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-7bfb1de691b23f6d99f10f99afb0c8d63b734841ab1b7e37d01c6fb700db3fe6 | A convex quadrilateral $ABCD$ is inscribed in a circle. Lines $AD$ and $BC$ intersect at point $E$. Points $M, N$ are chosen on the sides of $AD$, $BC$ respectively in such a way that $AM/MD = BN/NC$. Circumscribed circles of $\triangle EMN$ and the quadrilateral $ABCD$ intersect at points $X, Y$. If $AB$ and $CD$ inte... | [
"1"
] | [
"A convex quadrilateral $ABCD$ is inscribed in a circle. Lines $AD$ and $BC$ intersect at point $E$. Points $M, N$ are chosen on the sides of $AD$, $BC$ respectively in such a way that $AM/MD = BN/NC$. Circumscribed circles of $\\triangle EMN$ and the quadrilateral $ABCD$ intersect at points $X, Y$. If $AB$ and $CD... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-7399361101cdd6b2162cd643d45f8fb27c531d1772432ca147da581066edd38a | A convex quadrilateral $ABCD$ is such that $AB=AC=BD$. Diagonals $AC$ and $BD$ intersect at the point $O$. Circles circumscribed around triangles $ABC$ and $ADO$ intersect at the point $P$. The lines $AP$ and $BC$ intersect at the point $Q$. If $\angle COQ = \angle DOQ$, find the value of $\frac{AO}{OC}$ given that $AC... | [
"1"
] | [
"A convex quadrilateral $ABCD$ is such that $AB=AC=BD$. Diagonals $AC$ and $BD$ intersect at the point $O$. Circles circumscribed around triangles $ABC$ and $ADO$ intersect at the point $P$. The lines $AP$ and $BC$ intersect at the point $Q$. If $\\angle COQ = \\angle DOQ$, find the value of $\\frac{AO}{OC}$ given ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5eeb7e0bf2bfa238dca20bb8f804dae7de01d8011c08a1eb50cf6abe96a7a850 | A convex quadrilateral $ABCD$ undergoes a series of transformations by Vasya and Petya. Vasya changes vertex $B$ to a point symmetrical to $B$ with respect to the perpendicular bisector of $AC$, and Petya changes vertex $C$ to a point symmetrical to $C$ with respect to the perpendicular bisector of $BD$. They take turn... | [
"1"
] | [
"A convex quadrilateral $ABCD$ undergoes a series of transformations by Vasya and Petya. Vasya changes vertex $B$ to a point symmetrical to $B$ with respect to the perpendicular bisector of $AC$, and Petya changes vertex $C$ to a point symmetrical to $C$ with respect to the perpendicular bisector of $BD$. They take... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-163f16c30a22bcbbcd8d27980e0cc747d3a7d5c448a7861e05aa6bb3885275f4 | A convex quadrilateral $ABCD$ with $AB=1$, $BC=1$, $CD=2$, and $DA=3$ is inscribed in a circle $O$. Find the area of the circle that is tangent to the three sides $BC$, $CD$, and $DA$. | [
"\\(\\frac{\\pi}{3}\\)"
] | [
"A convex quadrilateral $ABCD$ with $AB=1$, $BC=1$, $CD=2$, and $DA=3$ is inscribed in a circle $O$. Find the area of the circle that is tangent to the three sides $BC$, $CD$, and $DA$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-8492fb365eed4cb7a6435c2bc99ef8a0c41997b5711f4066f2e6bbe1bb0a0f7d | A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB + \angle PCB = \angle PBC + \angle PDC = 90^\circ$, calculate the value of $\frac{OP}{OE}$ if $ABCD$ is a square with si... | [
"1"
] | [
"A convex quadrilateral $ABCD$ with $AC \\neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\\angle PAB + \\angle PCB = \\angle PBC + \\angle PDC = 90^\\circ$, calculate the value of $\\frac{OP}{OE}$ if $ABCD$ is a sq... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4bbfc2bdd34aa61ced5cb5e3529192ba039ad995f3c058834a3f9de1eb15de7 | A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45.$ Find the perimeter of $ABCD.$ | [
"144 + 4\\sqrt{113}"
] | [
"A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45.$ Find the perimeter of $ABCD.$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-63863f48eca6be52e847594b4749c1067f69ccab8525e5dc1ec65eed29a48ea2 | A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA=24$, $PB=32$, $PC=28$, and $PD=45$. Find the perimeter of $ABCD$. | [
"\\(144 + 4\\sqrt{113}\\)"
] | [
"A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA=24$, $PB=32$, $PC=28$, and $PD=45$. Find the perimeter of $ABCD$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 20
} | 58 | 0 | 0 |
omr-ws-9229c36ea6ce26b6ac07d6fed93d04942fc6132aaf9312ac86c2424fa6b7522d | A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. | [
"\\dfrac{7}{2}"
] | [
"A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-111f5eb1e6f29ea897d24dcd4835bf17dc379ac06a2783978559c1d013620323 | A convex quadrilateral ABCD is inscribed in a circle with center O. The diagonals AC and BD intersect at P. The circumcircles of triangles ABP and CDP intersect again at Q. If O, P, and Q are three distinct points, calculate the value of \(\frac{OQ^2}{PQ^2}\) given that \(OP = 1\) and \(PQ = \sqrt{2}\). | [
"\\dfrac{3}{2}"
] | [
"A convex quadrilateral ABCD is inscribed in a circle with center O. The diagonals AC and BD intersect at P. The circumcircles of triangles ABP and CDP intersect again at Q. If O, P, and Q are three distinct points, calculate the value of \\(\\frac{OQ^2}{PQ^2}\\) given that \\(OP = 1\\) and \\(PQ = \\sqrt{2}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-c8a5bbb36b4f85092233154295dcf09c288c26812bc39281804347bf026a10d3 | A convex quadrilateral \(ABCD\) has an area of 3852. The side lengths are \(AB = 13\), \(BC = 96\), \(CD = 72\), and \(DA = 109\). The sum of the lengths of the two diagonals is expressed as \(a + b\sqrt{c}\), where \(a\), \(b\), and \(c\) are integers. Minimize the value of \(a + b + c\). | [
"462"
] | [
"A convex quadrilateral \\(ABCD\\) has an area of 3852. The side lengths are \\(AB = 13\\), \\(BC = 96\\), \\(CD = 72\\), and \\(DA = 109\\). The sum of the lengths of the two diagonals is expressed as \\(a + b\\sqrt{c}\\), where \\(a\\), \\(b\\), and \\(c\\) are integers. Minimize the value of \\(a + b + c\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-9b8ae2085ad830de89fb7702e9c24719df81d7be98249cf68c9eeaddfd6b4896 | A convex quadrilateral \(ABCD\) has an inscribed circle touching its sides \(AB\), \(BC\), \(CD\), \(DA\) at the points \(M\), \(N\), \(P\), \(K\), respectively. Let \(O\) be the center of the inscribed circle, and the area of the quadrilateral \(MNPK\) is equal to 8. If \(S\) is the area of quadrilateral \(ABCD\), cal... | [
"16"
] | [
"A convex quadrilateral \\(ABCD\\) has an inscribed circle touching its sides \\(AB\\), \\(BC\\), \\(CD\\), \\(DA\\) at the points \\(M\\), \\(N\\), \\(P\\), \\(K\\), respectively. Let \\(O\\) be the center of the inscribed circle, and the area of the quadrilateral \\(MNPK\\) is equal to 8. If \\(S\\) is the area o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f9651ec8b41c5eb299303f1a5677c449587abb5793fcb78c0cf7f6522a0d507b | A convex quadrilateral \(ABCD\) is circumscribed about a circle \(\omega\). A tangent to \(\omega\) parallel to \(AC\) intersects \(BD\) at a point \(P\) outside of \(\omega\). The second tangent from \(P\) to \(\omega\) touches \(\omega\) at a point \(T\). If the radius of \(\omega\) is \(r\) and the radius of the cir... | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) is circumscribed about a circle \\(\\omega\\). A tangent to \\(\\omega\\) parallel to \\(AC\\) intersects \\(BD\\) at a point \\(P\\) outside of \\(\\omega\\). The second tangent from \\(P\\) to \\(\\omega\\) touches \\(\\omega\\) at a point \\(T\\). If the radius of \\(\\omega\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-3a7a62924dfa996836d96d6247d4a5e3aee87308f401d2c6c31635e850e225e4 | A convex quadrilateral \(ABCD\) is circumscribed about a circle \(\omega\). Let \(PQ\) be the diameter of \(\omega\) perpendicular to \(AC\). Suppose lines \(BP\) and \(DQ\) intersect at point \(X\), and lines \(BQ\) and \(DP\) intersect at point \(Y\). Calculate the ratio \(\frac{AX}{XC} + \frac{AY}{YC}\). | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) is circumscribed about a circle \\(\\omega\\). Let \\(PQ\\) be the diameter of \\(\\omega\\) perpendicular to \\(AC\\). Suppose lines \\(BP\\) and \\(DQ\\) intersect at point \\(X\\), and lines \\(BQ\\) and \\(DP\\) intersect at point \\(Y\\). Calculate the ratio \\(\\frac{AX}{XC}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86f09fe4f5a437447c1b1cca77d0d33668a6eacc8f6066f3b5b5cebe168cc4ed | A convex quadrilateral \(ABCD\) is circumscribed around a circle with radius \(R = \frac{1}{2}\). In the interior of the quadrilateral, there is a point \(P\) such that \(AP^2 + BP^2 + CP^2 + DP^2 = 2\). Calculate the area of quadrilateral \(ABCD\). | [
"1"
] | [
"A convex quadrilateral \\(ABCD\\) is circumscribed around a circle with radius \\(R = \\frac{1}{2}\\). In the interior of the quadrilateral, there is a point \\(P\\) such that \\(AP^2 + BP^2 + CP^2 + DP^2 = 2\\). Calculate the area of quadrilateral \\(ABCD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f8625b2810fac7f74ae87abeac5e0b0a8507691d69019ca34dbc468fd6acb50 | A convex quadrilateral \(ABCD\) is given. Hooligan Vasya changes vertex \(B\) to a point symmetric to \(B\) with respect to the perpendicular bisector of \(AC\). Hooligan Petya changes vertex \(C\) to a point symmetric to \(C\) with respect to the perpendicular bisector of \(BD\). They perform these operations in the o... | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) is given. Hooligan Vasya changes vertex \\(B\\) to a point symmetric to \\(B\\) with respect to the perpendicular bisector of \\(AC\\). Hooligan Petya changes vertex \\(C\\) to a point symmetric to \\(C\\) with respect to the perpendicular bisector of \\(BD\\). They perform these ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-047b41fd67e66ec9f34755e337cf667e8b66aaf88aa4cc3be4116aa444641097 | A convex quadrilateral \(ABCD\) is inscribed in a circle \(\omega\). Lines \(BA\) and \(CD\) intersect at \(P\), and lines \(AD\) and \(BC\) intersect at \(Q\). Points \(E\) and \(F\) are the midpoints of \(BC\) and \(AD\) respectively. Lines \(AC\) and \(BD\) intersect at \(K\), and lines \(QK\) and \(EF\) intersect a... | [
"1"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle \\(\\omega\\). Lines \\(BA\\) and \\(CD\\) intersect at \\(P\\), and lines \\(AD\\) and \\(BC\\) intersect at \\(Q\\). Points \\(E\\) and \\(F\\) are the midpoints of \\(BC\\) and \\(AD\\) respectively. Lines \\(AC\\) and \\(BD\\) intersect at \\(K\\), and... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2ac6eb90571d353524491c34685767607c64220fdc2df5f7e85892ce9330ecde | A convex quadrilateral \(ABCD\) is inscribed in a circle \(\omega\). Suppose there is a point \(X\) on the segment \(AC\) such that \(XB\) and \(XD\) are tangents to the circle \(\omega\). The tangent to \(\omega\) at \(C\) intersects \(XD\) at \(Q\). Let \(E\) (where \(E \ne A\)) be the intersection of the line \(AQ\)... | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle \\(\\omega\\). Suppose there is a point \\(X\\) on the segment \\(AC\\) such that \\(XB\\) and \\(XD\\) are tangents to the circle \\(\\omega\\). The tangent to \\(\\omega\\) at \\(C\\) intersects \\(XD\\) at \\(Q\\). Let \\(E\\) (where \\(E \\ne A\\)) be ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3a80294df5cfdbbcb07342f953709e894971ed591c5d212370313f231903eeb7 | A convex quadrilateral \(ABCD\) is inscribed in a circle \(k\), and \(DA\) and \(CB\) meet at \(E\), where \(CD^2 = AD \cdot ED\). The perpendicular to \(ED\) at \(A\) intersects \(k\) again at \(F\). Given that \(\overline{AD}\) and \(\overline{CF}\) are congruent, find the length of \(ED\) if \(AD = 6\) and \(CD = 6\... | [
"12"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle \\(k\\), and \\(DA\\) and \\(CB\\) meet at \\(E\\), where \\(CD^2 = AD \\cdot ED\\). The perpendicular to \\(ED\\) at \\(A\\) intersects \\(k\\) again at \\(F\\). Given that \\(\\overline{AD}\\) and \\(\\overline{CF}\\) are congruent, find the length of \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8bda630a7cbeba9205b8482b5be45935784fc9c13abc672b4ed449c418199a9 | A convex quadrilateral \(ABCD\) is inscribed in a circle \(k\). The rays \(DA\) and \(CB\) meet at a point \(E\) such that \(CD^2 = AD \cdot ED\). The perpendicular to \(ED\) at \(A\) intersects \(k\) again at a point \(F\). If the circumcenter of \(\triangle ABE\) lies on \(ED\), calculate the ratio \(\frac{CF}{AD}\). | [
"1"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle \\(k\\). The rays \\(DA\\) and \\(CB\\) meet at a point \\(E\\) such that \\(CD^2 = AD \\cdot ED\\). The perpendicular to \\(ED\\) at \\(A\\) intersects \\(k\\) again at a point \\(F\\). If the circumcenter of \\(\\triangle ABE\\) lies on \\(ED\\), calcula... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-d4bfcac7d630f28511fc0504a804bd98c648b8158f23b8fe17035513086fa9bd | A convex quadrilateral \(ABCD\) is inscribed in a circle with center \(O\) inside the quadrilateral. Let \(MNPQ\) be the quadrilateral whose vertices are the projections of the intersection point of the diagonals \(AC\) and \(BD\) onto the sides of \(ABCD\). If the area of \(ABCD\) is \(200\), find the maximum possible... | [
"100"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\) inside the quadrilateral. Let \\(MNPQ\\) be the quadrilateral whose vertices are the projections of the intersection point of the diagonals \\(AC\\) and \\(BD\\) onto the sides of \\(ABCD\\). If the area of \\(ABCD\\) is \\(200\\), find... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-63fa6678aeb165f774cbaf9120f603f47530c714df37eb42c32194e828d74aff | A convex quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Let \(E = AB \cap CD\) and \(F = AC \cap BD\). Let \(M\) be the midpoint of \(EF\). Suppose \(MK\) and \(MQ\) are tangents to the circle \(O\). If the radius of circle \(O\) is \(r\), and the length of \(EF\) is \(d\), find the length of \(MK\)... | [
"\\dfrac{d}{2}"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). Let \\(E = AB \\cap CD\\) and \\(F = AC \\cap BD\\). Let \\(M\\) be the midpoint of \\(EF\\). Suppose \\(MK\\) and \\(MQ\\) are tangents to the circle \\(O\\). If the radius of circle \\(O\\) is \\(r\\), and the length of \\(EF\\) is \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-843997416ffbde6e95f4c6847de6ca1c3560c9ac0b6b1bbfdc675b05debb6a93 | A convex quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Let \(E\) be the intersection of \(AD\) and \(BC\), and let \(M\) and \(N\) be points on \(AD\) and \(BC\) respectively such that \(\frac{AM}{MD} = \frac{BN}{NC}\). The circumcircle of \(\triangle EMN\) intersects the circumcircle of \(ABCD\) a... | [
"\\dfrac{3}{2}"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). Let \\(E\\) be the intersection of \\(AD\\) and \\(BC\\), and let \\(M\\) and \\(N\\) be points on \\(AD\\) and \\(BC\\) respectively such that \\(\\frac{AM}{MD} = \\frac{BN}{NC}\\). The circumcircle of \\(\\triangle EMN\\) intersects ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1e6756a10faeff82d7c8bc4cd5825eed08ee03e6e3473a1e65fcd0552ba976b6 | A convex quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). The angles \(AOB\), \(BOC\), \(COD\), and \(DOA\), taken in some order, are of the same size as the angles of the quadrilateral \(ABCD\). Calculate the area of \(ABCD\) if the radius of the circle is \(5\) units. | [
"50"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with center \\(O\\). The angles \\(AOB\\), \\(BOC\\), \\(COD\\), and \\(DOA\\), taken in some order, are of the same size as the angles of the quadrilateral \\(ABCD\\). Calculate the area of \\(ABCD\\) if the radius of the circle is \\(5\\) units."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2bcd057ed97b9ebb72830463748625288bff2cb4e00173f8865d4f483688071 | A convex quadrilateral \(ABCD\) is inscribed in a circle with diameter \(AB\). Let \(S\) be the intersection point of \(AC\) and \(BD\), and let \(T\) be the foot of the perpendicular from \(S\) to \(AB\). If the measure of angle \(CTD\) is \(120^\circ\), find the measure of angle \(CTS\). | [
"30^\\circ"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with diameter \\(AB\\). Let \\(S\\) be the intersection point of \\(AC\\) and \\(BD\\), and let \\(T\\) be the foot of the perpendicular from \\(S\\) to \\(AB\\). If the measure of angle \\(CTD\\) is \\(120^\\circ\\), find the measure of angle \\(CTS\\)."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-cf91ae85ce7a48963e5612188b30d685bc6db6465ba5f2e0f40e93bd9ae74907 | A convex quadrilateral \(ABCD\) is inscribed in a circle with radius \(R\). Given that \( \text{arc}(BC) + \text{arc}(DA) = 90^\circ \) and \( AB = x \) and \( CB = y \), find \( R \) as a function of \( x \) and \( y \). | [
"\\( R = \\frac{\\sqrt{x^2 + y^2}}{\\sqrt{2}} \\)"
] | [
"A convex quadrilateral \\(ABCD\\) is inscribed in a circle with radius \\(R\\). Given that \\( \\text{arc}(BC) + \\text{arc}(DA) = 90^\\circ \\) and \\( AB = x \\) and \\( CB = y \\), find \\( R \\) as a function of \\( x \\) and \\( y \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd380bf22cdd9f4cab5e4af08ed2403a861d011e82ab071ec56fd46973666ad3 | A convex quadrilateral \(ABCD\) is placed on the Cartesian plane. Vertices \(A\) and \(D\) belong to the negative branch of the hyperbola \(y = \frac{1}{x}\), vertices \(B\) and \(C\) belong to the positive branch of the hyperbola, and point \(B\) lies to the left of \(C\). The segment \(AC\) passes through the origin ... | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) is placed on the Cartesian plane. Vertices \\(A\\) and \\(D\\) belong to the negative branch of the hyperbola \\(y = \\frac{1}{x}\\), vertices \\(B\\) and \\(C\\) belong to the positive branch of the hyperbola, and point \\(B\\) lies to the left of \\(C\\). The segment \\(AC\\) pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-69943c567a6a583ec66f8b2232b195f8df4577189b2d3ec940841ef8a48c2e0c | A convex quadrilateral \(ABCD\) with \(AB \neq CD\) is cyclic. Quadrilaterals \(AKDL\) and \(CMBN\) are rhombs with side \(a\). Let \(O\) be the center of the circle circumscribing quadrilateral \(KLMN\). If \(r\) is the radius of this circle, find the value of \(r^2\) in terms of \(a\) and the circumradius \(R\) of qu... | [
"R^2 + a^2"
] | [
"A convex quadrilateral \\(ABCD\\) with \\(AB \\neq CD\\) is cyclic. Quadrilaterals \\(AKDL\\) and \\(CMBN\\) are rhombs with side \\(a\\). Let \\(O\\) be the center of the circle circumscribing quadrilateral \\(KLMN\\). If \\(r\\) is the radius of this circle, find the value of \\(r^2\\) in terms of \\(a\\) and th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-1bd62a3ea43adb6aed18f29604848de12e467f7ec9efabd7c7e4fe5514233138 | A convex quadrilateral \(ABCD\) with \(AC \neq BD\) is inscribed in a circle with center \(O\). Let \(E\) be the intersection of diagonals \(AC\) and \(BD\). If \(P\) is a point inside \(ABCD\) such that \(\angle PAB + \angle PCB = \angle PBC + \angle PDC = 90^\circ\), find the value of \(\frac{OE}{EP}\) if \(OE = 10\)... | [
"2"
] | [
"A convex quadrilateral \\(ABCD\\) with \\(AC \\neq BD\\) is inscribed in a circle with center \\(O\\). Let \\(E\\) be the intersection of diagonals \\(AC\\) and \\(BD\\). If \\(P\\) is a point inside \\(ABCD\\) such that \\(\\angle PAB + \\angle PCB = \\angle PBC + \\angle PDC = 90^\\circ\\), find the value of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3107074e9e7bb3f64ac05da3acdbbef8c634e8cb6eb43859ba589c7fdcf30a2e | A convex quadrilateral \(ABCD\) with \(AC \neq BD\) is inscribed in a circle with center \(O\). Let \(E\) be the intersection of diagonals \(AC\) and \(BD\). If \(P\) is a point inside \(ABCD\) such that \(\angle PCB + \angle PAB = \angle PDC + \angle PBC = 90^\circ\), find the value of \(\frac{OP}{OE}\) given that \(A... | [
"\\dfrac{1}{2}"
] | [
"A convex quadrilateral \\(ABCD\\) with \\(AC \\neq BD\\) is inscribed in a circle with center \\(O\\). Let \\(E\\) be the intersection of diagonals \\(AC\\) and \\(BD\\). If \\(P\\) is a point inside \\(ABCD\\) such that \\(\\angle PCB + \\angle PAB = \\angle PDC + \\angle PBC = 90^\\circ\\), find the value of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-9fc132b9d6e8c2c24eecbdb0e51d80c62b63f40dbeb6eff73ac6d61aef78de04 | A convex quadrilateral \(ABCD\) with area 2002 contains a point \(P\) in its interior such that \(PA = 24\), \(PB = 32\), \(PC = 28\), and \(PD = 45\). Find the perimeter of \(ABCD\). Options: A. \(4 \sqrt{2002}\) B. \(2 \sqrt{8465}\) C. \(2(48 + \sqrt{2002})\) D. \(2(\sqrt{8633})\) E. \(4(36 + \sqrt{113})\) | [
"\\(4(36 + \\sqrt{113})\\)"
] | [
"A convex quadrilateral \\(ABCD\\) with area 2002 contains a point \\(P\\) in its interior such that \\(PA = 24\\), \\(PB = 32\\), \\(PC = 28\\), and \\(PD = 45\\). Find the perimeter of \\(ABCD\\).\n\nOptions:\nA. \\(4 \\sqrt{2002}\\)\nB. \\(2 \\sqrt{8465}\\)\nC. \\(2(48 + \\sqrt{2002})\\)\nD. \\(2(\\sqrt{8633})\\... | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a93a4bb4010a8ff6b7a16a162500af36b8258479f8a9b975956c8da5f0e39803 | A convex quadrilateral \(ABCD\) with area \(S\) is given. Inside each side of the quadrilateral, a point is selected. These points are consecutively linked by segments, splitting \(ABCD\) into a smaller quadrilateral and 4 triangles. Let \(T\) be the maximum area among these 4 triangles. Find the minimum possible value... | [
"\\dfrac{1}{4}"
] | [
"A convex quadrilateral \\(ABCD\\) with area \\(S\\) is given. Inside each side of the quadrilateral, a point is selected. These points are consecutively linked by segments, splitting \\(ABCD\\) into a smaller quadrilateral and 4 triangles. Let \\(T\\) be the maximum area among these 4 triangles. Find the minimum p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-08a5b2e9cc481bf672b96e87f7b9049e6d3e39223622646c0b13bf0b851042c7 | A convex quadrilateral has an area of $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +... | [
"47"
] | [
"A convex quadrilateral has an area of $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-dfcb5b663295464020b6f15b21b45c01d349cae3d62e0f5ba98fe9db8e4d9faf | A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | [
"047"
] | [
"A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-9524d4242c776eae437003c5a643e3d9c57fb5281593d88b061ad2abbd953117 | A convex quadrilateral has area 30 and side lengths 5, 6, 9, and 7, in that order. Denote by \(\theta\) the measure of the acute angle formed by the diagonals of the quadrilateral. Then \(\tan \theta\) can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n... | [
"47"
] | [
"A convex quadrilateral has area 30 and side lengths 5, 6, 9, and 7, in that order. Denote by \\(\\theta\\) the measure of the acute angle formed by the diagonals of the quadrilateral. Then \\(\\tan \\theta\\) can be written in the form \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime positive int... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-89c684ea40006e1a74416a53801a98df6bc15590fa2d508c053c8e68d5bdf865 | A convex quadrilateral has its four vertices at the points $A (2, 0), B (2, 2), C (3, 2), D (5, -2)$, respectively. Reflect quadrilateral $ABCD$ across the $x$-axis to get quadrilateral $AB'C'D'$. If $C'D'$ and $CD$ meet at $E$, find the area of quadrilateral $C A C' E$. | [
"4"
] | [
"A convex quadrilateral has its four vertices at the points $A (2, 0), B (2, 2), C (3, 2), D (5, -2)$, respectively. Reflect quadrilateral $ABCD$ across the $x$-axis to get quadrilateral $AB'C'D'$. If $C'D'$ and $CD$ meet at $E$, find the area of quadrilateral $C A C' E$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90186e374fd2c1480816bd067668b8c8347b6aa43e54d4f0f25e745a016b0502 | A convex quadrilateral has three sides of length \( k \) and diagonals measuring \( k\sqrt{3} \). The area of the quadrilateral is \( k^2 x \). Compute \( x \). | [
"\\(\\frac{3\\sqrt{3}}{4}\\)"
] | [
"A convex quadrilateral has three sides of length \\( k \\) and diagonals measuring \\( k\\sqrt{3} \\). The area of the quadrilateral is \\( k^2 x \\). Compute \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-d9d7324b8cea901aa0c603311efea4af1594f02821285506d9de97a42b5c8d7a | A convex quadrilateral intersects a circle at points $A_1, A_2, B_1, B_2, C_1, C_2, D_1,$ and $D_2$ such that $A_1B_2 = B_1C_2 = C_1D_2 = D_1A_2$. If the quadrilateral formed by these four segments has an area of $K$, and the lengths of $A_1B_2$, $B_1C_2$, $C_1D_2$, and $D_1A_2$ are each equal to $s$, find the value of... | [
"1"
] | [
"A convex quadrilateral intersects a circle at points $A_1, A_2, B_1, B_2, C_1, C_2, D_1,$ and $D_2$ such that $A_1B_2 = B_1C_2 = C_1D_2 = D_1A_2$. If the quadrilateral formed by these four segments has an area of $K$, and the lengths of $A_1B_2$, $B_1C_2$, $C_1D_2$, and $D_1A_2$ are each equal to $s$, find the val... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afaa69153002677cdb6fa48fcf7d5c30378c1a711900722db9e8855d42c48e4c | A convex quadrilateral intersects a circle at points \(A_1, A_2, B_1, B_2, C_1, C_2, D_1,\) and \(D_2\) in that specific order on the circle. Given that \(A_1B_2 = B_1C_2 = C_1D_2 = D_1A_2\), let \(P\) be the intersection of diagonals \(A_1C_1\) and \(B_2D_2\). If the area of the quadrilateral \(A_1B_2C_1D_2\) is \(k\)... | [
"48"
] | [
"A convex quadrilateral intersects a circle at points \\(A_1, A_2, B_1, B_2, C_1, C_2, D_1,\\) and \\(D_2\\) in that specific order on the circle. Given that \\(A_1B_2 = B_1C_2 = C_1D_2 = D_1A_2\\), let \\(P\\) be the intersection of diagonals \\(A_1C_1\\) and \\(B_2D_2\\). If the area of the quadrilateral \\(A_1B_... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-fb1d10014a49790fcd8e97368599004bd8e84a8847e9101da255a5352f07c88b | A convex quadrilateral is cut by its diagonals into four triangles of equal perimeter. If the sides of the quadrilateral are $a$, $b$, $c$, and $d$, find the value of $\frac{a + c}{b + d}$. | [
"1"
] | [
"A convex quadrilateral is cut by its diagonals into four triangles of equal perimeter. If the sides of the quadrilateral are $a$, $b$, $c$, and $d$, find the value of $\\frac{a + c}{b + d}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3a4aeeffa807cd3492d9891c0e536be11da8d31c9a6ff05b84fda3a0f4d2c96 | A convex quadrilateral is drawn in the coordinate plane such that each of its vertices \((r,s)\) satisfies the equations \(ry = 24\) and \(r^2 + s^2 = 73\). Find the area of this quadrilateral. | [
"110"
] | [
"A convex quadrilateral is drawn in the coordinate plane such that each of its vertices \\((r,s)\\) satisfies the equations \\(ry = 24\\) and \\(r^2 + s^2 = 73\\). Find the area of this quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-4e37af24a8d87cdd9b8a64bf782929517853f8eb82ecf76349d2c8714b838e42 | A convex quadrilateral with an area of $1$ is divided into $n$ polygons by $k$ lines. Each of these polygons can have a circle inscribed within it. Let $S(n)$ be the maximum possible sum of the inradii of these inscribed circles. Calculate $S(9)$. | [
"2"
] | [
"A convex quadrilateral with an area of $1$ is divided into $n$ polygons by $k$ lines. Each of these polygons can have a circle inscribed within it. Let $S(n)$ be the maximum possible sum of the inradii of these inscribed circles. Calculate $S(9)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-30ac194a471570babd31a03b9fe9392b2c78af0944b7df2f575eba2ca2b3a4fe | A convex quadrilateral with an area of $1$ is divided into $n$ polygons by $k$ lines. Each of these polygons has an inscribed circle. If $S$ is the sum of the inradii of these inscribed circles, find the maximum possible value of $S^2$ given $n = 16$. | [
"4"
] | [
"A convex quadrilateral with an area of $1$ is divided into $n$ polygons by $k$ lines. Each of these polygons has an inscribed circle. If $S$ is the sum of the inradii of these inscribed circles, find the maximum possible value of $S^2$ given $n = 16$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-55f3a7e7f50a8ab4855c436d61b3ec993e9b0652bbe0c96bd90e89209f09e13d | A convex quadrilateral with an area of one has two parallel sides. Find the smallest possible value of the length of the greater diagonal of this quadrilateral. | [
"\\(\\sqrt{2}\\)"
] | [
"A convex quadrilateral with an area of one has two parallel sides. Find the smallest possible value of the length of the greater diagonal of this quadrilateral."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-a4d6b763f6ab0e6c67f638932264fbc193cef11b1eaf1a6daa4a4f4054bc725d | A convex quadrilateral with vertices \( A, B, C, D \) has angles at \( A \) and \( B \) of 60 degrees. Let \( P \) be a point inside the quadrilateral such that: 1. The area of triangle \( \triangle ABP \) is 9. 2. The area of triangle \( \triangle BCP \) is 20. 3. The area of triangle \( \triangle DAP \) is 18. 4. Tri... | [
"66"
] | [
"A convex quadrilateral with vertices \\( A, B, C, D \\) has angles at \\( A \\) and \\( B \\) of 60 degrees. Let \\( P \\) be a point inside the quadrilateral such that:\n1. The area of triangle \\( \\triangle ABP \\) is 9.\n2. The area of triangle \\( \\triangle BCP \\) is 20.\n3. The area of triangle \\( \\trian... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-1e4af30418e04a1b4f15261d471a6b6e5be28a20bc81c9c3a8a84f80b4d21cae | A convex regular icosahedron has 20 faces that are all congruent equilateral triangles, and five faces meet at each vertex. A regular pentagonal pyramid is sliced off at each vertex so that the vertex lies directly above the center of the base. The icosahedron has edge length 1. If the sliced off pyramids are identical... | [
"\\dfrac{25 + 5\\sqrt{5}}{48}"
] | [
"A convex regular icosahedron has 20 faces that are all congruent equilateral triangles, and five faces meet at each vertex. A regular pentagonal pyramid is sliced off at each vertex so that the vertex lies directly above the center of the base. The icosahedron has edge length 1. If the sliced off pyramids are iden... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd782d886280ac5a653c553810a1ee06c3df3ba8e8d113ae4c777fe1955bec1f | A convex, closed figure lies inside a given circle with radius $R$. The figure is seen from every point of the circumference at a right angle. Let $P$ be a point on the circumference, and let $A$ and $B$ be the points where the rays from $P$ touch the figure. If $O$ is the center of the circle, find the value of $OA^2 ... | [
"R^2"
] | [
"A convex, closed figure lies inside a given circle with radius $R$. The figure is seen from every point of the circumference at a right angle. Let $P$ be a point on the circumference, and let $A$ and $B$ be the points where the rays from $P$ touch the figure. If $O$ is the center of the circle, find the value of $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6903bf46545f5f49b5374887caf06cb0c3633d159b32503f2ab67beb49cc309 | A conveyor belt rises $2.40$ m. What is the horizontal distance between the ends of the belt for each angle of inclination? a) $32^\circ$ b) $55^\circ$ | [
"a) \\(3.8408\\) meters, b) \\(1.6805\\) meters"
] | [
"A conveyor belt rises $2.40$ m. What is the horizontal distance between the ends of the belt for each angle of inclination?\na) $32^\\circ$\nb) $55^\\circ$"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-23c0b1c3ff8edcea4e661e415e39dd5bc7d5b88b506364cb4ce8b05d3f9b68f4 | A conveyor belt's motor consists of two gears connected by a metal belt. The diameter of gear #1 is 8 meters, and the radius of gear #2 is 12 meters. The centers of the gears are separated by 20 meters. If gear #2 runs at 15 revolutions per minute, what is the rate of gear #1 in revolutions per minute? | [
"45"
] | [
"A conveyor belt's motor consists of two gears connected by a metal belt. The diameter of gear #1 is 8 meters, and the radius of gear #2 is 12 meters. The centers of the gears are separated by 20 meters. If gear #2 runs at 15 revolutions per minute, what is the rate of gear #1 in revolutions per minute?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-2eedce08797b362e621751f4787eb7cb0e2f464b9dc2e06a900ea0b9069a31f2 | A convicted person will be released when he reaches the top of a 100-step staircase. He must go up one step each day of the odd-numbered months and go down one step each day of the even-numbered months. If it begins on January 1, 2001, what day will he be released? | [
"January 1, 2051"
] | [
"A convicted person will be released when he reaches the top of a 100-step staircase. He must go up one step each day of the odd-numbered months and go down one step each day of the even-numbered months. If it begins on January 1, 2001, what day will he be released?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd52af272351f36c1138eb581747c896ce881ac084514eabdd31f470803b1795 | A cookie parlor offers 100 different varieties of cookies. If 72 varieties contain chocolate, 39 contain nuts, 16 contain oatmeal, 13 contain both chocolate and oatmeal, 10 contain both oatmeal and nuts, 21 contain both chocolate and nuts, and 9 contain all three ingredients, how many varieties contain none of these in... | [
"8"
] | [
"A cookie parlor offers 100 different varieties of cookies. If 72 varieties contain chocolate, 39 contain nuts, 16 contain oatmeal, 13 contain both chocolate and oatmeal, 10 contain both oatmeal and nuts, 21 contain both chocolate and nuts, and 9 contain all three ingredients, how many varieties contain none of the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-3bce117148e92804e213f1d01c440accb497242b39c1813a64273cee1e9ce715 | A cooler contains 15 cans of Coke and 10 cans of Pepsi. Three cans are drawn from the cooler without replacement. What is the probability that all three are the same brand? | [
"\\(\\frac{23}{92}\\)"
] | [
"A cooler contains 15 cans of Coke and 10 cans of Pepsi. Three cans are drawn from the cooler without replacement. What is the probability that all three are the same brand?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f61e5923965626fcdd7dde0d502fb531e7e4e715811476eb5d3cbcd70339e8b0 | A coordinate system (two perpendicular axes) is drawn on the plane. Polina noted the origin of the coordinates \( O \), as well as the points with integer coordinates \( A \) and \( B \) such that \( O \), \( A \), and \( B \) are the vertices of a triangle inside and on the boundary of which there are no points with i... | [
"\\text{Construct the lines through } O \\text{ and the points } (1,0) \\text{ and } (0,1) \\text{ obtained by reducing } OA \\text{ and } OB \\text{ via the Euclidean algorithm.}"
] | [
"A coordinate system (two perpendicular axes) is drawn on the plane. Polina noted the origin of the coordinates \\( O \\), as well as the points with integer coordinates \\( A \\) and \\( B \\) such that \\( O \\), \\( A \\), and \\( B \\) are the vertices of a triangle inside and on the boundary of which there are... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-ed5f51b5060968bd3b6bb0277d9acef565ef67e3ba77ea108b80e8dbb39156d3 | A cop is patrolling a highway from a plane flying directly above the road at an altitude of 1 mile, on a course parallel to the road, at a constant speed of 120 mph. Radar aboard shows an oncoming car approaching the plane along the line of sight at 136 mph at a distance of 1.5 miles. Determine the speed of the car rel... | [
"62.5"
] | [
"A cop is patrolling a highway from a plane flying directly above the road at an altitude of 1 mile, on a course parallel to the road, at a constant speed of 120 mph. Radar aboard shows an oncoming car approaching the plane along the line of sight at 136 mph at a distance of 1.5 miles. Determine the speed of the ca... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 7
} | 34 | 0 | 0 |
omr-ws-c5141da0d3c593be635b1af78bebdb9a16fa3cc60514daf26b1b347664e79aa8 | A copsychus and a sparrow, each initially located at one of the vertices of a regular polygon with 103 edges, fly clockwise to another vertex each. The copsychus moves across $\ell$ edges each time while the sparrow moves through $d$ edges of the polygon, where $\ell \ne d$ are both integers less than 103. Assume that,... | [
"51"
] | [
"A copsychus and a sparrow, each initially located at one of the vertices of a regular polygon with 103 edges, fly clockwise to another vertex each. The copsychus moves across $\\ell$ edges each time while the sparrow moves through $d$ edges of the polygon, where $\\ell \\ne d$ are both integers less than 103. Assu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-a1e9f79e17721be2a51115e2d81bb4f160eec77d5ff9d167ff922abef754867b | A copying machine can make copies that are 80%, 100%, and 150% as large as the original. By making copies of copies, what is the smallest number of times you must use the machine if you want a copy of an original that is 324% as large as the original? | [
"6"
] | [
"A copying machine can make copies that are 80%, 100%, and 150% as large as the original. By making copies of copies, what is the smallest number of times you must use the machine if you want a copy of an original that is 324% as large as the original?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-c7f7c39e70f6c6f81562aa27b0c9c3881f8e3a66961329726e3bcd04e4c0ba8e | A corner lot has dimensions 50 by 100 yards. The city plans to take a strip of uniform width along the two sides bordering the streets to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 4851 square yards? Let \( w \) be the width of the strip. The remaining dimensions o... | [
"1"
] | [
"A corner lot has dimensions 50 by 100 yards. The city plans to take a strip of uniform width along the two sides bordering the streets to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 4851 square yards?\n\nLet \\( w \\) be the width of the strip. The remaining di... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b0a89e1e2c2833d00253578f182ae8949748249d1651b9c0444ee4aa29a4db47 | A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that two rectangles $1 \times (n-1)$ are connected to two adjacent sides of a $1 \times 1$ square. The squares of a $100 \times 100$ chessboard are colored in 15 colors. A corner with arm 8 is called "multicolored" if it contains each of the 15 colors.... | [
"8649"
] | [
"A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that two rectangles $1 \\times (n-1)$ are connected to two adjacent sides of a $1 \\times 1$ square. The squares of a $100 \\times 100$ chessboard are colored in 15 colors. A corner with arm 8 is called \"multicolored\" if it contains each of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-c7144dd04ac87aba27e9baac4228be12bd8b0fe975cf7fb36aec8a415e2c2e8d | A couch has 5262 units of wear and loses 3 units per second. The naughty boy jumps on the couch every second. The school principal will arrive in ten minutes (600 seconds). If the couch is ruined by then, there is a 2/3 probability that the principal will freak out. If it is not ruined, there is a 1/20 probability that... | [
"\\(\\frac{1}{20}\\)"
] | [
"A couch has 5262 units of wear and loses 3 units per second. The naughty boy jumps on the couch every second. The school principal will arrive in ten minutes (600 seconds). If the couch is ruined by then, there is a 2/3 probability that the principal will freak out. If it is not ruined, there is a 1/20 probability... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-3c256f138ebc20cc84cd89ce4f5843ed61dccf4e6fba2c4bf2205c8c189ad745 | A council of 111 mathematicians voted on whether to hold their conference in Beijing or Shanghai. The initial vote was 70 votes in favor of Beijing and 41 votes in favor of Shanghai. What is the minimum number of mathematicians that would have to change their votes in order for Shanghai to win a majority of votes? | [
"15"
] | [
"A council of 111 mathematicians voted on whether to hold their conference in Beijing or Shanghai. The initial vote was 70 votes in favor of Beijing and 41 votes in favor of Shanghai. What is the minimum number of mathematicians that would have to change their votes in order for Shanghai to win a majority of votes?... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ee2957200439bee8b7899bab8cf84e17d6056f725eb28a6676fbd8a75725293a | A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities ther... | [
"1920"
] | [
"A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-4438748a0f4cd2f8388f96029e3279f30acee2a2d70b53b7bd4450317700c56a | A country has 2001 towns. Each town has a road to at least one other town. If a subset of the towns is such that any other town has a road to at least one member of the subset, then it has at least \( k > 1 \) towns. Determine the maximum possible value of \( k \) such that the country can be partitioned into \( 2001 -... | [
"1000"
] | [
"A country has 2001 towns. Each town has a road to at least one other town. If a subset of the towns is such that any other town has a road to at least one member of the subset, then it has at least \\( k > 1 \\) towns. Determine the maximum possible value of \\( k \\) such that the country can be partitioned into ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-23b930a9f448301ad0e932583f5770d189bae4c1c5a418adc6838bc92963afe3 | A country has 5 cities, and the citizens want to build a network of roads such that any city can be reached from any other city. The annual cost of maintaining each road is equal to the product of the populations of the two cities it connects, and each city has at least 1 inhabitant. How many different road networks ca... | [
"21"
] | [
"A country has 5 cities, and the citizens want to build a network of roads such that any city can be reached from any other city. The annual cost of maintaining each road is equal to the product of the populations of the two cities it connects, and each city has at least 1 inhabitant. How many different road networ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-5276a6f60affb8d6f6a17f9c80b5b1125634b95044a3431e9a1f6d49e3411fe0 | A country has 50 cities, and some pairs of cities are connected directly by one-way roads. It is known that we can go from any city to any other city using these roads. Suppose we upgrade some roads such that every city has an odd number of upgraded roads starting from that city. Calculate the minimum number of roads t... | [
"50"
] | [
"A country has 50 cities, and some pairs of cities are connected directly by one-way roads. It is known that we can go from any city to any other city using these roads. Suppose we upgrade some roads such that every city has an odd number of upgraded roads starting from that city. Calculate the minimum number of ro... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8888a1fd5d67650df6437ea953cac32399af69afad0120cad4b69514ceef3c4e | A country has 50 cities, with some pairs of cities connected directly by one-way roads. It is known that it is possible to travel from any city to any other city using these roads. Determine the minimum number of roads that must be upgraded so that every city has an odd number of upgraded roads starting from it. | [
"50"
] | [
"A country has 50 cities, with some pairs of cities connected directly by one-way roads. It is known that it is possible to travel from any city to any other city using these roads. Determine the minimum number of roads that must be upgraded so that every city has an odd number of upgraded roads starting from it."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e314e6ef76f73f85a9dbe9e71a19c9496921b8156d2d3747097e0dbb710fb5d | A country has 8 cities arranged in a regular octagon. It wants to build 7 roads between pairs of cities so that every city is reachable from every other city via some sequence of roads. The roads must be straight, may not cross, and no city may have more than two roads leaving it. Find the number of ways to build the r... | [
"132"
] | [
"A country has 8 cities arranged in a regular octagon. It wants to build 7 roads between pairs of cities so that every city is reachable from every other city via some sequence of roads. The roads must be straight, may not cross, and no city may have more than two roads leaving it. Find the number of ways to build ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-9bb0be2398f207b184541638507afd085088d58a84226be247ea97fc8ea95101 | A country has 9 cities labelled from 1 to 9, with exactly one bidirectional road between each pair of cities. A complete road trip is defined as a path that starts and ends at the same city, visits every open road exactly once, and only uses the roads to travel between cities. If six roads are closed for repairs, deter... | [
"770"
] | [
"A country has 9 cities labelled from 1 to 9, with exactly one bidirectional road between each pair of cities. A complete road trip is defined as a path that starts and ends at the same city, visits every open road exactly once, and only uses the roads to travel between cities. If six roads are closed for repairs, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-65fa16eca464edfc5b3138b8f028d85afa0c9247fcdf994e3b3a6d98a67c433c | A country has \( n \ge 3 \) cities, with exactly one directed road between any two cities. Let \( M \) and \( m \) denote the largest and smallest number of roads starting from a city, respectively. Define \( f(n) \) as the number of distinct triples of cities \( (A, B, C) \) such that: (a) \( m \) roads start from \( ... | [
"no_valid_answer_found"
] | [
"A country has \\( n \\ge 3 \\) cities, with exactly one directed road between any two cities. Let \\( M \\) and \\( m \\) denote the largest and smallest number of roads starting from a city, respectively. Define \\( f(n) \\) as the number of distinct triples of cities \\( (A, B, C) \\) such that:\n(a) \\( m \\) r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bb44005649d9d6ba66bae8d82e16e7da36ca93aad2c13d938564e43472c37d1 | A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year, and its annual food production increases by 4% each year. Assuming the average food requirement per person remains constant, calculate the smallest integer \( n \) such that the country becomes self-sufficient in food after... | [
"12"
] | [
"A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year, and its annual food production increases by 4% each year. Assuming the average food requirement per person remains constant, calculate the smallest integer \\( n \\) such that the country becomes self-sufficient in foo... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b8613204128f80e2542091fa2f4cd2cae08cb9ae9e160f16c684684494c8fae | A county contains 2010 cities, some pairs of which are linked by roads. Suppose the county is divided into two states, S and T, each containing 1005 cities. Let $N$ be the minimum number of roads that connect a city in S with a city in T over all possible partitions of the cities into S and T. Calculate the value of $N... | [
"0"
] | [
"A county contains 2010 cities, some pairs of which are linked by roads. Suppose the county is divided into two states, S and T, each containing 1005 cities. Let $N$ be the minimum number of roads that connect a city in S with a city in T over all possible partitions of the cities into S and T. Calculate the value ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-d5b42d9dc440927bd086ae48c28b7e195747823e58432349075f40b8909bb3e4 | A couple decides to have children until they have a child of the same sex as the first one. What is the expected number of children per family if a large number of families adopts this strategy? Consider the probability $p$ that a female child is born, and test the formulae for the special cases of $p=0$ and $p=1$. | [
"3"
] | [
"A couple decides to have children until they have a child of the same sex as the first one. What is the expected number of children per family if a large number of families adopts this strategy? Consider the probability $p$ that a female child is born, and test the formulae for the special cases of $p=0$ and $p=1$... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-3ef6dcc2a2a52a6ebf012f651b95cfd3239a3acde1af846483ea1593088ed6f4 | A courier has to reach a point $B$ on one bank of a river from a point $A$ on the other bank. Knowing that the courier can walk along the bank $k$ times faster than he can row along the river, find the angle $\alpha $ at which the courier must cross the river in order to reach point $B$ in the shortest time. The width ... | [
"$\\alpha =\\max\\left\\{\\arccos \\frac{1}{k},\\arctan \\frac{h}{d}\\right\\}$"
] | [
"A courier has to reach a point $B$ on one bank of a river from a point $A$ on the other bank. Knowing that the courier can walk along the bank $k$ times faster than he can row along the river, find the angle $\\alpha $ at which the courier must cross the river in order to reach point $B$ in the shortest time. The ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f2f81194eba84a5ef9010633a564c5b76fcc0af21f1a30d0cd5eb9caacbacc6 | A cow is grazing with a rope of length 10 meters tied to a pole. There are two boundary walls perpendicular to each other: one 5 meters to the east of the pole and another \(5\sqrt{2}\) meters to the north of the pole. Find the area the cow can graze on. | [
"\\frac{275\\pi}{4} - 25\\pi(2\\sqrt{2} + \\sqrt{3})"
] | [
"A cow is grazing with a rope of length 10 meters tied to a pole. There are two boundary walls perpendicular to each other: one 5 meters to the east of the pole and another \\(5\\sqrt{2}\\) meters to the north of the pole. Find the area the cow can graze on."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-4195e29dc5b844ccbf9e4a36b34e6d99fdeac0d84e5a17e740005c301967cad1 | A cow is inside a square field. Its distances from the nearest three corners are $30$ m, $40$ m, and $50$ m. Determine the distance from the cow to the furthest corner and the size of the field. | [
"Distance to the furthest corner = \\( 40\\sqrt{2} \\) m, Size of the field = \\( 2050 + 150\\sqrt{119} \\) m\\(^2\\)"
] | [
"A cow is inside a square field. Its distances from the nearest three corners are $30$ m, $40$ m, and $50$ m. Determine the distance from the cow to the furthest corner and the size of the field."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-069ce788fd3cec1397aafd8ccdc37794e3545c5633c6c58d91a0214c571a8643 | A cow is shackled to a rope tied between two poles that are 15 meters apart. The rope is 17 meters long. What is the maximum area the cow can graze, assuming the rope and thickness of the pillars are negligible? | [
"693.74"
] | [
"A cow is shackled to a rope tied between two poles that are 15 meters apart. The rope is 17 meters long. What is the maximum area the cow can graze, assuming the rope and thickness of the pillars are negligible?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b445def57369dcebe8cc4589aa0f0a29b3ac190ffcecdb45c02bdba5027e2ba9 | A cow is tethered to one corner of a rectangular barn that measures 10 feet by 20 feet, with a rope 100 feet long. What is the maximum grazing area of the cow? | [
"12350\\pi"
] | [
"A cow is tethered to one corner of a rectangular barn that measures 10 feet by 20 feet, with a rope 100 feet long. What is the maximum grazing area of the cow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-0d85e2781a102d128f459472691b2bd7a9652d4c168a26f1fa735676a558290c | A cow is tethered to the outside wall of a circular silo of radius \( r \). The length of the tether is equal to half the circumference of the silo. What is the area of the land that the cow can graze, expressed as a fraction of the area the cow could graze if the silo weren't in the way? | [
"\\(\\frac{5}{6}\\)"
] | [
"A cow is tethered to the outside wall of a circular silo of radius \\( r \\). The length of the tether is equal to half the circumference of the silo. What is the area of the land that the cow can graze, expressed as a fraction of the area the cow could graze if the silo weren't in the way?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 6
} | 23 | 0 | 0 |
omr-ws-2a7e42f1a83c4736e1e78a5ff913978dfd7220932b940c1f1f1bd3dc594d53c0 | A cow is tied at point B on the circumference of a circle with radius R. The cow can graze exactly half of the area of the circle. If C is the length of the rope, find the value of C in terms of R and d, where C = R + d. | [
"C = R + d"
] | [
"A cow is tied at point B on the circumference of a circle with radius R. The cow can graze exactly half of the area of the circle. If C is the length of the rope, find the value of C in terms of R and d, where C = R + d."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-ac80e98f78d23efcae8d33e962c205ddaf0d30f2b492ee5baecb6150cc4955da | A cow is tied to a fixed point on the circumference of a circle with radius \( R \). If the cow can graze exactly 50% of the area of the circle, what is the length of the string in terms of \( R \)? | [
"\\(0.863R\\)"
] | [
"A cow is tied to a fixed point on the circumference of a circle with radius \\( R \\). If the cow can graze exactly 50% of the area of the circle, what is the length of the string in terms of \\( R \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-41528fcf0acdcb589053344bfb8e631ecb469f8b28cdee3cdd29b2b4eb6e0f96 | A cow is tied to a silo with radius \( r \) by a rope long enough to reach just the other side of the silo. Calculate the area available to the cow for grazing. | [
"\\(\\pi r^2(\\pi-1)\\)"
] | [
"A cow is tied to a silo with radius \\( r \\) by a rope long enough to reach just the other side of the silo. Calculate the area available to the cow for grazing."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4524ea620e2bb9b908846c2e7954bdd82f706e0671212a471edce45c017c1762 | A cow is tied to a square camp-site with sides of 100 meters using four strings, each 100 meters long. Find the area accessible by the cow. | [
"\\( 100 - 100\\sqrt{3} + \\frac{100\\pi}{3} \\)"
] | [
"A cow is tied to a square camp-site with sides of 100 meters using four strings, each 100 meters long. Find the area accessible by the cow."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35140a1ee6e991cd287e71f51666c24644d12a74889222645b07375825f37a67 | A cow is tied to the corner of a $20 \times 15$ foot shed with a $30$ ft rope. Find her total grazing area. | [
"\\(\\frac{3025\\pi}{4}\\)"
] | [
"A cow is tied to the corner of a $20 \\times 15$ foot shed with a $30$ ft rope. Find her total grazing area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-8183d9d47c4c5b1a67edf7d14ff27ce60c319d2384dc0fc34df1821fee3489e0 | A cow is tied to the edge of a circular barn with radius \( r \) by a rope of length \( r \). How much land does the cow have to graze? | [
"\\left( \\dfrac{\\pi}{3} + \\dfrac{\\sqrt{3}}{2} \\right) r^2"
] | [
"A cow is tied to the edge of a circular barn with radius \\( r \\) by a rope of length \\( r \\). How much land does the cow have to graze?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-b6e0da108853f1842f290a6b32cd1216cc2be9186dd8a3ed637aa6711c64ae61 | A cow is to be tied on the midpoint of the larger diagonal of a cyclic quadrilateral garden with sides 5, 6, 7, and 8 such that it can graze exactly 3/5th of the garden. Find the length of the rope. | [
"6"
] | [
"A cow is to be tied on the midpoint of the larger diagonal of a cyclic quadrilateral garden with sides 5, 6, 7, and 8 such that it can graze exactly 3/5th of the garden. Find the length of the rope."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-345de1a58164bbeeba50b1300e221d4f5ae0c30562ff9f94d79448d62db516e9 | A cowboy fills a water trough that is 1.6 m long, 64 cm wide, and 47 cm deep using a hose with a diameter of 2.6 cm. The water emerges from the hose at 3.0 m/s. How long does it take to fill the trough? Answer in units of seconds. | [
"296.08"
] | [
"A cowboy fills a water trough that is 1.6 m long, 64 cm wide, and 47 cm deep using a hose with a diameter of 2.6 cm. The water emerges from the hose at 3.0 m/s. How long does it take to fill the trough? Answer in units of seconds."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-df7dac15b992e87170f8b8f4c6de940f51bf8d71c4e9f90c17ad8d6675cd1d96 | A cowboy is 4 miles south of a river that flows east. He is also 8 miles west and 7 miles north of his cabin. If he wants to go to the river, bathe, and then go to his cabin, what is the shortest distance he can travel? | [
"17 miles"
] | [
"A cowboy is 4 miles south of a river that flows east. He is also 8 miles west and 7 miles north of his cabin. If he wants to go to the river, bathe, and then go to his cabin, what is the shortest distance he can travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-3da280d0e9d3b76631cc5611f36760c3cf2a8a72b8efb6c5b058a16081a6521c | A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance (in miles) he can travel to accomplish this? | [
"17"
] | [
"A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance (in miles) he can travel to accomplish this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-afe3328f194fd58b9560a6b41b7cd17e0ea563fb1bec01c87df4d82558a284f2 | A coyote is chasing a roadrunner, starting 50 feet apart. The roadrunner is 10 feet from safety and travels at 5 feet per second, while the coyote travels at 20 feet per second. If the roadrunner reaches safety in 2 seconds, how many feet did the coyote get closer to the roadrunner in that same amount of time? | [
"30 feet"
] | [
"A coyote is chasing a roadrunner, starting 50 feet apart. The roadrunner is 10 feet from safety and travels at 5 feet per second, while the coyote travels at 20 feet per second. If the roadrunner reaches safety in 2 seconds, how many feet did the coyote get closer to the roadrunner in that same amount of time?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75956a78933775774f8e0c161004f3489c92a8b541e8b0103e2479f3c81ba2dc | A crankshaft with a length of 5 centimeters rotates at 20 revolutions per second. It is connected to a piston via a connecting rod that is 15 centimeters long. If the crankshaft starts at the horizontal position, how far is the piston from the rim of the crankshaft after 0.01 second? | [
"15"
] | [
"A crankshaft with a length of 5 centimeters rotates at 20 revolutions per second. It is connected to a piston via a connecting rod that is 15 centimeters long. If the crankshaft starts at the horizontal position, how far is the piston from the rim of the crankshaft after 0.01 second?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-46b4f0b1565a6658fa5c712b13f67489b646fd131d29268dc720df63595abf4e | A crazy biologist has $n \ge 3$ Petri dishes arranged in a line. Initially, some of them contain one bacterium each, and others are empty. Then the crazy biologist performs the following experiments: First, he makes every bacterium divide into $2$, and in all resulting pairs, one bacterium remains in its Petri dish and... | [
"2^{n-2}"
] | [
"A crazy biologist has $n \\ge 3$ Petri dishes arranged in a line. Initially, some of them contain one bacterium each, and others are empty. Then the crazy biologist performs the following experiments: First, he makes every bacterium divide into $2$, and in all resulting pairs, one bacterium remains in its Petri di... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-d5799ddaaaf647ae7aa9eeb4910506902898a131dca85ebbbf86ac797c694b74 | A creature walks from the south pole to the north pole of a sphere with a radius of 60 cm, following a spiral curve that completes one full revolution around the sphere for every 5 cm of vertical rise. What is the length of the path that the creature walks? | [
"720\\pi^2"
] | [
"A creature walks from the south pole to the north pole of a sphere with a radius of 60 cm, following a spiral curve that completes one full revolution around the sphere for every 5 cm of vertical rise. What is the length of the path that the creature walks?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d98d820c8eb80adfefca2c77672052eb91a8660bba97bcbd60dd4bded073561f | A crew of 30 people can build a certain road in 60 days. After the tenth day, the company wants the road built in 30 days instead of 60 days. How many more people must be hired to meet the new deadline? | [
"45"
] | [
"A crew of 30 people can build a certain road in 60 days. After the tenth day, the company wants the road built in 30 days instead of 60 days. How many more people must be hired to meet the new deadline?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-645486b2ca8f9658268c695dd70512a4b2b08c00fe5a2dfc330abb7d77e25d86 | A cricket is located at a point 2006 units away from the center of a circle $\Gamma$ with radius 1. The cricket can jump such that the perpendicular bisector of the segment connecting its starting and ending positions intersects the circle $\Gamma$. Let $n$ be the minimum number of jumps required for the cricket to ent... | [
"1009019"
] | [
"A cricket is located at a point 2006 units away from the center of a circle $\\Gamma$ with radius 1. The cricket can jump such that the perpendicular bisector of the segment connecting its starting and ending positions intersects the circle $\\Gamma$. Let $n$ be the minimum number of jumps required for the cricket... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b894e4ac0a19effd2ce62ac439c726d87835eb62975477a987e79fde7b6b98f0 | A cricket moves across a $2n \times 2n$ board covered by dominoes according to specific jumping rules. The cricket starts on the first vertical line and finishes on the last vertical line. Let $h(n)$ be the difference in height between the starting and ending points for any valid path. Find $h(5)$. | [
"5"
] | [
"A cricket moves across a $2n \\times 2n$ board covered by dominoes according to specific jumping rules. The cricket starts on the first vertical line and finishes on the last vertical line. Let $h(n)$ be the difference in height between the starting and ending points for any valid path. Find $h(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d13093dac1939846f2e65b0121679266d699a0176e859b198e99c11c50ce66f | A crime has been committed, and the DNA left at the scene matches only five strands. Each innocent person has a probability of $10^{-5}$ of matching on all five strands. There are one million residents in the town, with ten thousand being recently released ex-convicts. The district attorney believes each ex-convict has... | [
"\\(\\frac{1}{0.9 + \\frac{10^{-5}}{a}}\\)"
] | [
"A crime has been committed, and the DNA left at the scene matches only five strands. Each innocent person has a probability of $10^{-5}$ of matching on all five strands. There are one million residents in the town, with ten thousand being recently released ex-convicts. The district attorney believes each ex-convic... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61dd07281199dec9d54af42b9f6b157c70076b9659f3f9602b8483b54353e23b | A criminal hides in a room with 99 innocent people. You have a lie detector that correctly classifies 95% of people. You randomly select someone, ask if they are the criminal, and the machine indicates they are lying. What is the probability that the selected person is the criminal? | [
"\\(\\frac{19}{118}\\)"
] | [
"A criminal hides in a room with 99 innocent people. You have a lie detector that correctly classifies 95% of people. You randomly select someone, ask if they are the criminal, and the machine indicates they are lying. What is the probability that the selected person is the criminal?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4846b1d399df50138f481eafc5c6768c843654d090d3f64e966dc764670477a2 | A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants, and broccoli croissants. How many ways are there to choose two dozen croissants with at least five chocolate croissants and at most three almond croissants? | [
"27,000"
] | [
"A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants, and broccoli croissants. How many ways are there to choose two dozen croissants with at least five chocolate croissants and at most three almond croissants?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-65d918cccb192d596ec08078dde627d6009bb6afef6681b158b63a3ec2635600 | A cross section is taken of a solid sphere with radius 34, and the area of the cross section is $256\pi$. The cross section divides the sphere into two pieces. Find the total surface area of the smaller piece. | [
"528\\pi"
] | [
"A cross section is taken of a solid sphere with radius 34, and the area of the cross section is $256\\pi$. The cross section divides the sphere into two pieces. Find the total surface area of the smaller piece."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d24d7b033ecf9eaea34dac01c328eaa39647bfd41a10d0eaa0163cd752692409 | A cross-section of a river is a trapezoid with bases 10 ft and 16 ft and slanted sides of length 5 ft. At this section, the water is flowing at π mph. A little ways downstream is a dam where the water flows through 4 identical circular holes at 16π mph. What is the radius of the holes? | [
"\\dfrac{\\sqrt{13}}{4\\sqrt{\\pi}}"
] | [
"A cross-section of a river is a trapezoid with bases 10 ft and 16 ft and slanted sides of length 5 ft. At this section, the water is flowing at π mph. A little ways downstream is a dam where the water flows through 4 identical circular holes at 16π mph. What is the radius of the holes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba87d40711432ad7a27ea1821b524488857879409f20edaf7b6e026d857ca03f | A cross-section of a river is an isosceles trapezoid with bases 10 and 16 and slanted sides of length 5. The water flows at $\pi$ mph. In a dam, the water flows through 4 identical circular holes at 16 mph. What is the radius of these holes? | [
"\\dfrac{\\sqrt{13}}{4}"
] | [
"A cross-section of a river is an isosceles trapezoid with bases 10 and 16 and slanted sides of length 5. The water flows at $\\pi$ mph. In a dam, the water flows through 4 identical circular holes at 16 mph. What is the radius of these holes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-56077362704733b83b0153997245b873e3144c632ba4eb7293c09f121f2ceabf | A cross-section of a sphere is taken by slicing along a plane through the midpoint of a radius, perpendicular to that radius. If the radius of the sphere is 6, what is the area of the circular cross-section? | [
"\\( 27\\pi \\)"
] | [
"A cross-section of a sphere is taken by slicing along a plane through the midpoint of a radius, perpendicular to that radius. If the radius of the sphere is 6, what is the area of the circular cross-section?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
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