'$, and $C'$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars from $A'$ to $AC$ and $AB$, respectively, and let $A''$ be the midpoint of $PQ$. Define $B''$ and $C''$ similarly. If the coordinates of $A$, $B$, and $C$ are $(... | [
"-\\dfrac{5}{16}"
] | [
"An arbitrary line $\\ell$ intersects sides $BC$, $CA$, and $AB$ at points $A'$, $B'$, and $C'$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars from $A'$ to $AC$ and $AB$, respectively, and let $A''$ be the midpoint of $PQ$. Define $B''$ and $C''$ similarly. If the coordinates of $A$, $B$, and $C$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b3020dcdcc151afa6145abcae706ac5c960409e67b234679f37c112b5f6d00f | An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle at point $M$. Find the locus of the circumcenters of triangles $AMK$. | [
"\\( x[(a-c)^{2}+b^{2}]+2b(a-c)y=a(a^{2}+b^{2}+c^{2})-2c(a^{2}+b^{2}) \\)"
] | [
"An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle at point $M$. Find the locus of the circumcenters of triangles $AMK$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4982eb368bfb10f48b2d809b6520cde166a9a63f77fb5c906c322dfc339eb64a | An arbitrary line segment \( AB \) is the diameter of a circle. A point \( C \) lies on this circle such that the ratio of the area of the circle to the area of triangle \( ABC \) is \( 2\pi \). Let \( a, b, c \) be the angles in triangle \( ABC \). Find \( L = \sqrt{\frac{abc}{2}} \). | [
"\\dfrac{\\pi \\sqrt{5\\pi}}{24}"
] | [
"An arbitrary line segment \\( AB \\) is the diameter of a circle. A point \\( C \\) lies on this circle such that the ratio of the area of the circle to the area of triangle \\( ABC \\) is \\( 2\\pi \\). Let \\( a, b, c \\) be the angles in triangle \\( ABC \\). Find \\( L = \\sqrt{\\frac{abc}{2}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-ba3a37f2d1e5cd8c68f173f1f3d30b87c466cd0c925a8d53107d8ab7b0083c5d | An arbitrary point $B$ is chosen on the line $AC$, and circles $k_1$, $k_2$, and $k$ are constructed with diameters $AB$, $BC$, and $CA$, respectively. An arbitrary line through point $B$ intersects circle $k$ at points $P$ and $Q$, and circles $k_1$ and $k_2$ (besides point $B$) at points $R$ and $S$, respectively. If... | [
"14"
] | [
"An arbitrary point $B$ is chosen on the line $AC$, and circles $k_1$, $k_2$, and $k$ are constructed with diameters $AB$, $BC$, and $CA$, respectively. An arbitrary line through point $B$ intersects circle $k$ at points $P$ and $Q$, and circles $k_1$ and $k_2$ (besides point $B$) at points $R$ and $S$, respectivel... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-39966ebc8e419a0460bb251fabc60f0099f46b17283863a4072670c38f5b0748 | An arbitrary point $K$ is chosen inside the square $ABCD$. Perpendiculars to the lines $BK$, $CK$, $DK$, and $AK$ are drawn through the vertices $A$, $B$, $C$, and $D$. Let $P$ be the point of intersection of these four perpendiculars. If the side length of the square $ABCD$ is $s$, find the sum of the squares of the d... | [
"no_valid_answer_found"
] | [
"An arbitrary point $K$ is chosen inside the square $ABCD$. Perpendiculars to the lines $BK$, $CK$, $DK$, and $AK$ are drawn through the vertices $A$, $B$, $C$, and $D$. Let $P$ be the point of intersection of these four perpendiculars. If the side length of the square $ABCD$ is $s$, find the sum of the squares of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f01e128ae72f6fc46b76f94fcc7b505eea22efba8680d121f19ffbd75e8df78b | An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. L... | [
"y = \\dfrac{5}{2}"
] | [
"An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-56c9cb535b53e59d57f18239b321cbcfb8416e887e49bf896ed634a765e160ce | An arbitrary point $M$ is taken on the base of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. The straight line passing through the points $M$ and $P$ intersects the height of the p... | [
"\\cot^2 a"
] | [
"An arbitrary point $M$ is taken on the base of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. The straight line passing through the points $M$ and $P$ intersects the height of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-edb91076ffa3600cec1b298c9dba5392ac31c63200415bab442396ed0c51ff71 | An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and another point N. Find the locus of the midpoints of the ... | [
"\\text{The line segment parallel to } AB \\text{ at one-fourth the distance from } AB \\text{ between the points one-fourth and three-fourths of the way from } A \\text{ to } B"
] | [
"An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and another point N. Find the locus of the midpoints of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-756494ef1abd5255ebda9272a833a214ca14257e1b28b177babe1e74c647784a | An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and another point N. Let N′ denote the point of intersection... | [
"\\left( \\dfrac{40}{13}, \\dfrac{60}{13} \\right)"
] | [
"An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and another point N. Let N′ denote the point of interse... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02e510a4e99074b2fcc308e9c50635e6591fc2f1b891a8e13010c63100febc65 | An arbitrary point \( M \) is chosen inside the square \( ABCD \). Rays \( AM \), \( BM \), \( CM \), and \( DM \) intersect the circumscribed circle of the square at \( A_1 \), \( B_1 \), \( C_1 \), and \( D_1 \) respectively. Calculate the value of \( \frac{A_1B_1 \cdot C_1D_1}{D_1A_1 \cdot B_1C_1} \). | [
"1"
] | [
"An arbitrary point \\( M \\) is chosen inside the square \\( ABCD \\). Rays \\( AM \\), \\( BM \\), \\( CM \\), and \\( DM \\) intersect the circumscribed circle of the square at \\( A_1 \\), \\( B_1 \\), \\( C_1 \\), and \\( D_1 \\) respectively. Calculate the value of \\( \\frac{A_1B_1 \\cdot C_1D_1}{D_1A_1 \\cd... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdba6ddc54d4d0bd8c2f298b9d76ab1bf3d4d9e16eca9724b5be0bfef4ef7fcf | An arbitrary point is selected on each of the twelve face diagonals of a cube. Determine the locus of the centroids of these twelve points. | [
"|x| + |y| + |z| \\leq 1"
] | [
"An arbitrary point is selected on each of the twelve face diagonals of a cube. Determine the locus of the centroids of these twelve points."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cfc23b097d5a35ff52638ff1747525e85c19ba565168f76ee9db4a8f79e05a4 | An arbitrary point is selected on each of twelve diagonals of the faces of a cube. The centroid of these twelve points is determined. Find the locus of all these centroids. | [
"|x| + |y| + |z| \\leq 1"
] | [
"An arbitrary point is selected on each of twelve diagonals of the faces of a cube. The centroid of these twelve points is determined. Find the locus of all these centroids."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3cf9213d0f35e13299410e3b0a2df8e000396f8f67651127ab426cf4ea7922f6 | An arbitrary rectangle is split into right triangles, and a square is inscribed in each right triangle with its side lying on the hypotenuse. What is greater: the area of the largest square or the sum of the areas of the remaining three squares? | [
"\\text{The sum of the areas of the remaining three squares}"
] | [
"An arbitrary rectangle is split into right triangles, and a square is inscribed in each right triangle with its side lying on the hypotenuse. What is greater: the area of the largest square or the sum of the areas of the remaining three squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-867c19e4508c012f5630b50186ca1653a29e964a795feeea18b5782b7b3612fd | An arbitrary triangle \(ABC\) is given. A sequence of triangles is built such that the sides of triangle \(A_1B_1C_1\) are equal to the sines of the angles of triangle \(ABC\), the sides of triangle \(A_2B_2C_2\) are equal to the sines of the angles of triangle \(A_1B_1C_1\), and so on. Find the ratio of the area of tr... | [
"1"
] | [
"An arbitrary triangle \\(ABC\\) is given. A sequence of triangles is built such that the sides of triangle \\(A_1B_1C_1\\) are equal to the sines of the angles of triangle \\(ABC\\), the sides of triangle \\(A_2B_2C_2\\) are equal to the sines of the angles of triangle \\(A_1B_1C_1\\), and so on. Find the ratio of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-efa9b2e049a619a18eb2a316081984e1ac768ab8d42b904d837cb3175137aaf9 | An arbitrary triangle is partitioned into several smaller triangles, each homothetic to the original triangle with homothety ratios $r_1, r_2, \ldots, r_n$. Calculate the value of $r_1 + r_2 + \cdots + r_n$ if the sum of the areas of the smaller triangles equals the area of the original triangle. | [
"2"
] | [
"An arbitrary triangle is partitioned into several smaller triangles, each homothetic to the original triangle with homothety ratios $r_1, r_2, \\ldots, r_n$. Calculate the value of $r_1 + r_2 + \\cdots + r_n$ if the sum of the areas of the smaller triangles equals the area of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5ed202c5dc91515f3547e2c421cf168b2cd2521de8d68cb38d51cd5e23e25e9d | An arc flower grows in a garden such that each night, new circular arcs of radius $r$ grow outward from an initial point, covering a distance $s$. If the total length of the new circular arcs that grow in one night is denoted by $L$, find the minimum value of $\frac{L}{s}$ as $r$ approaches infinity. | [
"2\\pi"
] | [
"An arc flower grows in a garden such that each night, new circular arcs of radius $r$ grow outward from an initial point, covering a distance $s$. If the total length of the new circular arcs that grow in one night is denoted by $L$, find the minimum value of $\\frac{L}{s}$ as $r$ approaches infinity."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-15774538ce3746652d81a9ad5b822db6d72921f4c6cfc7707092efa5ad3089bd | An arc is drawn in a square with center D at one of the vertices of the square and is tangent to the opposite two sides of the square. The arc is also tangent to the hypotenuse of $ \triangle BFE$, where $ \overline{BF} = 1$. What is the radius of the circle? | [
"\\(\\frac{3 + \\sqrt{3}}{2}\\)"
] | [
"An arc is drawn in a square with center D at one of the vertices of the square and is tangent to the opposite two sides of the square. The arc is also tangent to the hypotenuse of $ \\triangle BFE$, where $ \\overline{BF} = 1$. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50ce3c19e5a4474ca803b470a66c7d12ed28e02c08c54fecde1823cf6ceaf1eb | An arch is built in the form of an arc of a circle and is subtended by a chord 30 ft long. If a cord 17 ft long subtends half that arc, what is the radius of the circle? | [
"18 ft 3/4 in",
"\\( 18 \\frac{1}{16} \\) feet"
] | [
"An arch is built in the form of an arc of a circle and is subtended by a chord 30 ft long. If a cord 17 ft long subtends half that arc, what is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 49
} | 80 | 0 | 0 |
omr-ws-a25a611ee346f2be4ef349973c04c5dc5d5fc7c16698ca0ff7d1024188db43bd | An arch is in the shape of a parabola with a vertical axis. The arch is 15 ft high at the center and 40 ft wide at the base. At which height above the base is the width 20 ft? | [
"11.25 ft"
] | [
"An arch is in the shape of a parabola with a vertical axis. The arch is 15 ft high at the center and 40 ft wide at the base. At which height above the base is the width 20 ft?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-165dadbededf875edff8cc98242d68d6f7625255d5e4327d52266beda7a6a26b | An arch supports a pipeline across a river that is 20 meters wide. Midway across the river, the suspending cable is 5 meters long. Find the radius of the arch. | [
"$\\frac{25}{2}$"
] | [
"An arch supports a pipeline across a river that is 20 meters wide. Midway across the river, the suspending cable is 5 meters long. Find the radius of the arch."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-b9bbbe2300e6fd74b9f41eaef060bc1b5ed8548d2a3659894898f8c8ef84ad9f | An archbishop in pseudo-chess can move to any number of squares vertically or diagonally and is said to attack another piece if the archbishop has the ability to move to the other piece's square. Find the number of ways to place 6 archbishops on a \(6 \times 6\) pseudo-chess board such that no archbishops attack each o... | [
"240"
] | [
"An archbishop in pseudo-chess can move to any number of squares vertically or diagonally and is said to attack another piece if the archbishop has the ability to move to the other piece's square. Find the number of ways to place 6 archbishops on a \\(6 \\times 6\\) pseudo-chess board such that no archbishops attac... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-30eb3beb6ebef31a133fa7129f17039877424820dbf66314d99fc2586a6cd881 | An archer shoots $2023$ arrows, each with a $\frac{1}{2}$ probability of hitting the target. The probability that the total number of hits is divisible by $4$ can be expressed in the form $\frac{1}{4} + x^{-y}$, where $x$ and $y$ are positive integers and $x$ is not a positive power of an integer other than itself. Fin... | [
"993"
] | [
"An archer shoots $2023$ arrows, each with a $\\frac{1}{2}$ probability of hitting the target. The probability that the total number of hits is divisible by $4$ can be expressed in the form $\\frac{1}{4} + x^{-y}$, where $x$ and $y$ are positive integers and $x$ is not a positive power of an integer other than itse... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-630d29587d223e0b57cf5b540952f71a01e2a15c24251916221a9e435433ef9b | An archery target consists of three concentric circles with radii $3$, $2$, and $1$, which split the target into three regions. What is the area of Region $1$ plus the area of Region $3$? | [
"\\(6\\pi\\)"
] | [
"An archery target consists of three concentric circles with radii $3$, $2$, and $1$, which split the target into three regions. What is the area of Region $1$ plus the area of Region $3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-e7e3e210f7544f1fa722846296cb6bba631ae4559a04b61b4c9b0c435a25427c | An archipelago consists of $n$ islands, where each island is connected to other islands by bridges. It is known that from any island one can travel to any other island, and the number of bridges connected to each island is as follows: the first island has 1 bridge, the second has 4 bridges, ..., the $n$-th island has $... | [
"\\( n = 3, 4, 7, 8, 11, 12, \\ldots \\)"
] | [
"An archipelago consists of $n$ islands, where each island is connected to other islands by bridges. It is known that from any island one can travel to any other island, and the number of bridges connected to each island is as follows: the first island has 1 bridge, the second has 4 bridges, ..., the $n$-th island ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-a9969fba34e16493ac9176de01483509423a382ffda03e3c3855fbd2f7337db5 | An archipelago consists of $n$ islands: $I_1, I_2, \ldots, I_n$ and $a_1, a_2, \ldots, a_n$ are the number of roads on each island. Given $a_1 = 55$ and $a_k = a_{k-1} + (k-1)$ for $k = 2, 3, \ldots, n$. a) Does there exist an island with 2008 roads? b) Calculate $a_1 + a_2 + \cdots + a_n$. | [
"\\(55n + \\frac{n(n+1)(n-1)}{6}\\)"
] | [
"An archipelago consists of $n$ islands: $I_1, I_2, \\ldots, I_n$ and $a_1, a_2, \\ldots, a_n$ are the number of roads on each island. Given $a_1 = 55$ and $a_k = a_{k-1} + (k-1)$ for $k = 2, 3, \\ldots, n$.\na) Does there exist an island with 2008 roads?\nb) Calculate $a_1 + a_2 + \\cdots + a_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 9
} | 17 | 0 | 0 |
omr-ws-4d12a3e5f6e5fd7107657519f14990e82a333f1eb5fffbb23db0e4b215dd5e29 | An architect is building a structure that will place vertical pillars at the vertices of a regular hexagon $ABCDEF$, which is lying horizontally on the ground. The heights of the pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$? | [
"17"
] | [
"An architect is building a structure that will place vertical pillars at the vertices of a regular hexagon $ABCDEF$, which is lying horizontally on the ground. The heights of the pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 4
} | 68 | 0 | 0 |
omr-ws-498d3738c3832a40958478ec0c30ad4df5f9bc205810db74c0183688a9fd736b | An architect is building a structure that will place vertical pillars at the vertices of a regular hexagon \(ABCDEF\), which is lying horizontally on the ground. The heights of the pillars at \(A\), \(B\), and \(C\) are 12, 9, and 10 meters, respectively. What is the height, in meters, of the pillar at \(E\)? | [
"17"
] | [
"An architect is building a structure that will place vertical pillars at the vertices of a regular hexagon \\(ABCDEF\\), which is lying horizontally on the ground. The heights of the pillars at \\(A\\), \\(B\\), and \\(C\\) are 12, 9, and 10 meters, respectively. What is the height, in meters, of the pillar at \\(... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-43010a6a440de5ba25c27ae623b4aa9098762b9fa6a878d560cb36d802c228d3 | An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ m... | [
"17"
] | [
"An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A$, $B$, and $C$ are $12$, $9$, and $... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c002361249fa84b09f4bcdbc655547b76d9cc3d6ea05e44a2f28d04906cc1330 | An arithmetic progression (A.P.) contains an odd number of terms, specifically \(2n+1\) terms. The sum of its even terms equals the sum of its odd terms. What is the sum of all terms of the A.P.? | [
"0"
] | [
"An arithmetic progression (A.P.) contains an odd number of terms, specifically \\(2n+1\\) terms. The sum of its even terms equals the sum of its odd terms. What is the sum of all terms of the A.P.?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0968cae1bf0faa8ced21de0ad4dee204d3d0329796f1d1f3a0755b2f7914429 | An arithmetic progression \( a_1, a_2, a_3, a_4, \dots, a_{1296} \) with a common difference of one has a sum of 893592. Find the product of the first and last term. | [
"56154"
] | [
"An arithmetic progression \\( a_1, a_2, a_3, a_4, \\dots, a_{1296} \\) with a common difference of one has a sum of 893592. Find the product of the first and last term."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7beb07fc248ddd2edd4bb4a024594467fe3632c7e2daa7d20bab25241d5c2c87 | An arithmetic progression consists of 20 terms. The sum of the terms occupying even places is equal to 250, and that of the terms occupying odd places equals 220. Find the two medium terms of the progression. | [
"The medium terms are equal to 22 and 25, respectively"
] | [
"An arithmetic progression consists of 20 terms. The sum of the terms occupying even places is equal to 250, and that of the terms occupying odd places equals 220. Find the two medium terms of the progression."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f38b66ffa871c00e19470657ee18a3a66eb4fae03e6e33881291dd09f9e0a220 | An arithmetic progression of nonnegative integers has \( n \) terms that sum to a prime \( p \). Find the maximum possible value of \( n - p \). | [
"0"
] | [
"An arithmetic progression of nonnegative integers has \\( n \\) terms that sum to a prime \\( p \\). Find the maximum possible value of \\( n - p \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6995a2463d2337f5f1cf7b33d1beedb6523d9dd526e3fb78de502cab6b04c484 | An arithmetic progression of positive integers has a sum of 2021. Find the smallest possible value of the largest term in such a progression. | [
"66"
] | [
"An arithmetic progression of positive integers has a sum of 2021. Find the smallest possible value of the largest term in such a progression."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-ec44648ef5ed629cafc52df9bb9ef1c96aa49fdad003aa8c5a51b1ba016e7ec2 | An arithmetic sequence $\{x_n\}_{n \geq 1}$ has the first term $x_1 \in \mathbb{N}^*$ and common difference $d \in \mathbb{N}^*$. It is given that for all $n \in \{2x_1 + d, 2x_1 + d + 1, \ldots, 4x_1 + 2d\}$, the sum $S_n = x_1 + x_2 + \cdots + x_n$ is a perfect square. Find the smallest possible value of $x_1 + d$ su... | [
"3"
] | [
"An arithmetic sequence $\\{x_n\\}_{n \\geq 1}$ has the first term $x_1 \\in \\mathbb{N}^*$ and common difference $d \\in \\mathbb{N}^*$. It is given that for all $n \\in \\{2x_1 + d, 2x_1 + d + 1, \\ldots, 4x_1 + 2d\\}$, the sum $S_n = x_1 + x_2 + \\cdots + x_n$ is a perfect square. Find the smallest possible valu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c26c5525153d8c0b7cd54708ade012428a1f7a3b886a6e1b5961d319d384679b | An arithmetic sequence $a_1, a_2, a_3,...$ is said to be Kodvick-spooky if the following list, $$a_1, a_2 + a_3, a_4 + a_5 + a_6, a_7 + a_8 + a_9 + a_{10}, ...$$ is an increasing sequence of consecutive perfect cubes. Find the sum of all possible values of $\sum^{100}_{k=1}a_k$ over all possible Kodvick-spooky sequence... | [
"10000"
] | [
"An arithmetic sequence $a_1, a_2, a_3,...$ is said to be Kodvick-spooky if the following list, $$a_1, a_2 + a_3, a_4 + a_5 + a_6, a_7 + a_8 + a_9 + a_{10}, ...$$ is an increasing sequence of consecutive perfect cubes. Find the sum of all possible values of $\\sum^{100}_{k=1}a_k$ over all possible Kodvick-spooky se... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bee8d6131e8353a5cfc9340bb9aa28f754457abe57a1bc9b977a7ad5c0d3e3b | An arithmetic sequence \( a_1, a_2, a_3, \ldots, a_{19} \) has 19 terms. Find \( a_1 + a_2 + \cdots + a_{18} + a_{19} \) given that \( a_1 + a_7 + a_{14} + a_{18} = 200 \). | [
"950"
] | [
"An arithmetic sequence \\( a_1, a_2, a_3, \\ldots, a_{19} \\) has 19 terms. Find \\( a_1 + a_2 + \\cdots + a_{18} + a_{19} \\) given that \\( a_1 + a_7 + a_{14} + a_{18} = 200 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a84ab0aed3f58a65abf551736520bf80164eaa1e49d6436b70f5a963021a008 | An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$? | [
"\\frac{8080}{199}"
] | [
"An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-9e03eb3d599c4c449541ec8864773e59baa82684fd091ea1ce1e105feeade958 | An arithmetic sequence consists of 200 numbers, each between 10 and 100 inclusive, with a sum of 10,000. Let \( L \) be the least possible value of the 50th term and \( G \) be the greatest possible value of the 50th term. What is the value of \( G - L \)? Express your answer as a fraction in reduced form. | [
"\\(\\frac{8080}{199}\\)"
] | [
"An arithmetic sequence consists of 200 numbers, each between 10 and 100 inclusive, with a sum of 10,000. Let \\( L \\) be the least possible value of the 50th term and \\( G \\) be the greatest possible value of the 50th term. What is the value of \\( G - L \\)? Express your answer as a fraction in reduced form."
... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c2786e2c1306ac528649cd8bea4e2f033d29132303411616b1d82053d23ceff6 | An arithmetic sequence has 50 terms. The sum of the first 30 terms is 40, and the sum of the last 30 terms is 70. Find the sum of the first and last terms. | [
"\\(13/5\\)"
] | [
"An arithmetic sequence has 50 terms. The sum of the first 30 terms is 40, and the sum of the last 30 terms is 70. Find the sum of the first and last terms."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-269a3d8a0bf029760b57ea525bc4ac18b8ab967b89ce0b831be0aeb0db854e37 | An arithmetic sequence has \( n \geq 3 \) terms, initial term \( a \), and common difference \( d > 1 \). Carl wrote down all the terms correctly except for one term which was off by 1. The sum of the terms was 222. What is \( a + d + n \)? | [
"20"
] | [
"An arithmetic sequence has \\( n \\geq 3 \\) terms, initial term \\( a \\), and common difference \\( d > 1 \\). Carl wrote down all the terms correctly except for one term which was off by 1. The sum of the terms was 222. What is \\( a + d + n \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-3a99ab341cea7111a22c8f5f50fe2d43e3d196d28e1b0e79510ddc96087b36c6 | An arithmetic sequence has first term $a$ and common difference $d$. If the first three terms of the sequence add to $15$ and multiply to $105$, determine the value of the product $ad$. | [
"6"
] | [
"An arithmetic sequence has first term $a$ and common difference $d$. If the first three terms of the sequence add to $15$ and multiply to $105$, determine the value of the product $ad$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b3e92484db01fbd5bac219a2da4a8d6c0ad70aeb4464a512785eca8af04f3253 | An arithmetic sequence of 10 consecutive positive integers has the property that the $n^{th}$ term in the sequence is a multiple of $11-n$. The smallest possible first term is $k$. Compute the remainder when $k$ is divided by 1000. | [
"510"
] | [
"An arithmetic sequence of 10 consecutive positive integers has the property that the $n^{th}$ term in the sequence is a multiple of $11-n$. The smallest possible first term is $k$. Compute the remainder when $k$ is divided by 1000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e8f678a14c124895c342a697e9f195efb5b537bf7376cc70c57b9305e9d08f2a | An arithmetic sequence of five terms is considered "good" if it contains both 19 and 20. For example, the sequence $18.5, 19.0, 19.5, 20.0, 20.5$ is a "good" sequence. For every "good" sequence, the sum of its terms is totaled. What is the total sum of all "good" sequences? | [
"1950"
] | [
"An arithmetic sequence of five terms is considered \"good\" if it contains both 19 and 20. For example, the sequence $18.5, 19.0, 19.5, 20.0, 20.5$ is a \"good\" sequence. For every \"good\" sequence, the sum of its terms is totaled. What is the total sum of all \"good\" sequences?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 8,
"tir": 12
} | 59 | 0 | 0 |
omr-ws-7ebe06d16046356b55467cf74eb4f6fa5d42b63b1fe9cef413039cadf0bbf2d0 | An arithmetic sequence of positive integers is called [i]gleaming[/i] if it has at least three terms, exactly one of which is larger than $2017$. Find the remainder when the number of [i]gleaming[/i] sequences is divided by $1000$. | [
"272"
] | [
"An arithmetic sequence of positive integers is called [i]gleaming[/i] if it has at least three terms, exactly one of which is larger than $2017$. Find the remainder when the number of [i]gleaming[/i] sequences is divided by $1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-d1bcbcd34b38c8c67f5f76c811cb88d66f506de2ccf1237d4e0c9c5e2a5e9e96 | An arithmetic sequence starts at $1$ and each subsequent term is the sum of the previous term and $9$. Add all the terms of the sequence starting at $1$ and finishing at $10^{2014}$. How many times does the digit $5$ occur in the final sum? | [
"4024"
] | [
"An arithmetic sequence starts at $1$ and each subsequent term is the sum of the previous term and $9$. Add all the terms of the sequence starting at $1$ and finishing at $10^{2014}$. How many times does the digit $5$ occur in the final sum?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-032c39d1c70a2887188786a07e3dd4e04631705d4a4d7910e6d7d0698bd123e4 | An arithmetic sequence with $10$ terms and a common difference of $1$ has the property that the $n^{th}$ term is a multiple of $11-n$. If the smallest possible first term is $a$, compute the remainder when $a$ is divided by $1000$. | [
"510"
] | [
"An arithmetic sequence with $10$ terms and a common difference of $1$ has the property that the $n^{th}$ term is a multiple of $11-n$. If the smallest possible first term is $a$, compute the remainder when $a$ is divided by $1000$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-543a01eefa20fc6ebbbc1cde2572c7fab8e08a622d2c825278c34b54359e258e | An arithmetic sequence with first term $1$ has a common difference of $6$. A second sequence begins with $4$ and has a common difference of $7$. In the range of $1$ to $100$, what is the largest number common to both sequences? | [
"67"
] | [
"An arithmetic sequence with first term $1$ has a common difference of $6$. A second sequence begins with $4$ and has a common difference of $7$. In the range of $1$ to $100$, what is the largest number common to both sequences?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af584e4b0da835aca940d2cd7be5fcec0b5d4a48465ed783147b5e560c19dcf1 | An army forms the largest square possible, but there are 12 people left over. If they were to form a triangle (one person in row 1, two people in row 2, etc.), there would be just enough people. What is the least possible number of people in the army? | [
"496"
] | [
"An army forms the largest square possible, but there are 12 people left over. If they were to form a triangle (one person in row 1, two people in row 2, etc.), there would be just enough people. What is the least possible number of people in the army?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-17781c9160e2ec128908130b4bdf14d64815d51b86063b5c1729c65655ca8e22 | An army marches at a steady pace. When the front of the army passes a suspected bomb, a runner runs at a steady pace from the front of the army to the rear, and then turns around and runs back to the front at the same steady pace, arriving back at the front just as the rear passes the suspected bomb. What is the ratio ... | [
"\\(1 + \\sqrt{2}\\)"
] | [
"An army marches at a steady pace. When the front of the army passes a suspected bomb, a runner runs at a steady pace from the front of the army to the rear, and then turns around and runs back to the front at the same steady pace, arriving back at the front just as the rear passes the suspected bomb. What is the r... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 31
} | 68 | 0 | 0 |
omr-ws-592901f6fc9fcefc153f9e72fcffb32ebd9dc63cc6a1a6a160cc29519f0c7d31 | An army of ants forms the largest square that it can, but twelve ants are left over. However, when they form a triangle ($1$ in the first row, $2$ in the second, $3$ in the third, etc.), no ants are left over. What is the smallest possible size of this army? | [
"21"
] | [
"An army of ants forms the largest square that it can, but twelve ants are left over. However, when they form a triangle ($1$ in the first row, $2$ in the second, $3$ in the third, etc.), no ants are left over. What is the smallest possible size of this army?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fc0c08b48642c63380fdb6c1ad1d740363da43b86ceff70c9e06a5b6e348cdff | An army of ants forms the largest square that it can, but twelve ants are left over. However, when they form a triangle (1 in the first row, 2 in the second, 3 in the third, etc.), no ants are left over. What is the smallest possible size of this army? | [
"496"
] | [
"An army of ants forms the largest square that it can, but twelve ants are left over. However, when they form a triangle (1 in the first row, 2 in the second, 3 in the third, etc.), no ants are left over. What is the smallest possible size of this army?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a92d56457cfc0b705fb2a715414be1502788e7c4d3debd07bd4bd50e97bca32e | An army plans to execute a prisoner who must drink 3 out of 12 bottles given to him. There are 6 bottles containing 0.5 moles of poison each, 2 bottles containing 1 mole of poison each, 1 bottle with an antidote that cures 0.5 moles of poison, 1 harmless bottle of wine, and 2 extremely potent antidotes that cure exactl... | [
"18"
] | [
"An army plans to execute a prisoner who must drink 3 out of 12 bottles given to him. There are 6 bottles containing 0.5 moles of poison each, 2 bottles containing 1 mole of poison each, 1 bottle with an antidote that cures 0.5 moles of poison, 1 harmless bottle of wine, and 2 extremely potent antidotes that cure e... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c89e1b503aae4f6c70f40a07bf53565ffbcd0492b8c1c5150dbb992b534a95e8 | An arrangement of 12 real numbers in a row is called "good" if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the t... | [
"45"
] | [
"An arrangement of 12 real numbers in a row is called \"good\" if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 6,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-f0d4737c7629bb083c5c6d17495bdeb70a292ffc6ae0acda6b2f7792797cc8ee | An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A move of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Find the number of distinct sequences of $n^2$ moves that allow the pawn to visit each unit square exactly on... | [
"82944"
] | [
"An array $n \\times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A move of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Find the number of distinct sequences of $n^2$ moves that allow the pawn to visit each unit square exac... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-41cf1e552dc4837e7bb651ef31a8c64848b6a2f80ab89cffe7761027fcba65b9 | An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. The pawn can move from a square of the $k$-th column to any square of the $k$-th row. Construct a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited exactly once, and t... | [
"(n, 1)"
] | [
"An array $n \\times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. The pawn can move from a square of the $k$-th column to any square of the $k$-th row. Construct a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited exactly once,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f790f7dc9dedc7f8884a7efb9304a78e5e4c9e4d593484b4bf833aadd0c14b7b | An array of squares has $m$ rows and $n$ columns where $m \le n$. Some of the squares are painted black such that any two rows have different patterns. Find the largest integer $k$ (in terms of $m$ and $n$) for which, regardless of the original painting, we can paint $k$ of the columns all red while still ensuring that... | [
"\\( n - m + 1 \\)"
] | [
"An array of squares has $m$ rows and $n$ columns where $m \\le n$. Some of the squares are painted black such that any two rows have different patterns. Find the largest integer $k$ (in terms of $m$ and $n$) for which, regardless of the original painting, we can paint $k$ of the columns all red while still ensurin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 88,
"genselect": 8
} | 96 | 0 | 0 |
omr-ws-3d5482f8bc48c5b2a2ee60eaaad6d3ee3082ab1e373926295a891b42d4622219 | An array of squares has \( m \) rows and \( n \) columns where \( m \le n \). Some of the squares are painted black in such a way that any two rows are painted in different patterns. Find the largest integer \( k \) (in terms of \( m \) and \( n \)) for which, regardless of how the array was originally painted, we can ... | [
"\\( n - m \\)"
] | [
"An array of squares has \\( m \\) rows and \\( n \\) columns where \\( m \\le n \\). Some of the squares are painted black in such a way that any two rows are painted in different patterns. Find the largest integer \\( k \\) (in terms of \\( m \\) and \\( n \\)) for which, regardless of how the array was originall... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-481a1a6d3096e5b85e76767c510acb4b16ab839952916ae0a72374b7d583cfdf | An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of 25 m? | [
"\\( t = \\frac{245 \\pm 5\\sqrt{1911}}{49} \\)"
] | [
"An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of 25 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-90609190013e6ac78d320aefa0400fe6f5d0f2dcac80345bf9f4326fb36f0255 | An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of 40 m? | [
"\\( t = \\frac{35 \\pm 5\\sqrt{33}}{7} \\)"
] | [
"An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of 40 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-3d54d4a8968d95a462d2b2f5f99262545d72d7bd3496358162e9066b86e6d572 | An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of h meters? | [
"\\( t = \\frac{245 \\pm 5\\sqrt{2401 - 19.6h}}{49} \\)"
] | [
"An arrow is shot straight up in the air with an initial velocity of 49 m/s. The height of the arrow above the ground is given by the function $ h(t) = 49t - 4.9t^2$. How long did it take the arrow to reach a height of h meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-f9236bb72528d77d31afe5e23c55f72dbfbdb287cdd8b2791f1debc23f55f3ce | An arrow is spun once and lands on one of three equally likely regions, numbered 1, 2, and 3. List all possible events for one spin of the arrow. | [
"\\{1, 2, 3\\}"
] | [
"An arrow is spun once and lands on one of three equally likely regions, numbered 1, 2, and 3. List all possible events for one spin of the arrow."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-30abf28768d58a008d709aef7074e5fd456aae84b360d0fe6b8bce94cb7c5d43 | An artificial intelligence dataset consists of 200 photos of airplanes, 300 photos of water skis, 250 photos of boats, 300 photos of trains, and 400 photos of spacecraft. What is the minimum number of images that must be selected to guarantee that at least one photo of each type of object is present? | [
"1251"
] | [
"An artificial intelligence dataset consists of 200 photos of airplanes, 300 photos of water skis, 250 photos of boats, 300 photos of trains, and 400 photos of spacecraft. What is the minimum number of images that must be selected to guarantee that at least one photo of each type of object is present?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-edd9e7fca7648c7884e0c8b3d29cac2229da96eb8bf32aba2ebfc3afd50ff7d0 | An artificial reservoir is a rectangle with a 1 km difference between the sides. Two fishermen simultaneously leave one vertex of the rectangle for a point located at the opposite vertex. One fisherman crosses the river in a boat, the other walks along the bank. Find the size of the reservoir if each of them has the sp... | [
"$3\\cdot 4$ km"
] | [
"An artificial reservoir is a rectangle with a 1 km difference between the sides. Two fishermen simultaneously leave one vertex of the rectangle for a point located at the opposite vertex. One fisherman crosses the river in a boat, the other walks along the bank. Find the size of the reservoir if each of them has t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9838a62d422b1800a7bd63332619d7aaec049ecfba04f5d400ce0b6f24015a37 | An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each time two red dots are adjacent is one fault, and each time two blue dots are exactly two apart (with exactly one dot in between them) is another fault. What is the smallest number of faults the crit... | [
"250"
] | [
"An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each time two red dots are adjacent is one fault, and each time two blue dots are exactly two apart (with exactly one dot in between them) is another fault. What is the smallest number of faults the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-d0eb25b3be8958aa2fd25e365da5b5ac783985cc2c1d74f96115693011af8e35 | An artist named Dean sold a painting to Rick for $100$ dollars and later bought it back from Rick for $80$ dollars. Dean then sold the painting to Morty for $90$ dollars. Calculate Dean's total profit from these transactions. | [
"110"
] | [
"An artist named Dean sold a painting to Rick for $100$ dollars and later bought it back from Rick for $80$ dollars. Dean then sold the painting to Morty for $90$ dollars. Calculate Dean's total profit from these transactions."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93041f0cb9d66b44522e67645b34bdfb3e7137eb3ff896bb7c2bf72932e11e00 | An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number: (a) The sum of any two digits of the number is divisible by the remaining digit. (b) The number is divisible by 3, but if the first digit is removed, the remaining two-di... | [
"246"
] | [
"An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number:\n(a) The sum of any two digits of the number is divisible by the remaining digit.\n(b) The number is divisible by 3, but if the first digit is removed, the remaining... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-d21b0dbfaeb6ccd566c20f3ef9e706e5e379fcd7423538da91f32fc5156d3b50 | An assignment of either a \( 0 \) or a \( 1 \) to each unit square of an \( m \times n \) chessboard is called fair if the total numbers of \( 0 \)s and \( 1 \)s are equal. A real number \( a \) is called beautiful if there are positive integers \( m, n \) and a fair assignment for the \( m \times n \) chessboard such ... | [
"50"
] | [
"An assignment of either a \\( 0 \\) or a \\( 1 \\) to each unit square of an \\( m \\times n \\) chessboard is called fair if the total numbers of \\( 0 \\)s and \\( 1 \\)s are equal. A real number \\( a \\) is called beautiful if there are positive integers \\( m, n \\) and a fair assignment for the \\( m \\times... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84fbe4f683baed71e4a71c8e209635654ee4490d599a18e68c00867f89a7e31c | An assistant holds a standard deck of 52 cards. An audience member selects five cards and gives them to the assistant. The assistant then reveals four of the five cards. Determine a method by which the assistant and the magician can collaborate so that the magician can always correctly identify the fifth card. | [
"\\text{The method uses suit matching and permutation encoding to determine the fifth card.}"
] | [
"An assistant holds a standard deck of 52 cards. An audience member selects five cards and gives them to the assistant. The assistant then reveals four of the five cards. Determine a method by which the assistant and the magician can collaborate so that the magician can always correctly identify the fifth card."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-d10e6f5307191cb9e7c31ca39d86ee5d6759d80a279be8682f1762b9c2cf0633 | An assortment of 200 pencils is sold through a catalog for $\$19.90$. Shipping is an additional $\$6.95$. Including the charges for both the pencils and the shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole number. | [
"13"
] | [
"An assortment of 200 pencils is sold through a catalog for $\\$19.90$. Shipping is an additional $\\$6.95$. Including the charges for both the pencils and the shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c48e1fa6dca0e26492f9df75bc42b2e5241d73c7529422d2a6c12f60a3a550ac | An asteroid has the shape of a parallelepiped with two identical wolves at opposite peaks. Each wolf controls the part of the surface within which it can run to any point faster than its antipode. At what ratio of the dimensions of the parallelepiped will one wolf control an entire face? | [
"1:1:\\sqrt{2}"
] | [
"An asteroid has the shape of a parallelepiped with two identical wolves at opposite peaks. Each wolf controls the part of the surface within which it can run to any point faster than its antipode. At what ratio of the dimensions of the parallelepiped will one wolf control an entire face?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-9f239d8fee3f8bdd74dd15744fa03f6235136c670190acfd1250b7e39e0eb1d5 | An astronaut at point A, M miles above the surface of the earth, measures the angle OAH between the directions to the horizon (H) and the center of the earth (O) to be 45 degrees. What is the radius of the earth? | [
"\\( r = \\frac{M}{\\sqrt{2} - 1} \\)"
] | [
"An astronaut at point A, M miles above the surface of the earth, measures the angle OAH between the directions to the horizon (H) and the center of the earth (O) to be 45 degrees. What is the radius of the earth?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-3d86fc7c2fa2fe4e0773fa9be83a7040c958198a4d1c1941554a10a1dd4cb3ef | An astronaut is tethered to the vertex of a rectangular prism in space by a string that is 15 units long. The dimensions of the rectangular prism are 10, 20, and 20 units. What is the volume of the region of space that the astronaut can cover? | [
"\\dfrac{24125}{6} \\pi"
] | [
"An astronaut is tethered to the vertex of a rectangular prism in space by a string that is 15 units long. The dimensions of the rectangular prism are 10, 20, and 20 units. What is the volume of the region of space that the astronaut can cover?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-d9c1e3f3f8d1a7abef70c3876d380f7aa64c31395897209113bf27d4478b3233 | An astronaut lands on the equator of a spherical asteroid. He travels due north 100 km, then east 100 km, then south 100 km. He finds out that he is due east of his original starting point by 200 km. How many kilometers more would he now need to travel by continuing in an easterly direction in order to reach his origin... | [
"400"
] | [
"An astronaut lands on the equator of a spherical asteroid. He travels due north 100 km, then east 100 km, then south 100 km. He finds out that he is due east of his original starting point by 200 km. How many kilometers more would he now need to travel by continuing in an easterly direction in order to reach his o... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5d0ee76221b8e85d32d650c706d83352f133bc33b7709e9c40c5ee4eb5a71ff4 | An astronaut lands on the equator of a spherical asteroid. She travels due north 100km, then east 100km, then south 100km. She finds that she is due east of her original starting point by 200km. How many kilometers would she need to travel by continuing her journey in an easterly direction in order to reach her origina... | [
"400",
"400km"
] | [
"An astronaut lands on the equator of a spherical asteroid. She travels due north 100km, then east 100km, then south 100km. She finds that she is due east of her original starting point by 200km. How many kilometers would she need to travel by continuing her journey in an easterly direction in order to reach her or... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-c6cbcdf44ea741fa3ba4be156f269df3a155c035e517665ba70f97735040fef8 | An astronaut plants a flagpole on the surface of the Moon such that the top of the flagpole is at eye-level height, which is $1.5$ metres. The astronaut then walks off over level ground until the top of the flagpole is just visible on the horizon. The straight-line distance from the astronaut's line of sight to the top... | [
"\\( 7053332.5833 \\) meters"
] | [
"An astronaut plants a flagpole on the surface of the Moon such that the top of the flagpole is at eye-level height, which is $1.5$ metres. The astronaut then walks off over level ground until the top of the flagpole is just visible on the horizon. The straight-line distance from the astronaut's line of sight to th... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-b1d121cb4e82eea6fe1a93f538b9765ad727aff965fdbd3910d40d852ddc3665 | An astronomer added all the distances between 50 stars, resulting in a sum S. If a cloud obscured 25 of these stars, calculate the maximum possible sum of the distances between the remaining 25 visible stars, expressed as a fraction of S. | [
"1"
] | [
"An astronomer added all the distances between 50 stars, resulting in a sum S. If a cloud obscured 25 of these stars, calculate the maximum possible sum of the distances between the remaining 25 visible stars, expressed as a fraction of S."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-d4c66c784e6d17433829c7e67752f259d5cb9c3ef6485dcf35e9fb5559dd98ae | An astronomer added all the distances between 50 stars, resulting in a sum \( S \). Suddenly, a cloud obscured 25 of the stars. Calculate the maximum possible sum of the distances between the 25 visible stars, expressed as a fraction of \( S \). Express your answer as a simplified fraction. | [
"\\dfrac{12}{49}"
] | [
"An astronomer added all the distances between 50 stars, resulting in a sum \\( S \\). Suddenly, a cloud obscured 25 of the stars. Calculate the maximum possible sum of the distances between the 25 visible stars, expressed as a fraction of \\( S \\). Express your answer as a simplified fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-51bf007bf197c6278dd3178b71968379115a6eeed7b4fcf538ba6d1516b7c0fd | An astronomer is at the bottom of a circular well with depth \( H = 15 \) meters and radius \( r = 1 \) meter. The astronomer's eyes are at height \( h = 1.75 \) meters from the bottom of the well. Ignoring refraction, what percentage of the sky can the astronomer see at one time? | [
"0.1418%"
] | [
"An astronomer is at the bottom of a circular well with depth \\( H = 15 \\) meters and radius \\( r = 1 \\) meter. The astronomer's eyes are at height \\( h = 1.75 \\) meters from the bottom of the well. Ignoring refraction, what percentage of the sky can the astronomer see at one time?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-5c8f9a9c28795b307c12ed3784f4e4b3e04254bb504500d4bff00d035c7a52b3 | An astronomer is at the bottom of a circular well with depth \( H = 15 \) meters and radius \( r = 1 \) meter. The astronomer's eyes are at height \( h = 1.75 \) meters from the bottom of the well. Ignoring refraction, what percentage of the sky can the astronomer see during the whole year? | [
"0.28\\%"
] | [
"An astronomer is at the bottom of a circular well with depth \\( H = 15 \\) meters and radius \\( r = 1 \\) meter. The astronomer's eyes are at height \\( h = 1.75 \\) meters from the bottom of the well. Ignoring refraction, what percentage of the sky can the astronomer see during the whole year?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-d122568a4115dd5ca2d7c27622d468e948f83bb458c5793a90becc883e0fbd03 | An athlete ran a distance of 2000 meters in 10 minutes. Let \( t \) be the time in minutes from the start of the run such that the athlete ran exactly 1000 meters between time \( t \) and \( t+5 \). Find the sum of all possible values of \( t \). | [
"5"
] | [
"An athlete ran a distance of 2000 meters in 10 minutes. Let \\( t \\) be the time in minutes from the start of the run such that the athlete ran exactly 1000 meters between time \\( t \\) and \\( t+5 \\). Find the sum of all possible values of \\( t \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-0592898726e16ed35e54c0144106489a1710c56f88306abf266a76dbd38835d3 | An athlete travels from point $A$ directly to point $B.$ If he runs $3/4$ of the way, walks $1/9$ of the way, and jogs the last $10$ miles, what is the distance from point $A$ to point $B?$ | [
"72 miles"
] | [
"An athlete travels from point $A$ directly to point $B.$ If he runs $3/4$ of the way, walks $1/9$ of the way, and jogs the last $10$ miles, what is the distance from point $A$ to point $B?$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1bf3aef5e52c7e4f0f5c9a40c0f14a512cf0d708c445c4478df7f8c746e7a14b | An athlete's heart beats an average of 150 times per minute while running. How many times does the athlete's heart beat during a 26-mile race if the athlete runs at a pace of 5 minutes per mile? | [
"19500"
] | [
"An athlete's heart beats an average of 150 times per minute while running. How many times does the athlete's heart beat during a 26-mile race if the athlete runs at a pace of 5 minutes per mile?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66c4384f4a711b265601d96c7c32a06c62307760b39554e041228a878c6d6703 | An atom of radioactive element A decays (with mean lifetime 5 years) to an atom of radioactive element B, which in turn decays (with mean lifetime 2 years) to an atom of stable element C. Let \( Y_1 \) be the time at which the first decay occurs and let \( Y_2 \) be the time at which the second decay occurs. \( Y_1 \) ... | [
"\\( f(y_1, y_2) = \\frac{1}{10} e^{(3/10)y_1 - y_2/2} \\) for \\( y_1 > 0 \\) and \\( y_2 > y_1 \\)."
] | [
"An atom of radioactive element A decays (with mean lifetime 5 years) to an atom of radioactive element B, which in turn decays (with mean lifetime 2 years) to an atom of stable element C. Let \\( Y_1 \\) be the time at which the first decay occurs and let \\( Y_2 \\) be the time at which the second decay occurs. \... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-9b65dfdd246911016cc3beef92f38647fb7c8adbee7d780ec167a29b4d525987 | An attacker shoots a laser beam into a smart card, causing a fault in the circuit implementing the computation \( c = m^e \mod p \). As a result, one random digit of \( c \) is incorrect, leading to a new incorrect value \( c^* \). Given \( n = pq \) where \( p \) and \( q \) are primes with \( \gcd(p, q) = 1 \), \( p ... | [
"48611"
] | [
"An attacker shoots a laser beam into a smart card, causing a fault in the circuit implementing the computation \\( c = m^e \\mod p \\). As a result, one random digit of \\( c \\) is incorrect, leading to a new incorrect value \\( c^* \\). Given \\( n = pq \\) where \\( p \\) and \\( q \\) are primes with \\( \\gcd... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-da50b771fbd08b6b333b7d9d03973d7bb3c2402dcfd25f30da1b6867d2067fba | An auditorium has $a$ chairs arranged in rows, the number of chairs in each row being the same. If $b$ chairs are added to each row and the number of rows is reduced by $c$, then the total number of places in the hall will increase by one-tenth of their original number. How many chairs are there in each row? | [
"The number of chairs in each row is equal to $\\frac{\\sqrt {(a+10bc)^2+400abc}-(a+10bc)}{20c}$"
] | [
"An auditorium has $a$ chairs arranged in rows, the number of chairs in each row being the same. If $b$ chairs are added to each row and the number of rows is reduced by $c$, then the total number of places in the hall will increase by one-tenth of their original number. How many chairs are there in each row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c373ebfab36de5685e2dc1ff2763c8c1b6093f9482e08224a43cc7702135b503 | An auditorium in New York contains 500 seats. Ticket prices for a recent play were $50 for adults and $25 for children. If the proceeds totaled $19,250, how many adults and how many children were there? | [
"270 adults and 230 children."
] | [
"An auditorium in New York contains 500 seats. Ticket prices for a recent play were $50 for adults and $25 for children. If the proceeds totaled $19,250, how many adults and how many children were there?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 29
} | 33 | 0 | 0 |
omr-ws-9a884a0c63eff42161bffa7abd8383d84b5d083227e053452ed9335460617548 | An auto on the attraction "Merry parking" can move along an arc with radius $r_1$ or $r_2$. Starting from point $A$ and moving towards the North, the auto covers a distance $\ell$ and rotates by an angle $a < 2\pi$. Determine the locus of the possible endpoints of the auto's movement. | [
"The intersection of two discs centered at \\(O_1\\) and \\(O_2\\) with radii \\(2(r_1 - r_2) \\sin \\alpha_2\\) and \\(2(r_1 - r_2) \\sin \\alpha_1\\) respectively."
] | [
"An auto on the attraction \"Merry parking\" can move along an arc with radius $r_1$ or $r_2$. Starting from point $A$ and moving towards the North, the auto covers a distance $\\ell$ and rotates by an angle $a < 2\\pi$. Determine the locus of the possible endpoints of the auto's movement."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66ff7e13ddd9eccb1778f40de66e1f98f34c3cee69656a9ce6aea0d1f09f8b82 | An autobiographical number is a number $N$ such that the first digit of $N$ counts how many zeroes are in $N$, the second digit counts how many ones are in $N$, and so on. What is the sum of the digits of an autobiographical number? | [
"10"
] | [
"An autobiographical number is a number $N$ such that the first digit of $N$ counts how many zeroes are in $N$, the second digit counts how many ones are in $N$, and so on. What is the sum of the digits of an autobiographical number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a673c53f2b80afbfa5b27dda105b63d6d3efb668e99b41005bf41e2eb113f1c7 | An automobile gets 28 miles per gallon of gasoline for speeds up to 50 miles per hour. Over 50 miles per hour, the number of miles per gallon drops at the rate of 12 percent for each 10 miles per hour. Let \( s \) be the speed and \( y \) be the number of miles per gallon. Find \( y \) as a function of \( s \) by solvi... | [
"\\( y = 28e^{-0.012(s - 50)} \\)"
] | [
"An automobile gets 28 miles per gallon of gasoline for speeds up to 50 miles per hour. Over 50 miles per hour, the number of miles per gallon drops at the rate of 12 percent for each 10 miles per hour. Let \\( s \\) be the speed and \\( y \\) be the number of miles per gallon. Find \\( y \\) as a function of \\( s... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4768bdbd6a64aa215184c5a67c8de2aadcdc4182464a8763afa88c95690c8f55 | An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute along a straight road. If the speed of the car is limited to a maximum of $90$ miles per hour, calculate the minimum magnitude of the acceleration or deceleration, in feet per second squared, that the car must experience at... | [
"6.6"
] | [
"An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute along a straight road. If the speed of the car is limited to a maximum of $90$ miles per hour, calculate the minimum magnitude of the acceleration or deceleration, in feet per second squared, that the car must experien... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5fb70177b06d4f3ee56a8331a0d321fcd8c03bf7a3c32dbdc275f30bcebc3816 | An automobile's six-digit license plate number is recorded upside down by an officer. When the original number is subtracted from the upside-down number, the difference is 512,082. What was the original license plate number? | [
"106819"
] | [
"An automobile's six-digit license plate number is recorded upside down by an officer. When the original number is subtracted from the upside-down number, the difference is 512,082. What was the original license plate number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.