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-4eb31e67dc371fddd98869f69a9302fca900a3dc291d3be34b4c670959a8a966 | What is the area of the circumcircle of a triangle with side lengths $130$, $144$, and $194$? | [
"\\(9709\\pi\\)"
] | [
"What is the area of the circumcircle of a triangle with side lengths $130$, $144$, and $194$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-013b28601e35e6f0aace3e514ac2bdbff301601ede06d4add093a6714af541f7 | What is the area of the cyclic pentagon $ABCDE$ with side lengths $AB=3$, $BC=4$, $CD=5$, $DE=6$, and $EA=7$? | [
"42"
] | [
"What is the area of the cyclic pentagon $ABCDE$ with side lengths $AB=3$, $BC=4$, $CD=5$, $DE=6$, and $EA=7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-e960cbb0a6b0c3c5435d8669cc30d28d263c3ed9c209185de96327a6b3cb556e | What is the area of the equilateral triangle formed by the points \((x, y)\) that satisfy the equation \(x^3 + y^3 + 3xy = 1\)? | [
"\\(\\frac{3\\sqrt{3}}{2}\\)"
] | [
"What is the area of the equilateral triangle formed by the points \\((x, y)\\) that satisfy the equation \\(x^3 + y^3 + 3xy = 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-7d4006bec227186179e39724a1847f97ccd169e091ad9689d78f2df963cbf53d | What is the area of the intersection of Circle A, which has a radius of 2 and is centered at \((-1, 0)\), and Circle B, which has a radius of 2 and is centered at \((1, 0)\)? | [
"\\(\\frac{8\\pi}{3} - 2\\sqrt{3}\\)"
] | [
"What is the area of the intersection of Circle A, which has a radius of 2 and is centered at \\((-1, 0)\\), and Circle B, which has a radius of 2 and is centered at \\((1, 0)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 27
} | 48 | 0 | 0 |
omr-ws-e2d9be94da4a18d6cf9e62a5518b1f4f02b63ae1574e14151cd7715b16002d6c | What is the area of the intersection of the regions defined by $x^{2}+y^{2}\le9$ and $x^{2}+y^{2}-10x+9\le0$? | [
"9 \\arccos\\left(\\frac{3}{5}\\right) + 16 \\arccos\\left(\\frac{4}{5}\\right) - 12"
] | [
"What is the area of the intersection of the regions defined by $x^{2}+y^{2}\\le9$ and $x^{2}+y^{2}-10x+9\\le0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-e01f03b0ee74d35c1f3fb3a4aa2ea0fa74b8c2b29704947ae5d621eaaa807b85 | What is the area of the largest $n$-gon that can be inscribed in a square? | [
"1"
] | [
"What is the area of the largest $n$-gon that can be inscribed in a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5aee1dc916233c16aad1fd4e4ccd75cbbe9521436fc3bb09614a07cf1a212582 | What is the area of the largest circle that can be drawn tangent to both a quarter-circle (a $90^{\circ}$ arc) of length $2\pi$ and the line segment that connects the endpoints of that quarter-circle? | [
"\\((6 - 4\\sqrt{2})\\pi\\)"
] | [
"What is the area of the largest circle that can be drawn tangent to both a quarter-circle (a $90^{\\circ}$ arc) of length $2\\pi$ and the line segment that connects the endpoints of that quarter-circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-b948c021912357c406847511fdb1f79bd6b2487ea0730cbf394ad2332c92ac18 | What is the area of the largest circle that does not contain any lattice point in its interior? | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"What is the area of the largest circle that does not contain any lattice point in its interior?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-c3d640486c585a8ec896813f6a0053ebe05717c6fee6b4b41c13f61b7aa81ba3 | What is the area of the largest ellipse that can be inscribed in a 3-4-5 triangle? | [
"\\dfrac{2\\pi\\sqrt{3}}{3}"
] | [
"What is the area of the largest ellipse that can be inscribed in a 3-4-5 triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-65ac5b6b43f1672a42006f0815a8a206e1019deeeec55a2d696ea961016be93b | What is the area of the largest equilateral triangle that can fit inside a square with side length $a$? | [
"\\(2a^2\\sqrt{3} - 3a^2\\)"
] | [
"What is the area of the largest equilateral triangle that can fit inside a square with side length $a$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-33457511ac0dad35ee4a8d2467248ff46abed0464afa7d801d0363986d7395e2 | What is the area of the largest heptagon that can be inscribed in a square? | [
"\\dfrac{44}{49}"
] | [
"What is the area of the largest heptagon that can be inscribed in a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-16dbeaebde77ea960d31129c02fbdf8d70271ebb2f7bc525021e0031f1af2aad | What is the area of the largest hexagon that can be inscribed in a square? | [
"\\(\\frac{6\\sqrt{3} - 9}{2}\\)"
] | [
"What is the area of the largest hexagon that can be inscribed in a square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5903452642380953ed725e1a081378a44acc5b5c4ef4818b695589cb166bd3c2 | What is the area of the largest octagon that can be inscribed in a square of side length 1? | [
"\\(2\\sqrt{2}-2\\)"
] | [
"What is the area of the largest octagon that can be inscribed in a square of side length 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-6834fcdfa1bce5161dee6b0b04da5a253558608c8046f5754dc642f6736522b2 | What is the area of the largest pentagon that can be inscribed in a square? | [
"1"
] | [
"What is the area of the largest pentagon that can be inscribed in a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-cb5ada2b7509a62499b6eb32e8f7b14fa94649a29d975c776c9b7252b71af4c5 | What is the area of the largest possible Norman window with a perimeter of 25 feet, where the window has the shape of a semicircle atop a rectangle and the diameter of the semicircle is equal to the width of the rectangle? | [
"43.74"
] | [
"What is the area of the largest possible Norman window with a perimeter of 25 feet, where the window has the shape of a semicircle atop a rectangle and the diameter of the semicircle is equal to the width of the rectangle?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-cd4c3c3e965da29d3a1ce3fca3f7a67a73fa6c59592db16bb38130315a927d03 | What is the area of the largest rectangle inscribed in a \(6 - 8 - 10\) right triangle, if two of the sides of the rectangle correspond to the legs with lengths \(6\) and \(8\)? Solve using derivatives. | [
"\\(\\frac{288}{49}\\) or approximately \\(11.76\\)"
] | [
"What is the area of the largest rectangle inscribed in a \\(6 - 8 - 10\\) right triangle, if two of the sides of the rectangle correspond to the legs with lengths \\(6\\) and \\(8\\)? Solve using derivatives."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-541162529279b2683c6e188ecb107daac8954bacb417b13d5f5f12b1e823eafa | What is the area of the largest rectangle that can be inscribed in a right triangle with legs of length 3 and 4? | [
"3"
] | [
"What is the area of the largest rectangle that can be inscribed in a right triangle with legs of length 3 and 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a046178f452c7d429116819b2510c360715edb80b54ab908660fb4b861b79d6 | What is the area of the largest rectangle that can be inscribed in the ellipse $4x^2 + 9y^2 = 36$? | [
"12"
] | [
"What is the area of the largest rectangle that can be inscribed in the ellipse $4x^2 + 9y^2 = 36$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-7b8a7abe516b218f2bab427716a33457f94c1b0d86edb3c9d8208facce6e82fc | What is the area of the largest regular hexagon that can be inscribed in a unit square? | [
"\\( 3\\sqrt{3} - 4.5 \\)"
] | [
"What is the area of the largest regular hexagon that can be inscribed in a unit square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 6,
"tir": 21
} | 48 | 0 | 0 |
omr-ws-24e8218b957b97ae6f55dc17ab21a9eca2a39251cb9ddb8fb1e3ebc286d8cba3 | What is the area of the largest semicircle that can be drawn inside an isosceles right triangle with a hypotenuse of 4? | [
"\\(\\pi\\)"
] | [
"What is the area of the largest semicircle that can be drawn inside an isosceles right triangle with a hypotenuse of 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-82cf4b539321132763ddb1f387dec71c94c143d8ebf50f1321743e66193e6304 | What is the area of the largest semicircle that can be placed in a right triangle whose side lengths are 6, 8, and 10? | [
"\\(\\frac{288\\pi}{49}\\)"
] | [
"What is the area of the largest semicircle that can be placed in a right triangle whose side lengths are 6, 8, and 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 8
} | 13 | 0 | 0 |
omr-ws-4cc54a473c0749586b4e70b1e46fe5a938f3a903bbb907c0cbca41cf15c92b7b | What is the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle <ABC = 60 degrees? | [
"\\dfrac{\\pi}{8}"
] | [
"What is the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle <ABC = 60 degrees?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 19,
"genselect": 7
} | 26 | 0 | 0 |
omr-ws-6855ba9de6c1499e1517a67b54b8c0cc97de2a707dca16acdfb3cafc797c1bd1 | What is the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle <ABC = 60°? | [
"\\(\\frac{\\pi}{6}\\)"
] | [
"What is the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle <ABC = 60°?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9654aee70ccadda12c368937aaaa3290ee20453a092fbd7f115c252ee6c8338 | What is the area of the largest square that can fit into a unit cube? What will its orientation be like? | [
"\\(\\frac{3}{2}\\)"
] | [
"What is the area of the largest square that can fit into a unit cube? What will its orientation be like?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 6,
"tir": 1
} | 24 | 0 | 0 |
omr-ws-e1ac5bc92072468810b631de65780ee03bb0f069ae5e159f67fb40000795110a | What is the area of the largest triangle that can be inscribed inside a unit pentagon? | [
"\\dfrac{\\sqrt{5 + 2\\sqrt{5}}}{4}"
] | [
"What is the area of the largest triangle that can be inscribed inside a unit pentagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5bb313a8ef62b1e6b83446b1450620be33fcfd570bd04badf2781f50b8aeca02 | What is the area of the letter $O$ made by Dürer, where the two circles have a unit radius and the angle formed by an intersection point of the circles is $30^\circ$? | [
"$\\frac{5\\pi}{6} - \\frac{1}{2}$"
] | [
"What is the area of the letter $O$ made by Dürer, where the two circles have a unit radius and the angle formed by an intersection point of the circles is $30^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-96be9511b0e0f761ae674147862b2586e8d219f6b362dad691b1f16da9089490 | What is the area of the locus of points that satisfy both $x^2 + 4y^2 < 4$ and $4x^2 + y^2 < 4$? | [
"\\(2\\pi\\)"
] | [
"What is the area of the locus of points that satisfy both $x^2 + 4y^2 < 4$ and $4x^2 + y^2 < 4$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e0caba3167a773efa3ccb1bc5a93b5a4683958f1b44a1c46e41f9ea44bb24ed | What is the area of the maximum size square that can be inscribed in a right-angle triangle with sides of lengths 3, 4, and 5 units? | [
"\\(\\frac{144}{49}\\)"
] | [
"What is the area of the maximum size square that can be inscribed in a right-angle triangle with sides of lengths 3, 4, and 5 units?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f8e08f7f9c3b4e638176a49f341ab92ee277cf9c060904d17b81a39ad2bb8218 | What is the area of the pentagon $A_1A_2A_3A_4A_5$ in a regular dodecagon $A_1A_2A_3...A_{12}$ where each side has length 2? | [
"\\(5 + 2\\sqrt{3}\\)"
] | [
"What is the area of the pentagon $A_1A_2A_3A_4A_5$ in a regular dodecagon $A_1A_2A_3...A_{12}$ where each side has length 2?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 23
} | 40 | 0 | 0 |
omr-ws-8162ad3b0f64b7e557332399f1b5038df51edb60429c08572d3280b98da7e9b6 | What is the area of the polygon spanned by the roots of the polynomial $x^5 - 1 = 0$ on the complex plane, starting at $1 + 0i$ and moving counterclockwise? | [
"\\( 5\\sin(36^{\\circ})\\cos(36^{\\circ}) \\)"
] | [
"What is the area of the polygon spanned by the roots of the polynomial $x^5 - 1 = 0$ on the complex plane, starting at $1 + 0i$ and moving counterclockwise?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-4ad160bf90556151ede1712f68565c872f1299f67e6a3bc949a65ac8d0287d55 | What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$? | [
"27"
] | [
"What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75cf116df1b464ffce6cf834eebb063f299979dfabc411d53916fbb1fb2575bc | What is the area of the polygon with vertices $(\frac{\sqrt{3}-1}{2},\frac{\sqrt{3}+1}{2})$, $(\frac{3\sqrt{3}-1}{2},\frac{\sqrt{3}+3}{2})$, $(2\sqrt{3}-1,\sqrt{3}+2)$, $(\frac{3\sqrt{3}-3}{2},\frac{3\sqrt{3}+3}{2})$, $(\frac{\sqrt{3}-3}{2},\frac{3\sqrt{3}+1}{2})$, and $(-1,\sqrt{3})$? | [
"6"
] | [
"What is the area of the polygon with vertices $(\\frac{\\sqrt{3}-1}{2},\\frac{\\sqrt{3}+1}{2})$, $(\\frac{3\\sqrt{3}-1}{2},\\frac{\\sqrt{3}+3}{2})$, $(2\\sqrt{3}-1,\\sqrt{3}+2)$, $(\\frac{3\\sqrt{3}-3}{2},\\frac{3\\sqrt{3}+3}{2})$, $(\\frac{\\sqrt{3}-3}{2},\\frac{3\\sqrt{3}+1}{2})$, and $(-1,\\sqrt{3})$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-728df9650f645b85d938e22c413fb1f88b1b708d3200b30f4f965adf3867ea29 | What is the area of the portion of the circle defined by $x^2-12x+y^2=28$ that lies above the $x$-axis and to the right of the line $y=6-x$? | [
"24 \\pi"
] | [
"What is the area of the portion of the circle defined by $x^2-12x+y^2=28$ that lies above the $x$-axis and to the right of the line $y=6-x$?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-ee40f6351c953383f92a9f65a1713c645510d02dced9d56b62ddc9e3ea6e732a | What is the area of the quadrilateral ABCD with sides AB = 2, AC = 3, AD = 4, BC = 13, BD = 2√5, and CD = 5? | [
"10"
] | [
"What is the area of the quadrilateral ABCD with sides AB = 2, AC = 3, AD = 4, BC = 13, BD = 2√5, and CD = 5?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-6e72c43d4b167cadb05be43bfbc5100395a09083b7731e4bd5029a618f36847c | What is the area of the quadrilateral formed by the equation \(xy - 3x - 4y = -12\) and the x-axis and y-axis? | [
"12"
] | [
"What is the area of the quadrilateral formed by the equation \\(xy - 3x - 4y = -12\\) and the x-axis and y-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb7876f90db097af63c6422e92f86f68d2a8a6ba2bc17bddd588cb5a91c366ae | What is the area of the region between the graphs of \( y = \frac{4}{x} \) and \( y = 5 \) in the range \([-6, -2]\)? | [
"20 + 4 \\ln 3"
] | [
"What is the area of the region between the graphs of \\( y = \\frac{4}{x} \\) and \\( y = 5 \\) in the range \\([-6, -2]\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f0de1c75bd218c46ba3692430b0679df3464740364509330b6e7add9697c04bf | What is the area of the region bounded by the axes and the circle with center $(4,4)$ and radius $7$? | [
"107.27"
] | [
"What is the area of the region bounded by the axes and the circle with center $(4,4)$ and radius $7$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-6b948739d0f53dfaa4572050c1c3bb010c590463daf15a05f76686fbe641544b | What is the area of the region bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number? Express the area in terms of $a$, valid for all $a > 0$. | [
"\\(\\frac{8a^2}{a^2+1}\\)"
] | [
"What is the area of the region bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number? Express the area in terms of $a$, valid for all $a > 0$."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 32
} | 63 | 0 | 0 |
omr-ws-5129fc3b662e1424a8e60db2e160bb4173d12ad901662e09a901b91f9a989454 | What is the area of the region containing all points that are outside an equilateral triangle with side length 6 and not more than 3 units from a point of the triangle? | [
"\\( 54 + 9\\pi \\)"
] | [
"What is the area of the region containing all points that are outside an equilateral triangle with side length 6 and not more than 3 units from a point of the triangle?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 53
} | 116 | 0 | 0 |
omr-ws-f8448dd613dfabca08606e0017000a1376f939b875b950a3af23b75595bf29b0 | What is the area of the region covered by the circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$ in a regular dodecagon $A_1A_2...A_{12}$ with side length $1$ and center $O$? Here, $(ABC)$ denotes the circle passing through points $A$, $B$, and $C$. | [
"2\\pi + 3\\sqrt{3}"
] | [
"What is the area of the region covered by the circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$ in a regular dodecagon $A_1A_2...A_{12}$ with side length $1$ and center $O$? Here, $(ABC)$ denotes the circle passing through points $A$, $B$, and $C$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-4260359dfa5460012afe70c46ab8eb682837a336cb2f0118be641f9720bdf893 | What is the area of the region defined by \(x^2 + 3y^2 \leq 4\) and \(3x^2 + y^2 \leq 4\)? | [
"\\(\\frac{8\\sqrt{3}\\pi}{9}\\)"
] | [
"What is the area of the region defined by \\(x^2 + 3y^2 \\leq 4\\) and \\(3x^2 + y^2 \\leq 4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-d7bcc11b81e1cd43c2940aa3ef7b262a125e302fdbf5326591f825d4a0bfff95 | What is the area of the region defined by the equation $x^2+y^2 - 7 = 2y-8x+1$? | [
"25\\pi"
] | [
"What is the area of the region defined by the equation $x^2+y^2 - 7 = 2y-8x+1$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c27cb2481ec55ce118466e115ca10a59f917c7f66ef8fb9a513a981699e71447 | What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$? | [
"63\\pi"
] | [
"What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-937d504ddea7c4dab414b93c110ae6c433b23945cad0f0ac165d741bd1edfee8 | What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$? | [
"3"
] | [
"What is the area of the region defined by the inequality $|3x-18|+|2y+7|\\le3$?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-d0de8d635d2c1c8d0b004a8eee5c3638f36d0b6e3efd45814b8bc28307ffc72a | What is the area of the region defined by the inequality \( |3x - 18| + |2y + 7| \le 3 \)? | [
"3"
] | [
"What is the area of the region defined by the inequality \\( |3x - 18| + |2y + 7| \\le 3 \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6d17f6221c35d23f730d0e2e9bf0b8f982921356677e2fa09ae36f26e579160 | What is the area of the region determined by the points outside a triangle with perimeter length $\pi$ where none of these points has a distance greater than $1$ to any corner of the triangle? | [
"\\(2\\pi\\)"
] | [
"What is the area of the region determined by the points outside a triangle with perimeter length $\\pi$ where none of these points has a distance greater than $1$ to any corner of the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-e4c60ae725c709cc29ab912f6555423565c8ef491487ba0f029390e78726e0ed | What is the area of the region enclosed by the graph of the equation $x^2 + y^2 = |x| + |y|$? | [
"\\(\\pi + 2\\)"
] | [
"What is the area of the region enclosed by the graph of the equation $x^2 + y^2 = |x| + |y|$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 19
} | 56 | 0 | 0 |
omr-ws-250bb8b2e2c141bed47f3087bb3a48111567f094a2f576d3ab7b1c2113c934ca | What is the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=21+11y-y^2$ that lies below the line $y=x-3$? | [
"8 \\pi"
] | [
"What is the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=21+11y-y^2$ that lies below the line $y=x-3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0b42314c0c0e85a7def580a963c28f2de23a0afe174198dfb2d94a5da79f215 | What is the area of the region enclosed by the lines \( y = 1 \), \( y = x \), and \( y = \frac{x^2}{4} \)? | [
"\\(\\frac{5}{6}\\)"
] | [
"What is the area of the region enclosed by the lines \\( y = 1 \\), \\( y = x \\), and \\( y = \\frac{x^2}{4} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-bee6d133aeda8da76517eedbd4224f94d90f1100c0ab25be2b67045ddc400256 | What is the area of the region enclosed by the tangent points of Circle P with radius 1, Circle Q with radius 2, and Circle R with radius 3? | [
"6"
] | [
"What is the area of the region enclosed by the tangent points of Circle P with radius 1, Circle Q with radius 2, and Circle R with radius 3?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f54e14e3746f96fdfa12e301778365bf0bb7c71613908c523e10941e8c82edbd | What is the area of the region in the Cartesian plane whose points satisfy the inequality \(|x| + 2|y| \leq 2\)? | [
"4."
] | [
"What is the area of the region in the Cartesian plane whose points satisfy the inequality \\(|x| + 2|y| \\leq 2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-621014cf0c1e666a8d189974a837371db0a430dbb32a4976460edef27783dcb7 | What is the area of the region in the coordinate plane defined by $$||x|-1|+||y|-1| \le 1?$$ | [
"8"
] | [
"What is the area of the region in the coordinate plane defined by $$||x|-1|+||y|-1| \\le 1?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-36c3787b89708170c7fa52b883b349a7eddce21de79d9970a3fbe71c8db4ea95 | What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\] | [
"8"
] | [
"What is the area of the region in the coordinate plane defined by the inequality \\[\\left||x|-1\\right|+\\left||y|-1\\right|\\leq 1?\\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-44d355b59a9f0260e3aee81bc9cb4367cd9db92510130fdde96a82163fab4298 | What is the area of the region in the first quadrant bounded by the x-axis, the line \( y = 2x \), and the circles \( x^2 + y^2 = 20 \) and \( x^2 + y^2 = 30 \)? | [
"\\( 5 \\arctan 2 \\)",
"\\(5\\arctan{2}\\)"
] | [
"What is the area of the region in the first quadrant bounded by the x-axis, the line \\( y = 2x \\), and the circles \\( x^2 + y^2 = 20 \\) and \\( x^2 + y^2 = 30 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 50
} | 81 | 0 | 0 |
omr-ws-555ae30246f3aa1abf96ab54cbfe8dc2bc0b300955aaddc064d8f2c7a00023ac | What is the area of the region inside a larger circle but outside 13 smaller circles of radius 1, where all intersections occur at points of tangency? The configuration is such that the 13 smaller circles are arranged within the larger circle, with each smaller circle tangent to its neighbors and the larger circle. | [
"\\(4\\pi \\sqrt{3}\\)"
] | [
"What is the area of the region inside a larger circle but outside 13 smaller circles of radius 1, where all intersections occur at points of tangency?\n\nThe configuration is such that the 13 smaller circles are arranged within the larger circle, with each smaller circle tangent to its neighbors and the larger cir... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dfc6b85aaf98458141f2bf09146acd2e5324e75153467e085770a9c486c70e6 | What is the area of the region inside an equilateral triangle of side length $a$ that is closer to a vertex than to the inscribed circle? | [
"\\dfrac{a^2}{32} \\left( 3(\\sqrt{3} - 1) - 2\\sqrt{3} \\ln\\left( \\dfrac{\\sqrt{3} + 1}{2} \\right) \\right)"
] | [
"What is the area of the region inside an equilateral triangle of side length $a$ that is closer to a vertex than to the inscribed circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-671642a4995b81d2b3a77821efb66546f56bfb2290f39852a8ba3f63b973e2c8 | What is the area of the region on the Cartesian plane bordered by \( y = x \), \( y = -1 \), \( x = 2 \), and \( x^2 + 4 + y^2 = 4x + 4y \)? | [
"\\dfrac{\\pi}{2}"
] | [
"What is the area of the region on the Cartesian plane bordered by \\( y = x \\), \\( y = -1 \\), \\( x = 2 \\), and \\( x^2 + 4 + y^2 = 4x + 4y \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-b6d26168657c56fab562e295bdfc907c08a7d252d280aef1f06570f9916b75bf | What is the area of the regular hexagon that can be inscribed inside a square with sides of unit length? | [
"\\dfrac{3\\sqrt{3}}{8}"
] | [
"What is the area of the regular hexagon that can be inscribed inside a square with sides of unit length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-7545e7023874daf9e04c427dbf612f97a0880aa97ab6c928a56716ec69d5a4bd | What is the area of the set \( D \) defined by \( |3x| + |2y| \le 12 \)? | [
"48"
] | [
"What is the area of the set \\( D \\) defined by \\( |3x| + |2y| \\le 12 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a07b75a6c68863ce53c994ae9b1848a8f40e63fba716c80e0419a94a0355b60f | What is the area of the set of points inside a regular hexagon with side length 1 that are within $\dfrac{\sqrt{3}}{4}$ of at least one of the sides? | [
"$\\frac{9\\sqrt{3}}{8}$"
] | [
"What is the area of the set of points inside a regular hexagon with side length 1 that are within $\\dfrac{\\sqrt{3}}{4}$ of at least one of the sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-e048c2dd8554a5f3096b7c3829ff7508ea2e98c05ad3bab78f97e235b182a45e | What is the area of the shaded figure formed by the vertices (1,0), (3,5), (5,0), and (3,2)? | [
"6"
] | [
"What is the area of the shaded figure formed by the vertices (1,0), (3,5), (5,0), and (3,2)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ba744f0d77f3d259530ece04edb9e7f0c722f4ab6ccffa81102a48af82cbcc9d | What is the area of the shaded region between two concentric circles, where the radius of the smaller circle is 1 meter and the radius of the larger circle is 4 meters? | [
"\\(\\sqrt{15} + 1 + 16\\sin^{-1}\\left(\\frac{1}{4}\\right) + 4\\pi\\)"
] | [
"What is the area of the shaded region between two concentric circles, where the radius of the smaller circle is 1 meter and the radius of the larger circle is 4 meters?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f6b93dc964ad25f1f3d1a30fa0735bf51d95f8787586fab9e7e66eb17ac6025 | What is the area of the shaded region of the given $8 \times 5$ rectangle? The shaded region is formed by the vertices at $(7,0)$, $(8,0)$, $(8,1)$, $(0,4)$, $(0,5)$, and $(1,5)$. | [
"6.5"
] | [
"What is the area of the shaded region of the given $8 \\times 5$ rectangle?\n\nThe shaded region is formed by the vertices at $(7,0)$, $(8,0)$, $(8,1)$, $(0,4)$, $(0,5)$, and $(1,5)$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 9
} | 19 | 0 | 0 |
omr-ws-32a068d015547413a975040748ac65e178180e989cab0f23042cb04f2d961a51 | What is the area of the shape formed by connecting the solutions to the equation $x^3 - 5x + 12x - 7 = 0$ in the complex plane? | [
"\\(\\sqrt{3}\\)"
] | [
"What is the area of the shape formed by connecting the solutions to the equation $x^3 - 5x + 12x - 7 = 0$ in the complex plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c35757c52dacc4facd2fea6214464eec241d0e32f560dabc077b27c4dc054e1d | What is the area of the shape formed by the graph $y = -x^2 + 5$ and the x-axis? (Solve without using calculus.) | [
"\\dfrac{20\\sqrt{5}}{3}"
] | [
"What is the area of the shape formed by the graph $y = -x^2 + 5$ and the x-axis? (Solve without using calculus.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fcbd98c411f1998bb1d726bc3945aab4de1813e6c49b2e950cb8f62b606fbd4 | What is the area of the shark's fin falcata, which is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$? | [
"\\(\\dfrac{9\\pi}{8}\\)"
] | [
"What is the area of the shark's fin falcata, which is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 7,
"tir": 22
} | 71 | 0 | 0 |
omr-ws-e0cb5c1d5b20c167d6fe2ad266174893103c36198e0cc87e8f88d6fe1af23df4 | What is the area of the smallest $n \times m$ grid (with $n$ and $m$ both positive integers) such that the number of paths from the bottom left to the top right is divisible by $2021$? | [
"210"
] | [
"What is the area of the smallest $n \\times m$ grid (with $n$ and $m$ both positive integers) such that the number of paths from the bottom left to the top right is divisible by $2021$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-6ff2627d8096eb89aac641f0115213b53fb4985bdf977c558bc857bd4636fd79 | What is the area of the smallest ellipse that can be circumscribed around a 3-4-5 triangle? | [
"\\(\\frac{25\\pi}{4}\\)"
] | [
"What is the area of the smallest ellipse that can be circumscribed around a 3-4-5 triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-c0ae3b232ffcd18599b597f689e3c9d118cba0aae0d463e7958ec80b491559fc | What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square? | [
"\\(8 + 4\\sqrt{3}\\)"
] | [
"What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-aa086a75d5fe0c06d2e6822ec5f43a2e8a1151998c5ccd3bcb829e8891bc4ad9 | What is the area of the square formed by the points (1,i), (1,-i), (-1,i), and (-1,-i) on the complex plane? | [
"4"
] | [
"What is the area of the square formed by the points (1,i), (1,-i), (-1,i), and (-1,-i) on the complex plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e47c69d6ec038340ef925da98785c254c52c2c1277f6fe7952351084ab8bf69e | What is the area of the triangle bounded by the lines $y=x,$ $y=-x,$ and $y=6$? | [
"36"
] | [
"What is the area of the triangle bounded by the lines $y=x,$ $y=-x,$ and $y=6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb9ac3ea91e6cc0b273d7750e6aed5ff8974decf8f17b5b5405cf69af7450d52 | What is the area of the triangle if the areas of three inscribed squares are respectively $4$, $25$, and $9$? | [
"\\(\\frac{9025}{156}\\)"
] | [
"What is the area of the triangle if the areas of three inscribed squares are respectively $4$, $25$, and $9$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11779b3e92587d71f8b6a4433c2b0bc7ee3522d0eb84c6fc6a8be207141d09bd | What is the area of the triangle with vertices (2, 1, 3), (-1, 0, 4), and (4, 2, 2)? | [
"\\(\\frac{1}{\\sqrt{2}}\\)"
] | [
"What is the area of the triangle with vertices (2, 1, 3), (-1, 0, 4), and (4, 2, 2)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ce7c26ff25c3c58cc925210aaaa4628368bb122729c839caf9b600ff4691a3cc | What is the area of the triangle with vertices \( (x, 0, 0) \), \( (0, y, 0) \), and \( (0, 0, z) \)? | [
"\\dfrac{1}{2} \\sqrt{x^2 y^2 + y^2 z^2 + z^2 x^2}"
] | [
"What is the area of the triangle with vertices \\( (x, 0, 0) \\), \\( (0, y, 0) \\), and \\( (0, 0, z) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0dac58e3d8f923bda8a7eb3298204e128391e35b300daecb9124304519fa1b8 | What is the area of the triangular region enclosed by the lines with equations \( y = \frac{1}{2}x + 3 \), \( y = -2x + 6 \), and \( y = 1 \)? Express your answer as a decimal to the nearest hundredth. | [
"8.45"
] | [
"What is the area of the triangular region enclosed by the lines with equations \\( y = \\frac{1}{2}x + 3 \\), \\( y = -2x + 6 \\), and \\( y = 1 \\)? Express your answer as a decimal to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-7e41d424d3ca16bc69a84f2c252d6b4cde1c1a428aa016a1ee0c22f44673e370 | What is the area of the twelve-sided figure drawn on $1 \text{ cm} \times 1 \text{ cm}$ graph paper in $\text{cm}^2$? The vertices of the figure are given by the coordinates: $(1,3)$, $(2,4)$, $(2,5)$, $(3,6)$, $(4,5)$, $(5,5)$, $(6,4)$, $(5,3)$, $(5,2)$, $(4,1)$, $(3,2)$, and $(2,2)$. | [
"13"
] | [
"What is the area of the twelve-sided figure drawn on $1 \\text{ cm} \\times 1 \\text{ cm}$ graph paper in $\\text{cm}^2$?\n\nThe vertices of the figure are given by the coordinates:\n$(1,3)$, $(2,4)$, $(2,5)$, $(3,6)$, $(4,5)$, $(5,5)$, $(6,4)$, $(5,3)$, $(5,2)$, $(4,1)$, $(3,2)$, and $(2,2)$."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-8bf351fb56c2b4227d203efebcdbd38b23e88c2a6bb03299496d95bd60c697b3 | What is the area of the unbounded region below the curves $y=e^{2x}$ and $y=e^{-2x}$, and above the line $y=0$ over the interval $(-\infty, \infty)$? | [
"1"
] | [
"What is the area of the unbounded region below the curves $y=e^{2x}$ and $y=e^{-2x}$, and above the line $y=0$ over the interval $(-\\infty, \\infty)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61cefd889dd8b517da70b15126cf68aec4b814d76b01ebe4671f0aa67f65222f | What is the area of triangle \(ABC\) with sides of lengths 15, 16, and 17? | [
"\\(24\\sqrt{21}\\)"
] | [
"What is the area of triangle \\(ABC\\) with sides of lengths 15, 16, and 17?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-39b96baf575aca45ba08ad9690f0edcc9529058c4f3049e22e826ba792391aa6 | What is the area that is swept out by a regular hexagon of side length $1$ as it rotates $30^\circ$ about its center? | [
"\\dfrac{\\pi}{8} \\left(4 + 3 \\ln 3\\right)"
] | [
"What is the area that is swept out by a regular hexagon of side length $1$ as it rotates $30^\\circ$ about its center?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-3746ddbfa0e45cb782786d1f7d614db4ad14293cbd8e3a0a71efcf44131ba7b7 | What is the area under the curve of \( \sin(x) \) from \( 0 \) to \( 2\pi \)? | [
"4"
] | [
"What is the area under the curve of \\( \\sin(x) \\) from \\( 0 \\) to \\( 2\\pi \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-081a1be989d37dc6429b3c2b1e02bfe76cc6bca1fb4be8eae2eb3fef752a5106 | What is the area, in cm$^2$, of an octagon with side lengths $3, 3, 11, 11, 15, 15, 15, 15$ cm that is inscribed in a circle? | [
"840"
] | [
"What is the area, in cm$^2$, of an octagon with side lengths $3, 3, 11, 11, 15, 15, 15, 15$ cm that is inscribed in a circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-868cf0defafd5dfe3918f1218976ef7de3f00b750e5c8ed3918dd50bc0b3437f | What is the area, in square inches, of a right triangle with a 24-inch leg and a 25-inch hypotenuse? | [
"84"
] | [
"What is the area, in square inches, of a right triangle with a 24-inch leg and a 25-inch hypotenuse?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6d87e84e76e76abd6e3ab916371538d0cd67ed45a787fd996b742af6656fcb2 | What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form. | [
"486 \\sqrt{3}"
] | [
"What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\\pi$ square units? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89501979418551084d36c260d01fc93bcfd58c409f224d0dce83fcf86bee9372 | What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 10$, $y = 5$ and the $y$-axis? Express your answer as a decimal to the nearest tenth. | [
"37.5"
] | [
"What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 10$, $y = 5$ and the $y$-axis? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b691e922fdd3efce75de5f59d1f3b277f8f85d15604be05e6000081259917a65 | What is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form. | [
"2\\sqrt{5}"
] | [
"What is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14e6cd08a3820196dccbf9115bea89f8956666725388927b12efae412f0bc6af | What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$? | [
"18"
] | [
"What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1ba63feb1b60f4a51a1eb7085748f452dd2bbaa0bbb4608740ce9856d0916aa8 | What is the area, in square units, of a triangle with vertices at $(0,0)$, $(0,5)$, and $(7,12)$? Express your answer as a decimal to the nearest tenth. | [
"17.5"
] | [
"What is the area, in square units, of a triangle with vertices at $(0,0)$, $(0,5)$, and $(7,12)$? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f4e725ee13e9894016bab305b1220472e31b40eb89ba0e192ce68cb4f045ca9 | What is the area, in square units, of a triangle with vertices at $A(1, 1), B(6, 1), C(3, 7)$? | [
"15"
] | [
"What is the area, in square units, of a triangle with vertices at $A(1, 1), B(6, 1), C(3, 7)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88b4faea38d18470cb02b7e46c5fbef144a46f64aac5cb07dbc0c72a0ca54906 | What is the area, in square units, of the figure defined by the set of points \((x, y)\) that satisfy the inequalities: \[ |x + 1| + |y + 1| \le 2 \] \[ x^2 + y^2 \ge 2 \] Express your answer as \(a - b\pi\), where \(a\) and \(b\) are integers. | [
"\\(8 - \\pi\\)"
] | [
"What is the area, in square units, of the figure defined by the set of points \\((x, y)\\) that satisfy the inequalities:\n\\[ |x + 1| + |y + 1| \\le 2 \\]\n\\[ x^2 + y^2 \\ge 2 \\]\nExpress your answer as \\(a - b\\pi\\), where \\(a\\) and \\(b\\) are integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-073d5c78af404a63e90599ebcdce096afc156a082f3d6e92a49657f1279c07d8 | What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis? | [
"40"
] | [
"What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd8f844e577125925d85317fa5fdb3619187fdcb02171795dcab4a258903429e | What is the area, in square units, of the square with the four vertices at $A\ (0, 0)$, $B\ (-5, -1)$, $C\ (-4, -6)$ and $D\ (1, -5)$? | [
"26"
] | [
"What is the area, in square units, of the square with the four vertices at $A\\ (0, 0)$, $B\\ (-5, -1)$, $C\\ (-4, -6)$ and $D\\ (1, -5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40b8be56282967f1c4a8b3007c924892f9f919ed51d9bb7da041cb563ebda497 | What is the arithmetic mean of $\frac{2}{5}$ and $\frac{4}{7}$ ? Express your answer as a common fraction. | [
"\\frac{17}{35}"
] | [
"What is the arithmetic mean of $\\frac{2}{5}$ and $\\frac{4}{7}$ ? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df8914a85f181e87714700e5f00ebf00311a02e74aa45949146c0b6300ff5f12 | What is the arithmetic mean of 14, 22 and 36? | [
"24"
] | [
"What is the arithmetic mean of 14, 22 and 36?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-150946cc7adeb9bc2078897829a3bdd249468b8d4e8282f9066efa5b7459ffec | What is the arithmetic mean of all positive integers less than \( N \) and co-prime to \( N \), where \( N > 1 \) is a positive integer? | [
"\\(\\frac{\\varphi(N)}{2}\\)"
] | [
"What is the arithmetic mean of all positive integers less than \\( N \\) and co-prime to \\( N \\), where \\( N > 1 \\) is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6dbf6563aedd4f1eb6fef824214840c69470188738781f839b4021a4f81c2606 | What is the arithmetic mean of all positive two-digit multiples of 7? | [
"56"
] | [
"What is the arithmetic mean of all positive two-digit multiples of 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f8001413d5c862159bf7b8320d2b36d8e66fd39ec525525205059e3de3a6639 | What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $, where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set $\{1,2,3,4\}$? | [
"2.5"
] | [
"What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $, where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set $\\{1,2,3,4\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffb4af3e2e45b5596d77e1ceda7b70d388968214a0b9c37335a16ffb1fb57fe5 | What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. | [
"0.5"
] | [
"What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2f12840f410ad561e1c4c039a18525cf9b6364af8ff9d216c2f52aeb69cbf8f | What is the arithmetic mean of the smallest elements of $r$-subsets of the set $\{1, 2, \dots , n\}$ where $1 \leq r \leq n$? | [
"\\(\\frac{n+1}{r+1}\\)"
] | [
"What is the arithmetic mean of the smallest elements of $r$-subsets of the set $\\{1, 2, \\dots , n\\}$ where $1 \\leq r \\leq n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-8a0e63f12346115804283ced3349acf80bd94a5356806e8630e84d547b3657a0 | What is the asymptotic behavior of \( \#\{k : k \text{ is even, } \phi^{-1}(k) = 0, k \le n\} \) as \( n \to \infty \)? | [
"\\dfrac{n}{2}"
] | [
"What is the asymptotic behavior of \\( \\#\\{k : k \\text{ is even, } \\phi^{-1}(k) = 0, k \\le n\\} \\) as \\( n \\to \\infty \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e30467a522b7398850a019c62616c8d7de86f9c48cc13b67fa2325a34e4592c0 | What is the asymptotic behavior of \( \sup_{k \le n} \phi^{-1}(k) \) as \( n \to \infty \)? | [
"e^{\\gamma} n \\log \\log n"
] | [
"What is the asymptotic behavior of \\( \\sup_{k \\le n} \\phi^{-1}(k) \\) as \\( n \\to \\infty \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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