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-f81cc2316a8024a860c32ef30cc99e57d3366f19d36732f4782f1a1ef3d09e5b | A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC... | [
"\\dfrac{13 + \\sqrt{85}}{9}"
] | [
"A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\\overline{AD}$ at point $C'$, and edge $\\overline{AB}$ at point $E$. Suppose that $C'D = \\frac{1}{3}$. What is the perimeter of triangle $\\bigtrian... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-95709067067cc4b9956cab93c324cd13ba888578ed61329e14a74242297b3f2c | A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along... | [
"871"
] | [
"A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\\circ}$ (see the figure below). The paper is then folded u... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21a2adb17a47c828037c0ab0f778c7ee18b40de9c831edaee7b116e05365d16d | A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along... | [
"871"
] | [
"A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\\circ}$ (see the figure below). The paper is then folded u... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ae446f9251e5318d55530f3fa0cc3d41e69c36e7fc5f7d06a8060c4a1ebc5cc | A square piece of paper has sides of length 100. From each corner, a wedge is cut such that each cut starts at a distance of $\sqrt{17}$ from the corner and they meet on the diagonal at an angle of $60^\circ$. The paper is then folded up along the lines joining the vertices of adjacent cuts, and the two edges of each c... | [
"871"
] | [
"A square piece of paper has sides of length 100. From each corner, a wedge is cut such that each cut starts at a distance of $\\sqrt{17}$ from the corner and they meet on the diagonal at an angle of $60^\\circ$. The paper is then folded up along the lines joining the vertices of adjacent cuts, and the two edges of... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-496640d9be14c921c1ce0dfa75af237fc517bf09a5928787f684679b6e6a8e8b | A square piece of paper is cut into two parts using a sequence of straight cuts. If one of the smaller pieces has 10 vertices, what is the maximum number of vertices the other piece could have? | [
"10"
] | [
"A square piece of paper is cut into two parts using a sequence of straight cuts. If one of the smaller pieces has 10 vertices, what is the maximum number of vertices the other piece could have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
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omr-ws-814da977874697bed0c5a76d41353773939ca43d529aa8e50d6af58f84aef47a | A square piece of toast \( ABCD \) with side length 1 and center \( O \) is cut in half to form two equal pieces \( ABC \) and \( CDA \). Find the length and location of the shortest straight cut which divides the triangle \( ABC \) into two parts of equal area. | [
"\\sqrt{\\sqrt{2} - 1}"
] | [
"A square piece of toast \\( ABCD \\) with side length 1 and center \\( O \\) is cut in half to form two equal pieces \\( ABC \\) and \\( CDA \\). Find the length and location of the shortest straight cut which divides the triangle \\( ABC \\) into two parts of equal area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-23a298d16777d8eeef6a2a2a6920076cf62df66c6aedd1e19ba42197c9a813ad | A square pyramid ABCDE has a base with side length 12. The lengths of the edges from the vertex E to the base vertices are AE = $\sqrt{193}$, BE = 11, CE = $\sqrt{97}$, and DE = 13. What is the volume of the pyramid? | [
"\\( 144 \\sqrt{7} \\)"
] | [
"A square pyramid ABCDE has a base with side length 12. The lengths of the edges from the vertex E to the base vertices are AE = $\\sqrt{193}$, BE = 11, CE = $\\sqrt{97}$, and DE = 13. What is the volume of the pyramid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b9197daff11a1cba049c3f95667050e22fc3623410df7ba9d49df81b2a22a940 | A square pyramid ABCDE with vertex E has a base of side length 12. If AE=193, BE=11, CE=97, and DE=13, what is the volume of the pyramid? | [
"720"
] | [
"A square pyramid ABCDE with vertex E has a base of side length 12. If AE=193, BE=11, CE=97, and DE=13, what is the volume of the pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-837b53c2833adf44754e629ef3b6f65b7eb1b9f5598851528a2af40c01939302 | A square pyramid circumscribed about a sphere with radius 10 has minimum surface area. What is the height of this pyramid? | [
"40"
] | [
"A square pyramid circumscribed about a sphere with radius 10 has minimum surface area. What is the height of this pyramid?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9bf1451386e88efa5bad1b6cb687b1e08cef27cbf1f0bd7ad5c7b5c43e1d9fbf | A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the original pyramid? | [
"\\frac{63}{64}"
] | [
"A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the original pyramid?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a6adf874dd13fa096b76dc6fad8169c5710e666b7610e127a75fb7d0933baf85 | A square pyramid has all edges of equal length, and point $A$ is the apex of the pyramid. A point $P$ is randomly and uniformly chosen in the interior of the pyramid. The probability that $P$ is closer to $A$ than any of the other four vertices of the pyramid is equal to $\frac{m}{n}$ for relatively prime positive inte... | [
"9"
] | [
"A square pyramid has all edges of equal length, and point $A$ is the apex of the pyramid. A point $P$ is randomly and uniformly chosen in the interior of the pyramid. The probability that $P$ is closer to $A$ than any of the other four vertices of the pyramid is equal to $\\frac{m}{n}$ for relatively prime positiv... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-d8dacda8096a404b9354d0b9f6470b485539c8bf889624500ca4b962a41b69c7 | A square pyramid is built around a pile of spheres arranged in $n$ layers, with the $k$-th layer containing $k^2$ spheres. Each sphere is tangent to adjacent spheres and those directly below it. As $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sq... | [
"6"
] | [
"A square pyramid is built around a pile of spheres arranged in $n$ layers, with the $k$-th layer containing $k^2$ spheres. Each sphere is tangent to adjacent spheres and those directly below it. As $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\\fr... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-477339a79327c8eb13e3e46c7058b0088e66c3512596bfe503c931552067ac5c | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | [
"080"
] | [
"A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-84786978ddc5c7feb00b7318b525340df0117bae1ea7317043a7beed79804c07 | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | [
"80"
] | [
"A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $\\overline{AE}$, $\\overline{BC}$, and $\\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 6
} | 39 | 0 | 0 |
omr-ws-750dd64ae9ada99f90d5b350e677d4b44c1b6d05f191a4642130fbb84a118c2a | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | [
"48"
] | [
"A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f2fb391418aa2d6bca6412aa046636856086e5c844132bcce900f12009239bf | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | [
"80"
] | [
"A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\\overline{AE}$, $\\overline{BC}$, and $\\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
] | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 55,
"genselect": 6,
"tir": 6
} | 67 | 0 | 0 |
omr-ws-c9266d36d45828767eadd6184ba99cbe493f90b8f7b45e2ff48e75452677527c | A square pyramid with base ABCD and vertex E has eight edges of length 4. A plane passes through the midpoints of AE, BC, and CD. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | [
"80"
] | [
"A square pyramid with base ABCD and vertex E has eight edges of length 4. A plane passes through the midpoints of AE, BC, and CD. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd35620ca66843bffda123189b49529ee365974a373410d2648329358db56866 | A square pyramid with base \(ABCD\) and vertex \(E\) has eight edges of length 2. A plane passes through the midpoints of \(AE\), \(BC\), and \(CD\). The plane's intersection with the pyramid has an area that can be expressed as \(\sqrt{p}\). Find \(p\). | [
"5"
] | [
"A square pyramid with base \\(ABCD\\) and vertex \\(E\\) has eight edges of length 2. A plane passes through the midpoints of \\(AE\\), \\(BC\\), and \\(CD\\). The plane's intersection with the pyramid has an area that can be expressed as \\(\\sqrt{p}\\). Find \\(p\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a374d578ddfd57afdf95fdd2275797cc7771fbb1628eb65fca0f525beca604c6 | A square pyramid with edge length $1$ is cut by a plane crossing all its side edges, forming a convex quadrilateral $ABCD$ that is not a trapezoid. Lines $AB$ and $CD$ intersect at point $P$. Determine all possible values for the distance from point $P$ to the base of the pyramid. | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"A square pyramid with edge length $1$ is cut by a plane crossing all its side edges, forming a convex quadrilateral $ABCD$ that is not a trapezoid. Lines $AB$ and $CD$ intersect at point $P$. Determine all possible values for the distance from point $P$ to the base of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-5f67f261e065890650fcca12d42145302d0970d40e512a18d82dfc1b10ff1d1e | A square region $WXYZ$ is externally tangent to the circle with equation $x^2 + y^2 = 1$ at the point $(0,1)$ on the side $YZ$. Vertices $W$ and $X$ are on the circle with equation $x^2 + y^2 = 4$. What is the side length of this square? | [
"\\(\\frac{2\\sqrt{19}-4}{5}\\)"
] | [
"A square region $WXYZ$ is externally tangent to the circle with equation $x^2 + y^2 = 1$ at the point $(0,1)$ on the side $YZ$. Vertices $W$ and $X$ are on the circle with equation $x^2 + y^2 = 4$. What is the side length of this square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 23
} | 36 | 0 | 0 |
omr-ws-cf6c47743eefa9d6af4e4e24f5845d3963646cd6d50898b57d0e9b3ed46fbd39 | A square region \(ABCD\) is externally tangent to the circle with equation \(x^2 + y^2 = 1\) at the point \((0,1)\) on the side \(CD\). Vertices \(A\) and \(B\) are on the circle with equation \(x^2 + y^2 = 4\). What is the side length of this square? | [
"\\(\\frac{2\\sqrt{19}-4}{5}\\)"
] | [
"A square region \\(ABCD\\) is externally tangent to the circle with equation \\(x^2 + y^2 = 1\\) at the point \\((0,1)\\) on the side \\(CD\\). Vertices \\(A\\) and \\(B\\) are on the circle with equation \\(x^2 + y^2 = 4\\). What is the side length of this square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 26
} | 47 | 0 | 0 |
omr-ws-7b3ab177b3193f8c998cf57472775de1d0111fc9f0ebb597c91aa53afb7cd1d2 | A square region is bounded by the graphs of $|x| = 1$ and $|y| = 1$. Two points are randomly chosen inside the region. Given that these two points form the vertices of an equilateral triangle, what is the probability that the third vertex will be inside the square? | [
"\\dfrac{1}{4}"
] | [
"A square region is bounded by the graphs of $|x| = 1$ and $|y| = 1$. Two points are randomly chosen inside the region. Given that these two points form the vertices of an equilateral triangle, what is the probability that the third vertex will be inside the square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-ec10a090b92022bd3faeb07361caaf65cf6f667771af869f1318a61e162caabc | A square room with an area of $81$ m$^2$ is entirely covered by two non-overlapping rectangular rugs $A$ and $B$. Rug $B$ is displaced, and rug $A$ is rotated through $90^\circ$ and placed on rug $B$. Given that the area of rug $B$ is twice the area of rug $A$, calculate the area of the part of the floor that was left ... | [
"18"
] | [
"A square room with an area of $81$ m$^2$ is entirely covered by two non-overlapping rectangular rugs $A$ and $B$. Rug $B$ is displaced, and rug $A$ is rotated through $90^\\circ$ and placed on rug $B$. Given that the area of rug $B$ is twice the area of rug $A$, calculate the area of the part of the floor that was... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-847a7e97db937587afe385cf258f8e01964f5b2f6158032bf43c962806280db3 | A square shack 30 feet by 30 feet is in the middle of a large field. A goat is tethered to one corner of the shack by a chain 60 feet long. She cannot get into, onto, or under the shack, but can graze anywhere else she can reach on her chain. What is the area of the portion of the field she can graze? | [
"\\( 3150\\pi \\)"
] | [
"A square shack 30 feet by 30 feet is in the middle of a large field. A goat is tethered to one corner of the shack by a chain 60 feet long. She cannot get into, onto, or under the shack, but can graze anywhere else she can reach on her chain. What is the area of the portion of the field she can graze?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 26
} | 39 | 0 | 0 |
omr-ws-ad63e73c698be88037f10f735751d3a761cad653b57493a030adf7146e8608fb | A square sheet of paper $ABCD$ is dropped onto a square tile such that corner $A$ of the sheet falls into a corner of the tile, side $BC$ goes through another corner of the tile $E$, and corner $C$ lands on the edge of the tile. Find the measure of angle $BAE$. | [
"\\(22.5^\\circ\\) or \\(\\frac{\\pi}{8}\\) radians"
] | [
"A square sheet of paper $ABCD$ is dropped onto a square tile such that corner $A$ of the sheet falls into a corner of the tile, side $BC$ goes through another corner of the tile $E$, and corner $C$ lands on the edge of the tile. Find the measure of angle $BAE$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-606e28a545ebfad44df60fdc7162d21f2fcfb3675637e930513d06bfd585c145 | A square sheet of paper $ABCD$ is folded along the segment $MN$ so that the vertex $A$ comes to the position $A_1$ on the edge $CD$. If the side length of the square is $s$, and $B_1$ is the image of $B$ after the fold, calculate the length of $B_1K$ where $K$ is the point of tangency of the incircle of $\triangle KCA_... | [
"\\dfrac{s}{2}"
] | [
"A square sheet of paper $ABCD$ is folded along the segment $MN$ so that the vertex $A$ comes to the position $A_1$ on the edge $CD$. If the side length of the square is $s$, and $B_1$ is the image of $B$ after the fold, calculate the length of $B_1K$ where $K$ is the point of tangency of the incircle of $\\triangl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e5d856f133030cab02673b915b72754aa2a8d51e5931f1083f82ea644844336f | A square sheet of paper $ABCD$ is folded such that point $B$ falls on the midpoint $M$ of side $CD$. Let $P$ be the point where the crease intersects side $BC$. If $BP = 5x$ and $PC = 3x$, find the value of $x$ given that the side length of the square is $8$. | [
"1"
] | [
"A square sheet of paper $ABCD$ is folded such that point $B$ falls on the midpoint $M$ of side $CD$. Let $P$ be the point where the crease intersects side $BC$. If $BP = 5x$ and $PC = 3x$, find the value of $x$ given that the side length of the square is $8$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e45c42899c6fe5b5bafdafd87af904f504e490d6e09e73e6f92ddff0c28cdb1 | A square sheet of paper $ABCD$ is folded such that point $B$ hits the midpoint of side $CD$. In what ratio does the fold line divide side $BC$? | [
"\\( 5:3 \\)"
] | [
"A square sheet of paper $ABCD$ is folded such that point $B$ hits the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-3e5bf627c9d4b8a0fd3153cdbd7b8f32df1ec166c285f3bd998679d71938476d | A square sheet of paper $ABCD$ is folded with $A$ falling on $E$, which is on $BC$, with $T$ falling on $F$, and $GH$ is the crease. Suppose $\overline{AB} = 18$ (cm), $\overline{BE} = 6$ (cm). Find the area of the trapezoid $EFGH$. | [
"150"
] | [
"A square sheet of paper $ABCD$ is folded with $A$ falling on $E$, which is on $BC$, with $T$ falling on $F$, and $GH$ is the crease. Suppose $\\overline{AB} = 18$ (cm), $\\overline{BE} = 6$ (cm). Find the area of the trapezoid $EFGH$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-1fe29694524d64b29c3959aeafcd09a1c2fdaee6ba5b014491130be8804ef05e | A square sheet of paper $ABCD$ with an area of $12$ cm$^2$ is blue on one side and red on the other. The paper is laid out on a table, red-side up. Corner $A$ is folded over to a point $A'$ such that a blue isosceles triangle is now visible. The area of the blue triangle equals the area of the remaining red region. If ... | [
"44"
] | [
"A square sheet of paper $ABCD$ with an area of $12$ cm$^2$ is blue on one side and red on the other. The paper is laid out on a table, red-side up. Corner $A$ is folded over to a point $A'$ such that a blue isosceles triangle is now visible. The area of the blue triangle equals the area of the remaining red region... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ceb82e916dbfdb5546abd402f3f1b048bf9822520c4c3d845f264b3142165c61 | A square sheet of paper ABCD is folded such that B falls on the midpoint M of CD. Let the crease intersect BC at point P. If the length of BP is represented as \( \frac{5}{8} \) of the side length of the square, find the length of PC as a fraction of the side length of the square. | [
"\\dfrac{3}{8}"
] | [
"A square sheet of paper ABCD is folded such that B falls on the midpoint M of CD. Let the crease intersect BC at point P. If the length of BP is represented as \\( \\frac{5}{8} \\) of the side length of the square, find the length of PC as a fraction of the side length of the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c9205165ad4b990b3ef1c3b5bedb754028b82449f39fa8ebaafd698e41b981d | A square sheet of paper ABCD is folded such that point B falls on the midpoint M of side CD. Determine the ratio in which the crease divides side BC. | [
"\\( 3:5 \\)",
"\\(\\frac{3}{5}\\)"
] | [
"A square sheet of paper ABCD is folded such that point B falls on the midpoint M of side CD. Determine the ratio in which the crease divides side BC."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 32,
"genselect": 1,
"tir": 2
} | 35 | 0 | 0 |
omr-ws-766d9f03082e7fb7d565914a0684ba6bba8bfe7337a6e393a37b8b893d6171ad | A square sheet of paper has area $6 \text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest radic... | [
"2\\sqrt{2}"
] | [
"A square sheet of paper has area $6 \\text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92f90f9f397942481a9c4aa619777df76cb6bb205f66fb763a5ef8aeddcb2c11 | A square sheet of paper has been cut into eight square parts. The areas of seven of these parts are each 25 square centimeters. What is the area of the eighth square? | [
"225"
] | [
"A square sheet of paper has been cut into eight square parts. The areas of seven of these parts are each 25 square centimeters. What is the area of the eighth square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-1c8a5379e1e01b734d4f85613d1dc1a35c9df56ec807e10df9b2af58f9b0f95f | A square sheet of paper is folded along two parallel lines and then pierced at one place (not on the fold lines). After unfolding the sheet, how many different holes could there be on the sheet? | [
"3"
] | [
"A square sheet of paper is folded along two parallel lines and then pierced at one place (not on the fold lines). After unfolding the sheet, how many different holes could there be on the sheet?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a51ba6a9206156b792f2891766f711a54098857eaf7bddf045eaa911ec9f9302 | A square sheet of paper measuring $24$ cm $\times 24$ cm is folded in 8 steps to create a duck. After unfolding, crease lines are visible. The folds that opened towards Benedek are marked in blue, and those that opened towards the table are marked in red. Determine the difference in total length between the blue lines ... | [
"0"
] | [
"A square sheet of paper measuring $24$ cm $\\times 24$ cm is folded in 8 steps to create a duck. After unfolding, crease lines are visible. The folds that opened towards Benedek are marked in blue, and those that opened towards the table are marked in red. Determine the difference in total length between the blue ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7bea8cb598d52086e8e73a6d2e9944f96bb80bd5da9cd4eaf1913765f8c2ed01 | A square stool had all its legs sawed off, resulting in four pieces of different lengths. After sawing, the stool still touches the floor with all four ends of the legs. If three of the pieces have lengths of 8 cm, 9 cm, and 10 cm, what can be the length of the fourth piece? | [
"7 or 11"
] | [
"A square stool had all its legs sawed off, resulting in four pieces of different lengths. After sawing, the stool still touches the floor with all four ends of the legs. If three of the pieces have lengths of 8 cm, 9 cm, and 10 cm, what can be the length of the fourth piece?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b1261d47bbd2fe369fb72e73e6754ed326073b7ce6474b4f654747a87c161ec2 | A square table is covered with a square cloth without folds or wrinkles, leaving all four corners of the table uncovered and creating four triangular hanging parts. Given that two adjacent hanging parts have areas of 16 square units each, calculate the total area of the other two hanging parts. | [
"32"
] | [
"A square table is covered with a square cloth without folds or wrinkles, leaving all four corners of the table uncovered and creating four triangular hanging parts. Given that two adjacent hanging parts have areas of 16 square units each, calculate the total area of the other two hanging parts."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f4e44dfd068ed82c39a38f42f1b7ba253e7ebec89a06c806cb1527cd9f4d1e0 | A square table of dimensions \( n \times n \) is filled arbitrarily with the numbers \( 1, 2, \ldots, n^2 \) such that each number appears exactly once. From each row, the least number is chosen, and the maximum of these chosen numbers is denoted by \( x \). From each column, the least number is chosen, and the maximum... | [
"(n!)^2"
] | [
"A square table of dimensions \\( n \\times n \\) is filled arbitrarily with the numbers \\( 1, 2, \\ldots, n^2 \\) such that each number appears exactly once. From each row, the least number is chosen, and the maximum of these chosen numbers is denoted by \\( x \\). From each column, the least number is chosen, an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ab8c37b7b56f06119900c69dead1e7e62844cd7973ba894658d9f95956343edc | A square tile is divided into four smaller squares. Each small square must be painted using one of the colours yellow, red, or blue. How many different tiles can be painted if exactly two colours are used per tile and two tiles are considered the same if they can be rotated to look the same? | [
"18"
] | [
"A square tile is divided into four smaller squares. Each small square must be painted using one of the colours yellow, red, or blue. How many different tiles can be painted if exactly two colours are used per tile and two tiles are considered the same if they can be rotated to look the same?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93231e8f7cb5773c9b5e1664b30195871789c34cd3009e6c3530b1ea61448fa8 | A square train car is moving forward at $60$ miles per hour. An ant in a front corner of the train crawls toward the diagonally opposite corner on the floor of the car at $1$ mile per hour. At what speed in miles per hour is the ant moving relative to the ground? | [
"\\(\\sqrt{3601 - 60\\sqrt{2}}\\) or approximately \\(59.3\\) miles per hour."
] | [
"A square train car is moving forward at $60$ miles per hour. An ant in a front corner of the train crawls toward the diagonally opposite corner on the floor of the car at $1$ mile per hour. At what speed in miles per hour is the ant moving relative to the ground?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-115c19c5b9a2c55b57c06649cef1c96fd897546ca3cdc4513c8a8f021f5c6c13 | A square tray has nine houses arranged in a 3x3 grid. A piece on the tray can move to any adjacent house (up, down, left, or right). If the piece starts at the center house and moves randomly for 10 steps, what is the probability that it returns to the center house after 10 movements? | [
"\\(\\frac{3345}{8077}\\)"
] | [
"A square tray has nine houses arranged in a 3x3 grid. A piece on the tray can move to any adjacent house (up, down, left, or right). If the piece starts at the center house and moves randomly for 10 steps, what is the probability that it returns to the center house after 10 movements?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0af21e517e9dddced512eea8d7495afd4b82c46a1424033331b59ce9e19dbf4 | A square was divided into three equal parts by drawing two parallel lines. The vertical distance between these lines is 6. Find the area of the square. | [
"324"
] | [
"A square was divided into three equal parts by drawing two parallel lines. The vertical distance between these lines is 6. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5d093ec72637f8bc201af30adf1ef2198b4b079c8819b61ca97741cb3f72e7b | A square window has a central section made of plain glass and an outer section made of coloured glass. The curved lines in the outer section are arcs of circles with the same radius as the side length of the square, centered at the corners of the square. What is the ratio of the area of the coloured glass to the area o... | [
"3"
] | [
"A square window has a central section made of plain glass and an outer section made of coloured glass. The curved lines in the outer section are arcs of circles with the same radius as the side length of the square, centered at the corners of the square. What is the ratio of the area of the coloured glass to the a... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-77177db4680080bcae0005962584b9982fef09a2013992ae868049a8b1716a9e | A square with a circumradius of 6 units is rotated \( 45^\circ \) clockwise with its circumcenter \( O \) as the center of rotation. Both squares, before and after the rotation, are combined into one figure with a perimeter of \( K \) and an area of \( L \). The value of \( \left( \frac{L}{K} \right)^2 \) is ... | [
"\\(\\frac{153}{4} + 27\\sqrt{2}\\)"
] | [
"A square with a circumradius of 6 units is rotated \\( 45^\\circ \\) clockwise with its circumcenter \\( O \\) as the center of rotation. Both squares, before and after the rotation, are combined into one figure with a perimeter of \\( K \\) and an area of \\( L \\). The value of \\( \\left( \\frac{L}{K} \\right)^... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6ebe82faf03e191a0925f27b1097117b44610ebe4e696b65b66d1b775274325 | A square with a perimeter of 20 is inscribed in a square with a perimeter of 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? | [
"\\(\\sqrt{65}\\)"
] | [
"A square with a perimeter of 20 is inscribed in a square with a perimeter of 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-10732c49c726a86035a971844d99783567982dda8f1e0fef69b59cd5792bf1eb | A square with a side length of $100$ cm is divided into $9$ rectangles by two pairs of parallel lines. The central rectangle has side lengths of $40$ cm and $60$ cm. Find the sum of the areas of the four corner rectangles. | [
"\\(2400\\)"
] | [
"A square with a side length of $100$ cm is divided into $9$ rectangles by two pairs of parallel lines. The central rectangle has side lengths of $40$ cm and $60$ cm. Find the sum of the areas of the four corner rectangles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-08450f8d36fb88f6a8f24f5d868835507666e2dc00543419ea3c44ee609d2dce | A square with a side length of 10 cm contains a circle with a diameter of 1 cm, centered at the square's center. A coin with a diameter of 1 cm is tossed onto the square. Find the probability that the tossed coin intersects the circle. | [
"\\(\\frac{\\pi}{81}\\)"
] | [
"A square with a side length of 10 cm contains a circle with a diameter of 1 cm, centered at the square's center. A coin with a diameter of 1 cm is tossed onto the square. Find the probability that the tossed coin intersects the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 31
} | 83 | 0 | 0 |
omr-ws-6c400dfc20c7bb57da94c941c24eada7eca0a5b7751269bc6a426d9ca7d2ae8d | A square with a side length of 100 has a smaller square with a side length of 80 cut out from it. The remaining piece is then cut into single squares. Pavel wants to add a new square to these pieces. What will be the side length of this new square? | [
"60"
] | [
"A square with a side length of 100 has a smaller square with a side length of 80 cut out from it. The remaining piece is then cut into single squares. Pavel wants to add a new square to these pieces. What will be the side length of this new square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8742ae17350293887aa5b65e46b02915d852623889dfebc88ba262bd1118cf50 | A square with a side of 10 cm is divided into four smaller squares, and some of them are colored. Each uncolored square is then divided into four smaller squares and colored in the same manner. This process is repeated for five steps. What is the area of the colored region after the fifth step? | [
"\\(\\frac{775}{8}\\)"
] | [
"A square with a side of 10 cm is divided into four smaller squares, and some of them are colored. Each uncolored square is then divided into four smaller squares and colored in the same manner. This process is repeated for five steps. What is the area of the colored region after the fifth step?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48ec1a4575d896e89d135bdec3ea836bcf4b99bc20e908d08b22222001eb219e | A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius? | [
"100"
] | [
"A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-bed4629f1aff42076f120b390929cc03ab377907372f21a140f472896185b11f | A square with an area of 100 is attached to a circle. The vertex of the square is the center of the circle, which has a diameter of 20. Calculate the total area of the figure. | [
"\\(100 + 75\\pi\\)"
] | [
"A square with an area of 100 is attached to a circle. The vertex of the square is the center of the circle, which has a diameter of 20. Calculate the total area of the figure."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 29
} | 49 | 0 | 0 |
omr-ws-9a8e756bafcf9f006f81635f2b7e70c1fdef42c0f09fa9d3f85f4393933e2575 | A square with an area of 12 is placed within a rectangle. The areas of the plain white regions outside the square but within the rectangle are 3, 4, and 5 respectively. What is the area of the large rectangle? | [
"45"
] | [
"A square with an area of 12 is placed within a rectangle. The areas of the plain white regions outside the square but within the rectangle are 3, 4, and 5 respectively. What is the area of the large rectangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-810faae407a735bd3310d15faf6e40d0f5bddfbe196e7a314a7bceba8892ab59 | A square with an area of 40 square units is inscribed in a sphere such that the distance from the center of the square to the center of the sphere is 3 units. What is the area of a square that is inscribed in the same sphere, if the distance from the center of the square to the center of the sphere is 5 units? | [
"8"
] | [
"A square with an area of 40 square units is inscribed in a sphere such that the distance from the center of the square to the center of the sphere is 3 units. What is the area of a square that is inscribed in the same sphere, if the distance from the center of the square to the center of the sphere is 5 units?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-c97d9b7cdd0b2537d35afef1deaa37564b113d302c0b800a22b14982dea68da5 | A square with an area of \(100 \, \text{cm}^2\) is inscribed in a semicircular region. What is the radius of the semicircle in cm? | [
"\\(\\sqrt{125}\\)"
] | [
"A square with an area of \\(100 \\, \\text{cm}^2\\) is inscribed in a semicircular region. What is the radius of the semicircle in cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-363216485db34f258559e21df707c7d90f11acd381f0aaabfdd61f4d4d7955a3 | A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$? | [
"\\(\\dfrac{1}{2}\\)"
] | [
"A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-772c77270319b242009798227485b06b5f443c0e88df4f00d0e564e5119003da | A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N$ and $K$ of sides $BC$ and $AB$, respectively. Restore the triangle $ABC$. | [
""
] | [
"A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N$ and $K$ of sides $BC$ and $AB$, respectively. Restore the triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-8404e7dd0271a640d1390aa528601c80b24893dc95c4e013e907567ab397ae10 | A square with horizontal and vertical sides is divided into rectangles by several segments parallel to the sides, such that no two segments lie on the same line or intersect at an interior point. Any vertical line intersecting the square and not containing any segments intersects exactly $k$ rectangles, and any horizon... | [
"\\( k\\ell \\)"
] | [
"A square with horizontal and vertical sides is divided into rectangles by several segments parallel to the sides, such that no two segments lie on the same line or intersect at an interior point. Any vertical line intersecting the square and not containing any segments intersects exactly $k$ rectangles, and any ho... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-4f7cfbd019f5b14872be6c03edd749a4ee49002b23d752130e27b496cadd312f | A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? | [
"4"
] | [
"A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-0870e9b51050f77ba1cb1b885fe3bce06654a9e930ec90f03f6eb384e39c9eae | A square with opposite vertices at $(0,0,0)$ and $(1,1,0)$ lies on the $xy$-plane. An observer at $(-1,-1,1)$ looks at the point $\left(\frac12,\frac12,0\right)$. Determine the area of the quadrilateral that the square appears to form from this perspective. | [
"\\(\\frac{\\sqrt{22}}{11}\\)"
] | [
"A square with opposite vertices at $(0,0,0)$ and $(1,1,0)$ lies on the $xy$-plane. An observer at $(-1,-1,1)$ looks at the point $\\left(\\frac12,\\frac12,0\\right)$. Determine the area of the quadrilateral that the square appears to form from this perspective."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-55e1fa52b7b57528bb4e0dd3b7c57616d9925b2f625779573a4286c9339bdd4e | A square with side 1 is cut into rectangles, the perimeter of each of which is 2. How many parts can be obtained? | [
"4"
] | [
"A square with side 1 is cut into rectangles, the perimeter of each of which is 2. How many parts can be obtained?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1452cc43d25b857fef498938638c1dd478b494e1317dcba345693098a4df4438 | A square with side 1 is painted in two colors. For what maximum \(x\) are there guaranteed to be two points of the same color at a distance not less than \(x\)? | [
"\\( \\frac{\\sqrt{2}}{2} \\)"
] | [
"A square with side 1 is painted in two colors. For what maximum \\(x\\) are there guaranteed to be two points of the same color at a distance not less than \\(x\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-6e68bd52fcbb4afca01032d2ec600dcebd3710a610ca26c18aea32087e8593d5 | A square with side 8 cm is divided into 64 squares of 1 cm². 7 little squares are colored black and the rest white. Find the maximum area of a rectangle composed only of small white squares, independent of the distribution of the little black squares. | [
"56"
] | [
"A square with side 8 cm is divided into 64 squares of 1 cm². 7 little squares are colored black and the rest white. Find the maximum area of a rectangle composed only of small white squares, independent of the distribution of the little black squares."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-235623edc5f668d175b3ce7fd35b6b0e2f3bfb4d488e662887191bac716827ca | A square with side length $1$ is arbitrarily partitioned into $n$ rectangles $H_1, H_2, \ldots, H_n$. For each rectangle $H_i$, let $h_i$ be the side of minimum length. Calculate the minimum possible value of the sum $h_1 + h_2 + \cdots + h_n$ for a partition that includes exactly $5$ rectangles. | [
"1"
] | [
"A square with side length $1$ is arbitrarily partitioned into $n$ rectangles $H_1, H_2, \\ldots, H_n$. For each rectangle $H_i$, let $h_i$ be the side of minimum length. Calculate the minimum possible value of the sum $h_1 + h_2 + \\cdots + h_n$ for a partition that includes exactly $5$ rectangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d46c2974c8fa24d2f40aef9e2a832365504580f94023677b836c1012aa1d1ae | A square with side length $1$ is intersected by two parallel lines, with the distance between the lines also being $1$. Find the sum of the perimeters of the shaded triangles formed by these lines and the square. | [
"2"
] | [
"A square with side length $1$ is intersected by two parallel lines, with the distance between the lines also being $1$. Find the sum of the perimeters of the shaded triangles formed by these lines and the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-35d12c47a6b8d6aaf814a6d9d49e806bf5712972faa0e52645ac68feec0748d0 | A square with side length $12$ cm is cut into three rectangles of equal perimeter. Find the perimeter of each rectangle. | [
"32 cm or 30 cm"
] | [
"A square with side length $12$ cm is cut into three rectangles of equal perimeter. Find the perimeter of each rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-bc315cd57aad326601b0a3cee2acf3674f4144780dbefbd15d45000e3b022773 | A square with side length $2008$ is divided into smaller squares of side length $1$. Each smaller square contains either $0$ or $1$, with an equal number of $0$'s and $1$'s. The border between two adjacent smaller squares is removed, and their values are replaced by their arithmetic mean. After several such operations,... | [
"10^6"
] | [
"A square with side length $2008$ is divided into smaller squares of side length $1$. Each smaller square contains either $0$ or $1$, with an equal number of $0$'s and $1$'s. The border between two adjacent smaller squares is removed, and their values are replaced by their arithmetic mean. After several such operat... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-750f4fe56654d0a25248c5597f5f87be4aaab3b58aa09f291b33cd93c42a134a | A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle. What is the area of the triangle? | [
"\\( 20 \\frac{1}{4} \\)",
"\\(20\\frac{1}{4}\\)",
"\\dfrac{81}{4}"
] | [
"A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle. What is the area of the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 109,
"tir": 40
} | 149 | 0 | 0 |
omr-ws-bceedeae88c4b3bc976b5db5c83a09853af4e2587b8b2ef52c1838c373b5aa42 | A square with side length $5$ is divided into $25$ unit squares by lines parallel to the sides. Let $A$ be the set of the $16$ interior points, which are vertices of the unit squares but not on the sides of the initial square. What is the largest number of points in $A$ that can be chosen so that any three of them are ... | [
"4"
] | [
"A square with side length $5$ is divided into $25$ unit squares by lines parallel to the sides. Let $A$ be the set of the $16$ interior points, which are vertices of the unit squares but not on the sides of the initial square. What is the largest number of points in $A$ that can be chosen so that any three of them... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-967f05d285dc62f5374cd398716db2b9aae21dc52d306a3609838a352ffd570b | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt2$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square an... | [
"48"
] | [
"A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt2$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the squ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-62826d56356e246bd1d22cdd1ffbd125929e64cf011dc0b3217b94cb1af71b0d | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a random loc... | [
"64",
"68"
] | [
"A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a rand... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-a33d1fb024e92484c2802217ddd9cb4e19ea4f7865e26c61dfc5d6607f74353d | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a random loc... | [
"68"
] | [
"A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a rand... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-7a3d9574d5de8dbd17fa3a200c309a5b112832c19e7dac002e8e409edc9c9748 | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a random loc... | [
"64"
] | [
"A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square. A circular coin with diameter $1$ is dropped onto the square and lands in a rand... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-e688f757532e05ba1682941e9e5bc524a4451c7095ad35804807a1307facf1ed | A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger squa... | [
"2"
] | [
"A square with side length $\\sqrt{10}$ and two squares with side length $\\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larg... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-b406d1140b4b3ae2dee6ab9c589aa74b083e9618cf5accc35a7bd761102dee42 | A square with side length $a$ is revolved about an axis lying in its plane which intersects it at one of its vertices but no other points. Determine the position of the axis that yields the largest volume for the resulting solid of revolution. What is this largest volume? What is the corresponding surface area? | [
"\\( V = a^3 \\sqrt{2} \\pi \\) and \\( A = 4a^2 \\sqrt{2} \\pi \\)"
] | [
"A square with side length $a$ is revolved about an axis lying in its plane which intersects it at one of its vertices but no other points. Determine the position of the axis that yields the largest volume for the resulting solid of revolution. What is this largest volume? What is the corresponding surface area?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-aea7069d4965e40edb65d1b12bb579318fb3a0ce2724174f925fbcd2a9c6086e | A square with side length $n$ is divided into $n^2$ unit squares. What is the largest $n$ for which one can mark $n$ unit squares such that every rectangle, having sides at gridlines and area at least $n$, contains at least one marked square in its interior? | [
"4"
] | [
"A square with side length $n$ is divided into $n^2$ unit squares. What is the largest $n$ for which one can mark $n$ unit squares such that every rectangle, having sides at gridlines and area at least $n$, contains at least one marked square in its interior?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-6d02e627ecc200c3b29d88e41de74418d2dc25b637b8f10dac665140b35c51ef | A square with side length $r$ has all four vertices lying on the sides of a circumscribing triangle. The incircle of the triangle has radius $r$. If $x$ is the side length of the triangle, find the value of $x^2 - 4r^2$. | [
"8"
] | [
"A square with side length $r$ has all four vertices lying on the sides of a circumscribing triangle. The incircle of the triangle has radius $r$. If $x$ is the side length of the triangle, find the value of $x^2 - 4r^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e2a3d53805d32b1fa1a146de8bf62c3e7594132309d1c8cd4a34fec20270d0a | A square with side length 'a' is drawn. Four identical circles, each centered at a vertex of the square and with radius 'a', are also drawn. (a) Into how many regions is the square divided? (b) Find the area of each region. | [
"\\left( 1 + 2\\sqrt{3} - \\pi \\right)a^2"
] | [
"A square with side length 'a' is drawn. Four identical circles, each centered at a vertex of the square and with radius 'a', are also drawn. (a) Into how many regions is the square divided? (b) Find the area of each region."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61ef3f446ffa92ea8fbd331e5369da56aa3209719b069e092418914d4370330f | A square with side length 'a' is drawn. Four identical circles, each with radius 'a', are centered at the vertices of the square. Find the area of each region formed by the intersection of the square and the circles. | [
"\\dfrac{\\pi a^2}{4}"
] | [
"A square with side length 'a' is drawn. Four identical circles, each with radius 'a', are centered at the vertices of the square. Find the area of each region formed by the intersection of the square and the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-6e4e0676af3c42320440e23b2de23baed695a8c6d8cb51389918cf533315838b | A square with side length 'a' is drawn. Four identical circles, each with radius 'a', are centered at the vertices of the square. How many regions does the square get divided into? | [
"9"
] | [
"A square with side length 'a' is drawn. Four identical circles, each with radius 'a', are centered at the vertices of the square. How many regions does the square get divided into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-c94d0eb0f3b4609c2d2aa6e0f3a26ad60b86d2d9f2694e4f5e58114eb752e8f2 | A square with side length 1 has four circles of radius 1, each centered at a vertex of the square. Find the area of the shaded region. | [
"\\(\\sqrt{3} + \\frac{2\\pi}{3} - 3\\)"
] | [
"A square with side length 1 has four circles of radius 1, each centered at a vertex of the square. Find the area of the shaded region."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8571f1d96f67575417e38c6482a532a81e5826f9c5b3cda0322c49606f839ec0 | A square with side length 1 is divided into 100 rectangles. The perimeter of each rectangle is $p$. Find the largest possible value of $p$. | [
"\\(\\frac{101}{50}\\)"
] | [
"A square with side length 1 is divided into 100 rectangles. The perimeter of each rectangle is $p$. Find the largest possible value of $p$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3dda0879c88a776779065132835206d5eaf16583bc3fea6bd13b80940480ac16 | A square with side length 1 is divided into one hundred rectangles, each having the same perimeter $p$. What is the largest possible value of $p$? | [
"0.4"
] | [
"A square with side length 1 is divided into one hundred rectangles, each having the same perimeter $p$. What is the largest possible value of $p$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35561cd4a5c2898dcb2639d51d5b0c99c21e7bb838ca461d5ab4c3cc6b722f46 | A square with side length 1 is divided into polygons such that no two polygons share any interior points. If the maximum distance between any two points on any polygon is less than $\frac{1}{30}$, calculate the minimum number of adjacent polygons that one of these polygons must have. Two polygons are considered adjacen... | [
"8"
] | [
"A square with side length 1 is divided into polygons such that no two polygons share any interior points. If the maximum distance between any two points on any polygon is less than $\\frac{1}{30}$, calculate the minimum number of adjacent polygons that one of these polygons must have. Two polygons are considered a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-35218f5fb4aa28c320565a133e5c6b7437488bf2adcfce9b450ee7568ac2ad99 | A square with side length 1 is given. Each side is divided into three equal parts, and segments are drawn through the division points. Find the area of the hatched square formed by these segments. | [
"\\(\\frac{1}{10}\\) \n\nDespite some confusion and contradictory statements, the correct area of the hatched square, as supported by the geometric analysis and calculations in the posts, is \\(\\frac{1}{10}\\). Post 1 and Post 6 provide consistent and correct reasoning for this result. Post 8 and Post 7 contain er... | [
"A square with side length 1 is given. Each side is divided into three equal parts, and segments are drawn through the division points. Find the area of the hatched square formed by these segments."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee9d83555d00b1d34af76dc3194e0abaa971a1812c1a65e530bcfa27291991ca | A square with side length 100 is cut by 99 horizontal and 99 vertical lines into 10,000 rectangles (not necessarily with integer sides). How many of these rectangles have an area not exceeding 1? | [
"199"
] | [
"A square with side length 100 is cut by 99 horizontal and 99 vertical lines into 10,000 rectangles (not necessarily with integer sides). How many of these rectangles have an area not exceeding 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-65eeb3e13f753f08d0ed492049b288cdbc1778ce1212b670528cb2ed45d285a5 | A square with side length 1000000000 has a smaller square with side length 1 cut from one of its corners. The remaining area is partitioned into 10 rectangles. Let $r$ be the maximum value obtained by dividing the length of one side of any of these rectangles by the length of the other side. Calculate the minimum possi... | [
"\\dfrac{5}{2}"
] | [
"A square with side length 1000000000 has a smaller square with side length 1 cut from one of its corners. The remaining area is partitioned into 10 rectangles. Let $r$ be the maximum value obtained by dividing the length of one side of any of these rectangles by the length of the other side. Calculate the minimum ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c6b4b7a76f320747ebaaf43546031e9b7e19acd71b5f4756db8351f0bc652ccb | A square with side length 2 dm is divided into three regions \(A\), \(B\), and \(C\). Region \(C\) has two right angles, and point \(M\) is the midpoint of one of the sides of the square. What is the area of region \(C\)? | [
"\\(\\frac{11}{5}\\)"
] | [
"A square with side length 2 dm is divided into three regions \\(A\\), \\(B\\), and \\(C\\). Region \\(C\\) has two right angles, and point \\(M\\) is the midpoint of one of the sides of the square. What is the area of region \\(C\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d43e7d56293f247f753f8892451b4c201899251e271e2dbdb50f36b157e7f910 | A square with side length 2 is revolved around its diagonals as axes. What is the volume of the coinciding part? | [
"\\(\\frac{4}{3} \\pi\\)"
] | [
"A square with side length 2 is revolved around its diagonals as axes. What is the volume of the coinciding part?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c9cf42a93d19a5067df2289f23d563697acb48e7aaac4afa556d9edf2356b52 | A square with side length 2016 is cut into \( n \) rectangles that do not overlap. Denote the sides of these rectangles as \( a_i \) and \( b_i \) where \( a_i \le b_i \) for \( i = 1, 2, \ldots, n \). What is the smallest value of the sum \( a_1 + a_2 + \cdots + a_n \), and for which square cuts is this sum achieved? | [
"2016"
] | [
"A square with side length 2016 is cut into \\( n \\) rectangles that do not overlap. Denote the sides of these rectangles as \\( a_i \\) and \\( b_i \\) where \\( a_i \\le b_i \\) for \\( i = 1, 2, \\ldots, n \\). What is the smallest value of the sum \\( a_1 + a_2 + \\cdots + a_n \\), and for which square cuts is... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fcb15aefac265e934f594608ce7fcd2d4a7eae945fffd06f1683b136d5fdf320 | A square with side length 3 units has a 1x1 square cut from one of its corners. This new figure is inscribed in a circle. Find the exact radius of the circle. | [
"\\(\\frac{3\\sqrt{2}}{2}\\)"
] | [
"A square with side length 3 units has a 1x1 square cut from one of its corners. This new figure is inscribed in a circle. Find the exact radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 13
} | 18 | 0 | 0 |
omr-ws-705cc3523d06fb647ec1be19d8e6a44e5f34548bb97f1a3114ce903a23f6d3ca | A square with side length 49 is completely tiled with 600 $1 \times 4$ rectangles and a unit square. Find the number of possible locations of the unit square. | [
"169"
] | [
"A square with side length 49 is completely tiled with 600 $1 \\times 4$ rectangles and a unit square. Find the number of possible locations of the unit square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-7c1e75adce73b73a63f059827f4c5104958919e9666ef1f541317cbcf14a7a0f | A square with side length 6 inches is given. If P is a point inside the square such that the segments PA, PB, and PC are equal in length and PC is perpendicular to FD, what is the area, in square inches, of triangle APB? | [
"9"
] | [
"A square with side length 6 inches is given. If P is a point inside the square such that the segments PA, PB, and PC are equal in length and PC is perpendicular to FD, what is the area, in square inches, of triangle APB?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78d999c00c2a6d13d1ce9e5da163ef92238827b1950cd00cd74cf89a5ea97b62 | A square with side length \( n \) is divided into \( n^2 \) unit squares. What is the largest \( n \) for which one can mark \( n \) unit squares such that every rectangle, having sides at gridlines and area at least \( n \), contains at least one marked square in its interior? | [
"4"
] | [
"A square with side length \\( n \\) is divided into \\( n^2 \\) unit squares. What is the largest \\( n \\) for which one can mark \\( n \\) unit squares such that every rectangle, having sides at gridlines and area at least \\( n \\), contains at least one marked square in its interior?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-ce26343d4576b8e344bbc1fee878effc57cfccaf7150acceef1edd0b32425b80 | A square with side length \( n \ge 3 \) is split into a unit grid, and every square within the grid is filled with either an apple or a banana. At each step, you may choose any square, and it and its immediate neighbors (including diagonal) are switched (apple becomes banana and banana becomes apple). For which values ... | [
"\\( n \\equiv 0, 1 \\mod 3 \\)"
] | [
"A square with side length \\( n \\ge 3 \\) is split into a unit grid, and every square within the grid is filled with either an apple or a banana. At each step, you may choose any square, and it and its immediate neighbors (including diagonal) are switched (apple becomes banana and banana becomes apple). For which... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f0deb793f5992fdcc5f4750c994758d8d45e55c9d52afaa41dc2734a8e2db5a5 | A square with side length \( s \) is inscribed in an isosceles triangle with one side of the square along the base of the triangle. Another square with side length \( t \) has two vertices on the first square and the other two on the sides of the triangle. What is the area of the triangle? | [
"\\dfrac{s^4}{2t(s - t)}"
] | [
"A square with side length \\( s \\) is inscribed in an isosceles triangle with one side of the square along the base of the triangle. Another square with side length \\( t \\) has two vertices on the first square and the other two on the sides of the triangle. What is the area of the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ccc1964f506a5e73a7184d0e96af9e24bb1ea8b911bbaa223c431266d2b707b | A square with side length \( x \) is inscribed in a right triangle with sides of length \( 3 \), \( 4 \), and \( 5 \) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \( y \) is inscribed so that one side of the square lies on the hypotenuse of the triang... | [
"\\(\\frac{37}{35}\\)"
] | [
"A square with side length \\( x \\) is inscribed in a right triangle with sides of length \\( 3 \\), \\( 4 \\), and \\( 5 \\) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \\( y \\) is inscribed so that one side of the square lies on the hypotenus... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-74ccdc34346cdaafdf0599c77965967c51618dd3152058e15b73efcfdf3b7cd5 | A square with sides of 10 cm and a right triangle with sides of 20 cm, 21 cm, and 29 cm overlap such that the vertex of the right angle of the triangle is at the center of the square. What is the area of the overlap? | [
"25"
] | [
"A square with sides of 10 cm and a right triangle with sides of 20 cm, 21 cm, and 29 cm overlap such that the vertex of the right angle of the triangle is at the center of the square. What is the area of the overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-6a36c36664f567c8054e9051df696e5728dee112125f48f65f57806ad5d20733 | A square with sides of 10 cm contains a circle with a radius of 5 cm centered in the square. A smaller circle with a radius of 1 cm is placed randomly inside the square such that it does not extend outside the square. What is the probability that the smaller circle is entirely within the larger circle? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A square with sides of 10 cm contains a circle with a radius of 5 cm centered in the square. A smaller circle with a radius of 1 cm is placed randomly inside the square such that it does not extend outside the square. What is the probability that the smaller circle is entirely within the larger circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8e620b0de301493d1ea39f047cd56d0abba300a92b8d9f4e51eee247ddb69476 | A square with sides of length $1000$ is divided into $31$ smaller squares. Let $S$ be the sum of the side lengths of all squares with side lengths less than $1$. Calculate the minimum possible value of $1000S$. | [
"30000"
] | [
"A square with sides of length $1000$ is divided into $31$ smaller squares. Let $S$ be the sum of the side lengths of all squares with side lengths less than $1$. Calculate the minimum possible value of $1000S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
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