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-0027c2070d23ec32ecbb1124454d2874191acde1203067102103fb7fc90f1028 | Trapezoid $TRAP$ is inscribed in circle $\odot O$ such that points $T$, $O$, and $R$ are collinear, $TR = 8$, and $\angle PTR = 75^\circ$. The area of trapezoid $TRAP$ can be written in the form $k + w\sqrt{p}$. Find the value of $k + w + p$. | [
"15"
] | [
"Trapezoid $TRAP$ is inscribed in circle $\\odot O$ such that points $T$, $O$, and $R$ are collinear, $TR = 8$, and $\\angle PTR = 75^\\circ$. The area of trapezoid $TRAP$ can be written in the form $k + w\\sqrt{p}$. Find the value of $k + w + p$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-5c6acefb86e8867f95fad489d1f6f3802557e9b9a634846e8986d9d736a0771c | Trapezoid $WXYZ$ is circumscribed about a circle, with $\angle X$ and $\angle Y$ being right angles. Given that $XW = 16$ and $YZ = 7$, find the perimeter of trapezoid $WXYZ$. | [
"46"
] | [
"Trapezoid $WXYZ$ is circumscribed about a circle, with $\\angle X$ and $\\angle Y$ being right angles. Given that $XW = 16$ and $YZ = 7$, find the perimeter of trapezoid $WXYZ$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-593c337973d1dd55aff47486c54f2e67ddd175fde86976f723ee2425caea0908 | Trapezoid ABCD has AD || BC, BD=1, angle DBA=23 degrees, and angle BDC=46 degrees. The ratio BC:AD is 9:5. What is the length of CD? | [
"\\(\\frac{4}{5}\\)"
] | [
"Trapezoid ABCD has AD || BC, BD=1, angle DBA=23 degrees, and angle BDC=46 degrees. The ratio BC:AD is 9:5. What is the length of CD?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 6,
"tir": 9
} | 45 | 0 | 0 |
omr-ws-6bcc8fcc48e09a860172cf6df3024aeb3f17f756ce90f167fb749fad0000cfe3 | Trapezoid ABCD has bases AB and CD, with AB = 9 and DC = 12. The diagonals intersect at K, and the area of Triangle AKD is 24. What is the area of trapezoid ABCD? | [
"98"
] | [
"Trapezoid ABCD has bases AB and CD, with AB = 9 and DC = 12. The diagonals intersect at K, and the area of Triangle AKD is 24. What is the area of trapezoid ABCD?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-20835e97dca6b06be0facfd6245025b95479b9835dbd0c8a06fa001869e111d4 | Trapezoid ABCD is circumscribed about circle P. AB and CD are perpendicular to AD. The length of AB is 13 and the length of CD is 10. Find the perimeter of ABCD. | [
"46"
] | [
"Trapezoid ABCD is circumscribed about circle P. AB and CD are perpendicular to AD. The length of AB is 13 and the length of CD is 10. Find the perimeter of ABCD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-8d3fb9ecd703588ada00bcb7d934eaff3455e7ff248fac842ae7984cdc383d3d | Trapezoid ABCD is divided into four congruent trapezoids. Given AB = 4 and CD = 8, find the sum of the lengths of all line segments in the figure. | [
"30"
] | [
"Trapezoid ABCD is divided into four congruent trapezoids. Given AB = 4 and CD = 8, find the sum of the lengths of all line segments in the figure."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18a64e8cd6046aa7799c49607bd1edd496830a7ba789706ec25d729fd95e33d4 | Trapezoid ABCD is inscribed in a circle with bases AB = 10 and CD = 20, and side AD = $\sqrt{61}$. What is the area of triangle BCD? | [
"60"
] | [
"Trapezoid ABCD is inscribed in a circle with bases AB = 10 and CD = 20, and side AD = $\\sqrt{61}$. What is the area of triangle BCD?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-65515b6074b829f2ec99d46fd1b45c29d42ababf847d3ca52d5c4dc1fe179d86 | Trapezoid \( ABCD \) has \( AD \perp CD \) and \( \angle ADC = 90^\circ \). \( M \) is the midpoint of \( AB \) and \( CM = 6.5 \). Given that \( BC + CD + DA = 17 \), find the area of \( ABCD \). | [
"30"
] | [
"Trapezoid \\( ABCD \\) has \\( AD \\perp CD \\) and \\( \\angle ADC = 90^\\circ \\). \\( M \\) is the midpoint of \\( AB \\) and \\( CM = 6.5 \\). Given that \\( BC + CD + DA = 17 \\), find the area of \\( ABCD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-4f2af95adcf58487b9d32f32396bd7cafccf34995e02928efd6df215343ef8c6 | Trapezoid \( ABCD \) has sides \( AB = 92 \), \( BC = 50 \), \( CD = 19 \), \( AD = 70 \), and \( AB \) is parallel to \( CD \). A circle with center \( P \) on \( AB \) is drawn tangent to \( BC \) and \( AD \). Given that \( AP = \frac{m}{n} \) (where \( m \) and \( n \) are relatively prime), what is \( m + n \)? | [
"164"
] | [
"Trapezoid \\( ABCD \\) has sides \\( AB = 92 \\), \\( BC = 50 \\), \\( CD = 19 \\), \\( AD = 70 \\), and \\( AB \\) is parallel to \\( CD \\). A circle with center \\( P \\) on \\( AB \\) is drawn tangent to \\( BC \\) and \\( AD \\). Given that \\( AP = \\frac{m}{n} \\) (where \\( m \\) and \\( n \\) are relative... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-cd52b619cf930f6a9f374696857a6ed874ed01c47c3984b1c4506ddc10fb2cca | Trapezoid \(ABCD\) has \(AB \parallel CD\), \(AD = 17\), \(AB = 7\), \(BC = 18\), and \(CD = 14\). The median of \(ABCD\) is \(XY\), with \(X\) on side \(AD\) and \(Y\) on side \(BC\). The diagonals of the trapezoid intersect the median at points \(S\) and \(T\). What is the length of \(ST\)? | [
"\\(\\frac{7}{2}\\)"
] | [
"Trapezoid \\(ABCD\\) has \\(AB \\parallel CD\\), \\(AD = 17\\), \\(AB = 7\\), \\(BC = 18\\), and \\(CD = 14\\). The median of \\(ABCD\\) is \\(XY\\), with \\(X\\) on side \\(AD\\) and \\(Y\\) on side \\(BC\\). The diagonals of the trapezoid intersect the median at points \\(S\\) and \\(T\\). What is the length of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-9a37fffa04439266d29130d97a37ec531d0d18a652d3d2a74a7e5ffceba7a864 | Trapezoid \(ABCD\) has \(AB \parallel CD\). Suppose \(AC\) and \(BD\) intersect at point \(E\). If the area of triangle \(AEB\) is \(1\) and the area of triangle \(BEC\) is \(2\), compute the area of trapezoid \(ABCD\). | [
"9"
] | [
"Trapezoid \\(ABCD\\) has \\(AB \\parallel CD\\). Suppose \\(AC\\) and \\(BD\\) intersect at point \\(E\\). If the area of triangle \\(AEB\\) is \\(1\\) and the area of triangle \\(BEC\\) is \\(2\\), compute the area of trapezoid \\(ABCD\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3dadb270ed5a4834234a8cc9ffe516a8d365082370fafc27756f512cf8bbd5a4 | Trapezoid \(ABCD\) has \(AB \parallel DC\) and \(AB > DC\). A circle \(\Omega\) is tangent to \(AB\), \(BC\), and the arc \(BC\) of the circumcircle of \(\triangle ABC\). If the radius of \(\Omega\) is \(r\) and the radius of the incircle of \(\triangle ABC\) is \(r_I\), find the value of \(\frac{r}{r_I}\) given that \... | [
"\\dfrac{13}{9}"
] | [
"Trapezoid \\(ABCD\\) has \\(AB \\parallel DC\\) and \\(AB > DC\\). A circle \\(\\Omega\\) is tangent to \\(AB\\), \\(BC\\), and the arc \\(BC\\) of the circumcircle of \\(\\triangle ABC\\). If the radius of \\(\\Omega\\) is \\(r\\) and the radius of the incircle of \\(\\triangle ABC\\) is \\(r_I\\), find the value... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-7c43811eb28c258a76718e09a46df7a1add2db162b40f1bc0ce69f11b5f246b9 | Trapezoid \(ABCD\) has \(AB\) parallel to \(CD\), with \(\angle ADC = 90^\circ\). Given that \(AD = 5\), \(BC = 13\), and \(DC = 18\), compute the area of the trapezoid. | [
"60"
] | [
"Trapezoid \\(ABCD\\) has \\(AB\\) parallel to \\(CD\\), with \\(\\angle ADC = 90^\\circ\\). Given that \\(AD = 5\\), \\(BC = 13\\), and \\(DC = 18\\), compute the area of the trapezoid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1ed2dcc45883b53ed74f09af2a81201eb2a79089d68adad4276abdfe5639e403 | Trapezoid \(ABCD\) has \(AB\) parallel to \(DC\). \(E\) is the midpoint of \(BC\) and \(F\) is the midpoint of \(DA\). The area of quadrilateral \(ABEF\) is twice the area of quadrilateral \(FECD\). What is the ratio \(\frac{AB}{DC}\)? | [
"5"
] | [
"Trapezoid \\(ABCD\\) has \\(AB\\) parallel to \\(DC\\). \\(E\\) is the midpoint of \\(BC\\) and \\(F\\) is the midpoint of \\(DA\\). The area of quadrilateral \\(ABEF\\) is twice the area of quadrilateral \\(FECD\\). What is the ratio \\(\\frac{AB}{DC}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-0fc982d4f6970a0611664cd2d96ca2c6c9bce490be3e77222e4218b861257e27 | Trapezoid \(ABCD\) has \(\overline{AB} \parallel \overline{CD}\), \(BC = CD = 43\), and \(\overline{AD} \perp \overline{BD}\). Let \(O\) be the intersection of the diagonals \(\overline{AC}\) and \(\overline{BD}\), and let \(P\) be the midpoint of \(\overline{BD}\). Given that \(OP = 11\), the length \(AD\) can be writ... | [
"194"
] | [
"Trapezoid \\(ABCD\\) has \\(\\overline{AB} \\parallel \\overline{CD}\\), \\(BC = CD = 43\\), and \\(\\overline{AD} \\perp \\overline{BD}\\). Let \\(O\\) be the intersection of the diagonals \\(\\overline{AC}\\) and \\(\\overline{BD}\\), and let \\(P\\) be the midpoint of \\(\\overline{BD}\\). Given that \\(OP = 11... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-02c6033ff442758c42f9f1be6cf616dfa7af5357eb89c240bae6d52a76dd3959 | Trapezoid \(ABCD\) has \(\overline{AB} \parallel \overline{CD}\), \(BC = CD = 43\), and \(\overline{AD} \perp \overline{BD}\). Let \(O\) be the intersection of the diagonals \(\overline{AC}\) and \(\overline{BD}\), and let \(P\) be the midpoint of \(\overline{BD}\). Given that \(OP = 11\), the length of \(AD\) can be w... | [
"194"
] | [
"Trapezoid \\(ABCD\\) has \\(\\overline{AB} \\parallel \\overline{CD}\\), \\(BC = CD = 43\\), and \\(\\overline{AD} \\perp \\overline{BD}\\). Let \\(O\\) be the intersection of the diagonals \\(\\overline{AC}\\) and \\(\\overline{BD}\\), and let \\(P\\) be the midpoint of \\(\\overline{BD}\\). Given that \\(OP = 11... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b41f8c6d6ac3a01021f835bcee194b480b3cd227531d33ca2d1183800c88d9b | Trapezoid \(ABCD\) has bases \(AB\) and \(CD\) with \(AB < CD\), and non-parallel sides \(BC\) and \(DA\). Side \(BC\) is perpendicular to diagonal \(BD\). A perpendicular from \(A\) to diagonal \(BD\) intersects side \(CD\) at \(E\). If \(BD = DE = 36\) and \(BC = 27\), calculate \(AB\) and \(CD\). | [
"45"
] | [
"Trapezoid \\(ABCD\\) has bases \\(AB\\) and \\(CD\\) with \\(AB < CD\\), and non-parallel sides \\(BC\\) and \\(DA\\). Side \\(BC\\) is perpendicular to diagonal \\(BD\\). A perpendicular from \\(A\\) to diagonal \\(BD\\) intersects side \\(CD\\) at \\(E\\). If \\(BD = DE = 36\\) and \\(BC = 27\\), calculate \\(AB... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9972897a3a26a05081b27006492f97b49a98baad75d3c88ef3bf099d12abec13 | Trapezoid \(ABCD\) has bases \(AB\) and \(CD\), with \(DA = x\), \(BC = 2x\), and \(\angle DAB + \angle ABC = 120^\circ\). Calculate \(\angle DAB\). | [
"90^\\circ"
] | [
"Trapezoid \\(ABCD\\) has bases \\(AB\\) and \\(CD\\), with \\(DA = x\\), \\(BC = 2x\\), and \\(\\angle DAB + \\angle ABC = 120^\\circ\\). Calculate \\(\\angle DAB\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c9109d94aba7400afaccebebde65a55611b2e50f8ac5979473b26d4c523ad65 | Trapezoid \(ABCD\) has parallel sides \(AB\) and \(CD\), with \(BC\) perpendicular to them. Given \(AB = 13\), \(BC = 16\), and \(DC = 11\). Let \(E\) be the midpoint of \(AD\) and \(F\) the point on \(BC\) so that \(EF\) is perpendicular to \(AD\). Find the area of quadrilateral \(AEFB\). | [
"91"
] | [
"Trapezoid \\(ABCD\\) has parallel sides \\(AB\\) and \\(CD\\), with \\(BC\\) perpendicular to them. Given \\(AB = 13\\), \\(BC = 16\\), and \\(DC = 11\\). Let \\(E\\) be the midpoint of \\(AD\\) and \\(F\\) the point on \\(BC\\) so that \\(EF\\) is perpendicular to \\(AD\\). Find the area of quadrilateral \\(AEFB\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-fac459a54e7f549d9f0dd3fd2bcfa415695df5585ca26e13a529a5dace269faf | Trapezoid \(ABCD\) has sides \(AB = 92\), \(BC = 50\), \(CD = 19\), and \(AD = 70\) with \(AB\) parallel to \(CD\). A circle with center \(P\) on \(AB\) is drawn tangent to \(BC\) and \(AD\). Given that \(AP = \frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, find \(\frac{m+n}{4}\). | [
"41"
] | [
"Trapezoid \\(ABCD\\) has sides \\(AB = 92\\), \\(BC = 50\\), \\(CD = 19\\), and \\(AD = 70\\) with \\(AB\\) parallel to \\(CD\\). A circle with center \\(P\\) on \\(AB\\) is drawn tangent to \\(BC\\) and \\(AD\\). Given that \\(AP = \\frac{m}{n}\\), where \\(m\\) and \\(n\\) are coprime positive integers, find \\(... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e54fe73b37460e6b904b266d25beac74eea503a166b3018c5b2ab4da22e2eb0d | Trapezoid \(ABCD\) with \(AB \parallel CD\) has sides \(AB = 8\), \(BC = 5\), \(CD = 4\), and \(AD = 3\). Find the area of triangle \(CDE\), where \(E\) is the intersection point of the bisectors of angles \(ADC\) and \(BCD\). | [
"6"
] | [
"Trapezoid \\(ABCD\\) with \\(AB \\parallel CD\\) has sides \\(AB = 8\\), \\(BC = 5\\), \\(CD = 4\\), and \\(AD = 3\\). Find the area of triangle \\(CDE\\), where \\(E\\) is the intersection point of the bisectors of angles \\(ADC\\) and \\(BCD\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89875ea8c0f9346e5d71183a7f35c474c972cce06d287f3400897c6a0416fb67 | Trapezoids RSNM and MNPQ are similar with $\text{RS} = 3$, $\text{m}\angle\text{MRQ} = \text{m}\angle\text{NSP} = 90^\circ$, $\text{MQ} = \text{NP} = 5$. Find the area of quadrilateral $\text{RSNM}$ in square units. | [
"9"
] | [
"Trapezoids RSNM and MNPQ are similar with $\\text{RS} = 3$, $\\text{m}\\angle\\text{MRQ} = \\text{m}\\angle\\text{NSP} = 90^\\circ$, $\\text{MQ} = \\text{NP} = 5$. Find the area of quadrilateral $\\text{RSNM}$ in square units."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-842fc8ed6e7e3d85f6e8cc948600c431148d8fd0fe36fa6dc118595896554098 | Trapezoids \(ABCD\) and \(EFGH\) have side lengths satisfying the following equations: \[ AB = BC = CD = EH \] \[ BD = AC = EF = FG = GH \] \[ AD = EG = FH \] Given that the degree measure \(\angle BAD\) can be written in the form \(\frac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers, compute \(m... | [
"120"
] | [
"Trapezoids \\(ABCD\\) and \\(EFGH\\) have side lengths satisfying the following equations:\n\\[ AB = BC = CD = EH \\]\n\\[ BD = AC = EF = FG = GH \\]\n\\[ AD = EG = FH \\]\nGiven that the degree measure \\(\\angle BAD\\) can be written in the form \\(\\frac{m}{n}\\) where \\(m\\) and \\(n\\) are relatively prime p... | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-f9ba922da566bd09187b8f9d837f1b478f6920217aee8b4856208f13611d35d7 | Travis has to babysit the Thompson triplets and devises a counting game where Tadd starts by saying the number $1$, Todd says the next two numbers ($2$ and $3$), Tucker says the next three numbers ($4$, $5$, $6$), and so on, with each child saying one more number than the previous child did. The process continues to ro... | [
"5979"
] | [
"Travis has to babysit the Thompson triplets and devises a counting game where Tadd starts by saying the number $1$, Todd says the next two numbers ($2$ and $3$), Tucker says the next three numbers ($4$, $5$, $6$), and so on, with each child saying one more number than the previous child did. The process continues ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-bb2ea6115150b51f125091551eba04bf72fc12393c8d7374e47847d25c6c795f | Triangle $ \triangle MTE $ has vertices on the $ xy $-plane with $ MT = 9, ME = 12, TE = 15 $. Point $ P $ is in the first quadrant such that $ TP = EP = \alpha \cdot MP $. (a) Find the minimum value of $ \alpha $ so that this is possible. (b) Let $ \alpha = 1 $, so that $ TP = EP = MP $. The line through the incenter ... | [
"\\dfrac{27}{2} \\left( \\sqrt{5} - 2 \\right)"
] | [
"Triangle $ \\triangle MTE $ has vertices on the $ xy $-plane with $ MT = 9, ME = 12, TE = 15 $. Point $ P $ is in the first quadrant such that $ TP = EP = \\alpha \\cdot MP $.\n(a) Find the minimum value of $ \\alpha $ so that this is possible.\n(b) Let $ \\alpha = 1 $, so that $ TP = EP = MP $. The line through t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-61518df9226a335bd1b99d0098933e9a43ff1f0459e6f086a35ba185e2b0303c | Triangle $ABC$ ($AC > BC$) is inscribed in circle $\omega$. The angle bisector $CN$ of this triangle intersects $\omega$ at the point $M$ ($M \ne C$). An arbitrary point $T$ is marked on the segment $BN$. Let $H$ be the orthocenter of triangle $MNT$. The circumcircle of triangle $MNH$ intersects $\omega$ at point $R$ (... | [
"30"
] | [
"Triangle $ABC$ ($AC > BC$) is inscribed in circle $\\omega$. The angle bisector $CN$ of this triangle intersects $\\omega$ at the point $M$ ($M \\ne C$). An arbitrary point $T$ is marked on the segment $BN$. Let $H$ be the orthocenter of triangle $MNT$. The circumcircle of triangle $MNH$ intersects $\\omega$ at po... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-314315c50f5e2bcc6076b422a7455f31d2bfa01fd872e50caaea5ca2e6bdef62 | Triangle $ABC$ and $ABD$ are isosceles with $AB=AC=BD$, and $\overline{BD}$ intersects $\overline{AC}$ at $E$. If $\overline{BD}$ and $\overline{AC}$ are perpendicular, find $\angle{C}+\angle{D}$. | [
"\\(135^\\circ\\)"
] | [
"Triangle $ABC$ and $ABD$ are isosceles with $AB=AC=BD$, and $\\overline{BD}$ intersects $\\overline{AC}$ at $E$. If $\\overline{BD}$ and $\\overline{AC}$ are perpendicular, find $\\angle{C}+\\angle{D}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 8
} | 29 | 0 | 0 |
omr-ws-bdea335df6e61b7d238d6a67e5008e1f2a4911061a3ee5e7e813381fff87fa65 | Triangle $ABC$ and a straight line $l$ are given in the plane. Construct using a compass and a ruler the straight line which is parallel to $l$ and bisects the area of triangle $ABC$. | [
"\\text{A line parallel to } l \\text{ constructed by the geometric mean of the triangle's height perpendicular to } l"
] | [
"Triangle $ABC$ and a straight line $l$ are given in the plane. Construct using a compass and a ruler the straight line which is parallel to $l$ and bisects the area of triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92bcf09e7dcd35cb2f50532fda8fbe4a3896097b40ccebbfa587b00b2ac99c44 | Triangle $ABC$ circumscribed by circle $(O)$ has an $A$-excircle $(J)$ that touches $AB$, $BC$, and $AC$ at points $F$, $D$, and $E$, respectively. Let $EF \cap BC = \{G\}$ and $GJ$ intersects $AB$ and $AC$ at points $M$ and $N$, respectively. Points $P$ and $Q$ are on $JB$ and $JC$ such that $\angle PAB = \angle QAC =... | [
"90"
] | [
"Triangle $ABC$ circumscribed by circle $(O)$ has an $A$-excircle $(J)$ that touches $AB$, $BC$, and $AC$ at points $F$, $D$, and $E$, respectively. Let $EF \\cap BC = \\{G\\}$ and $GJ$ intersects $AB$ and $AC$ at points $M$ and $N$, respectively. Points $P$ and $Q$ are on $JB$ and $JC$ such that $\\angle PAB = \\a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1684546bb3d7b9348c3d8ecce99db7adee905c4b7f9eaf86a15c5d14ded7b770 | Triangle $ABC$ circumscribed by circle $(O)$ has an $A$-excircle $(J)$ that touches $AB$, $BC$, and $AC$ at points $F$, $D$, and $E$, respectively. Let $L$ be the midpoint of $BC$. The circle with diameter $LJ$ intersects $DE$ and $DF$ at points $K$ and $H$, respectively. Let $P$ be the intersection of circles $(BDK)$ ... | [
"r"
] | [
"Triangle $ABC$ circumscribed by circle $(O)$ has an $A$-excircle $(J)$ that touches $AB$, $BC$, and $AC$ at points $F$, $D$, and $E$, respectively. Let $L$ be the midpoint of $BC$. The circle with diameter $LJ$ intersects $DE$ and $DF$ at points $K$ and $H$, respectively. Let $P$ be the intersection of circles $(B... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-40dea05f6c941c0fcc128d228a32b6fc96632a10e147017d079e98e83f0c66c1 | Triangle $ABC$ has $AB < AC$, and its perimeter is equal to $7 \cdot BC$. The incircle of triangle $ABC$ touches $BC$ at $E$, and the diameter $DE$ of the incircle is drawn, cutting the median from $A$ to $BC$ at $F$. Find $\frac{DC}{FE}$ and express your answer as a reduced fraction. | [
"\\dfrac{12}{5}"
] | [
"Triangle $ABC$ has $AB < AC$, and its perimeter is equal to $7 \\cdot BC$. The incircle of triangle $ABC$ touches $BC$ at $E$, and the diameter $DE$ of the incircle is drawn, cutting the median from $A$ to $BC$ at $F$. Find $\\frac{DC}{FE}$ and express your answer as a reduced fraction."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45132f4d22fdbf8ef6cee7717af63112e030a6581c045cb8ac21fe8ed8f7c6ef | Triangle $ABC$ has $AB = 13$, $BC = 14$, and $AC = 15$. The points $D$, $E$, and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | [
"\\dfrac{195}{8}"
] | [
"Triangle $ABC$ has $AB = 13$, $BC = 14$, and $AC = 15$. The points $D$, $E$, and $F$ are the midpoints of $\\overline{AB}$, $\\overline{BC}$, and $\\overline{AC}$, respectively. Let $X \\neq E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. What is $XA + XB + XC$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6beae73901a882d05813f96403f41ce50cd4d4b709ea592171ee99ac45c3f231 | Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. Point $D$ is the foot of the altitude from $A$ to $BC$, and points $E$ and $F$ are the feet of the altitudes from $D$ to $AB$ and $AC$, respectively. Compute the length of $EF$. | [
"\\(\\frac{168\\sqrt{22}}{65}\\)"
] | [
"Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. Point $D$ is the foot of the altitude from $A$ to $BC$, and points $E$ and $F$ are the feet of the altitudes from $D$ to $AB$ and $AC$, respectively. Compute the length of $EF$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a57bbd5943bf6edf9900ca3efa080b20e7e118608da370324cbe7bbafd09440f | Triangle $ABC$ has $AB = 14\sqrt{3}$. Let $D$ be the intersection of the angle bisector from $A$ to $BC$. Given that $AD = 22$ and the perpendicular bisector of $AB$ intersects the altitude from $B$ to $AC$ on line $AD$, find $AC$. | [
"\\( AC = \\frac{77\\sqrt{3}}{5} \\)"
] | [
"Triangle $ABC$ has $AB = 14\\sqrt{3}$. Let $D$ be the intersection of the angle bisector from $A$ to $BC$. Given that $AD = 22$ and the perpendicular bisector of $AB$ intersects the altitude from $B$ to $AC$ on line $AD$, find $AC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-2deb83b76eb362fe4954fa9b956df6f1c31f3394684f123abb226a98fc05ee4c | Triangle $ABC$ has $AB = 17$, $BC = 25$, and $CA = 28$. Let $X$ be the point such that $AX \perp AC$ and $AC \parallel BX$. Let $Y$ be the point such that $BY \perp BC$ and $AY \parallel BC$. Find the area of $ABXY$. | [
"$\\frac{756}{10}$"
] | [
"Triangle $ABC$ has $AB = 17$, $BC = 25$, and $CA = 28$. Let $X$ be the point such that $AX \\perp AC$ and $AC \\parallel BX$. Let $Y$ be the point such that $BY \\perp BC$ and $AY \\parallel BC$. Find the area of $ABXY$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-01d551bdae4d325c179fdfe0723e7ff0429cd3075e7bd17a56f5ab8d755a107c | Triangle $ABC$ has $AB = 20\sqrt{2}$, $BC = 28$, and $\angle ABC = 135^\circ$. If $H$ is the foot of the altitude from $A$ and $M$ is the midpoint of $AC$, calculate $HM$. | [
"26"
] | [
"Triangle $ABC$ has $AB = 20\\sqrt{2}$, $BC = 28$, and $\\angle ABC = 135^\\circ$. If $H$ is the foot of the altitude from $A$ and $M$ is the midpoint of $AC$, calculate $HM$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c51c95055b20ca90c88da5da2945083e66e520abcf31aa11d9f5c70e05f51ca | Triangle $ABC$ has $AB = 25$, $BC = 17$, and $AC = 26$. An infinitely small object is launched from an arbitrary point in the triangle and bounces infinitely against the walls of the triangle, eventually converging to a triangle $A_1B_1C_1$ with points $A_1$, $B_1$, and $C_1$ on sides $BC$, $AC$, and $AB$ respectively.... | [
"11"
] | [
"Triangle $ABC$ has $AB = 25$, $BC = 17$, and $AC = 26$. An infinitely small object is launched from an arbitrary point in the triangle and bounces infinitely against the walls of the triangle, eventually converging to a triangle $A_1B_1C_1$ with points $A_1$, $B_1$, and $C_1$ on sides $BC$, $AC$, and $AB$ respecti... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-256bf46c425d62734708c238a4ff45f476693594276884113b28657947346b04 | Triangle $ABC$ has $AB = 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m$ and $n$. Find the sum of the digits of $m + n$... | [
"15"
] | [
"Triangle $ABC$ has $AB = 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $BMC$ at points $X$ and $Y$, then $XY^2 = \\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$. Find the sum of the digits of $... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0f561c1272ddd504129798c9bbfff85740532f0720b2424e9d250b771dfc1b0a | Triangle $ABC$ has $AB = 3$, $BC = 4$, and $AC = 6$. Let $M$ denote the midpoint of $\overline{BC}$. The incircle of $\triangle ABC$ is tangent to $\overline{AC}$ at $N$. Segments $\overline{AM}$ and $\overline{BN}$ intersect at $O$. What is the ratio of the area of quadrilateral $MONC$ to the area of $\triangle ABC$? | [
"\\(\\frac{35}{88}\\)"
] | [
"Triangle $ABC$ has $AB = 3$, $BC = 4$, and $AC = 6$. Let $M$ denote the midpoint of $\\overline{BC}$. The incircle of $\\triangle ABC$ is tangent to $\\overline{AC}$ at $N$. Segments $\\overline{AM}$ and $\\overline{BN}$ intersect at $O$. What is the ratio of the area of quadrilateral $MONC$ to the area of $\\tria... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77858c8fc8b616b04735a7196147143feceafd92214cf65fab1eace87ee51dd1 | Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$. | [
"78"
] | [
"Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-b7bce84ed6ac5efba35a4440b21973c9c0357cc059656201b9adc7001260eb1a | Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \frac{AM}{CM}$? | [
"100"
] | [
"Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \\frac{AM}{CM}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6059e150152df0bec274301c6d265f4ddf37a29f8700a0ba5d4b736d07f8e792 | Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$. | [
"744"
] | [
"Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\\sin A = \\tfrac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 17
} | 35 | 0 | 0 |
omr-ws-d339eb02edbd39add9401523266e07a5aa1979000371c069a79c518776249e49 | Triangle $ABC$ has $AB = 58$, $BC = 40$, and $AC = 42$. Point $R$ is the midpoint of $AC$, and $E$ is the foot of the altitude to $AB$ from $C$. Find the length $ER$. | [
"21"
] | [
"Triangle $ABC$ has $AB = 58$, $BC = 40$, and $AC = 42$. Point $R$ is the midpoint of $AC$, and $E$ is the foot of the altitude to $AB$ from $C$. Find the length $ER$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-406178fd4183d61d5a5a8e046ca1d57fff879b0d14d8b23de5dc1c61a8cfb416 | Triangle $ABC$ has $AB = 8$, $AC = 9$, and circumcircle $\omega$ centered at $O$. The circumcircle of triangle $BOC$ meets lines $AB$ and $AC$ for a second time at points $P$ and $Q$. If $PQ$ is tangent to $\omega$, find $BC$. | [
"\\(\\sqrt{145}\\)"
] | [
"Triangle $ABC$ has $AB = 8$, $AC = 9$, and circumcircle $\\omega$ centered at $O$. The circumcircle of triangle $BOC$ meets lines $AB$ and $AC$ for a second time at points $P$ and $Q$. If $PQ$ is tangent to $\\omega$, find $BC$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f99e77fbb31746789777cada103c1e49f417daa72c3b7b4cac2c1b69461312da | Triangle $ABC$ has $AB = AC = 12$. Let $P$ be a point on $BC$ such that $AP = 8$. Find the product $PB \cdot PC$. | [
"80"
] | [
"Triangle $ABC$ has $AB = AC = 12$. Let $P$ be a point on $BC$ such that $AP = 8$. Find the product $PB \\cdot PC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a56f8b4c422115b8eb4912d1cf16f8d6e9b6a9d4ed3ac80463115f2a90da42c | Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE$? | [
"\\( 12\\sqrt{2} \\)"
] | [
"Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. Let $P$ be the point on $\\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-c2a238508df5d23eb7927af032c914fccc1d97aef24649280d39310860927dba | Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | [
"\\(\\frac{195}{8}\\)"
] | [
"Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\\overline{AB}$, $\\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\ne E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. What is $XA + XB + XC$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-fbd99a8172f137c6f77a67f43cb5d89741161fc4198d84621290872ccb6fae13 | Triangle $ABC$ has $AB=13, BC=14,$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE$? | [
"12\\sqrt{2}"
] | [
"Triangle $ABC$ has $AB=13, BC=14,$ and $AC=15$. Let $P$ be the point on $\\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4fecd58c2175baf1a4d85907b60acf8e52f1174bb73940099a56a941d6b0b17e | Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$? | [
"\\( 90^\\circ \\)"
] | [
"Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-5126e5ad709abf354933037e8181f8e2d7a7d6fe5ffbc77b69fc1047f60d66db | Triangle $ABC$ has $AB=20$, $BC=15$, and $CA=7$. Let there be a point $D$ such that $C$ and $D$ are on opposite sides of $AB$, and that $AD=13$ and $BD=11$. Let there be a point $E$ such that $B$ and $E$ are on opposite sides of $AD$, and that $AE=12$ and $DE=5$. Call the incenters of triangles $ABC$, $ABD$, and $ADE$ ... | [
"18"
] | [
"Triangle $ABC$ has $AB=20$, $BC=15$, and $CA=7$. Let there be a point $D$ such that $C$ and $D$ are on opposite sides of $AB$, and that $AD=13$ and $BD=11$. Let there be a point $E$ such that $B$ and $E$ are on opposite sides of $AD$, and that $AE=12$ and $DE=5$. Call the incenters of triangles $ABC$, $ABD$, and $... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-2be89e89628efb67e51a53f2868fb8c225e2bfb7afeb184b2550f5df9a68c2c8 | Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive... | [
"923"
] | [
"Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4338ad667cd30d2afc3ee441e219034aa7d3d6a853e963c85171a6951e7319e | Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive... | [
"923"
] | [
"Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively p... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-4602994cbfee43e2950100c49bf3b62300b858f74a6fac538382966a5c53a8c8 | Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE = \frac{m}{n}$, where $m$ and $n$ are relatively pri... | [
"923"
] | [
"Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE = \\frac{m}{n}$, where $m$ and $n$ are rela... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-da525adc5101d47344e06b4f86efd0d37bb9be1806daf2a8bb431b92c4a27295 | Triangle $ABC$ has $AB=40$, $AC=31$, and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$. | [
"744"
] | [
"Triangle $ABC$ has $AB=40$, $AC=31$, and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 16
} | 34 | 0 | 0 |
omr-ws-12008885d13e2e8a3957b541c612113fde07d101fc65e1e7893c4d15ac6cf957 | Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$. | [
"744"
] | [
"Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$."
] | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-481f33e54c1a5de75f4cf8f6639634a0c8aaaa39e961ea64c3042282d6a083cb | Triangle $ABC$ has $AB=6$, $BC=8$, and $AC=10$. A semicircle is inscribed in $ABC$ such that it is tangent to $AC$ and the endpoints $E$ and $F$ of its diameter lie on $AB$ and $BC$ respectively. Let $X$ be the midpoint of segment $EF$. If $XA=XC$, what is the area of the semicircle? | [
"\\(\\frac{15625 \\pi}{4608}\\)"
] | [
"Triangle $ABC$ has $AB=6$, $BC=8$, and $AC=10$. A semicircle is inscribed in $ABC$ such that it is tangent to $AC$ and the endpoints $E$ and $F$ of its diameter lie on $AB$ and $BC$ respectively. Let $X$ be the midpoint of segment $EF$. If $XA=XC$, what is the area of the semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-f73a72dd9590f8a3ede86c45231ee17be05d6c91817ad3cf072cc0f4ddad3ed1 | Triangle $ABC$ has $AB=7$, $BC=8$, and $AC=9$. Point $D$ is on side $BC$ such that the sum of the incircles of triangles $ABD$ and $ADC$ is minimal. Determine the length of $AD$. | [
"4\\sqrt{3}"
] | [
"Triangle $ABC$ has $AB=7$, $BC=8$, and $AC=9$. Point $D$ is on side $BC$ such that the sum of the incircles of triangles $ABD$ and $ADC$ is minimal. Determine the length of $AD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-dabecaab8334612e47f792aeed41e086c5636c9476dad5d48b35d3b8a7401c0f | Triangle $ABC$ has $AB=7$, $BC=8$, and $AC=9$. Point $D$ is on side $BC$ such that the sum of the incircles of triangles $ABD$ and $ADC$ is minimal. Determine the minimal such sum of the two incircles. | [
"\\sqrt{5}"
] | [
"Triangle $ABC$ has $AB=7$, $BC=8$, and $AC=9$. Point $D$ is on side $BC$ such that the sum of the incircles of triangles $ABD$ and $ADC$ is minimal. Determine the minimal such sum of the two incircles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-ec7c079ebb54d13eba3a9aa66e4564c287b8782d3a5ae9a4ab4b34d7150c39df | Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What is the largest area that this triangle can have? | [
"820"
] | [
"Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What is the largest area that this triangle can have?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-871cf6651ba336923792950029ef77a3acbfa100d50b5f672d9ea0cd0ec93153 | Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have? | [
"820"
] | [
"Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ea257c6cceb98fc9278d6ad674893e6252e980f9c249b7cba90c33768ba96a53 | Triangle $ABC$ has $AB=9$ and the ratio $BC: AC=40: 41$. What is the largest possible area of this triangle? | [
"820"
] | [
"Triangle $ABC$ has $AB=9$ and the ratio $BC: AC=40: 41$. What is the largest possible area of this triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-83abb66417165b2aee7ff3b4de91dca9d5bf80e993f0ed97648d2839745d008a | Triangle $ABC$ has $AB=AC$, circumcircle $\Gamma$ with center $O$, and incircle $\omega$ with center $I$. Let $EF$ be a variable chord of $\Gamma$ parallel to $BC$. We construct $D=AI\cap \Gamma$, $G=DE\cap BC$, and $H=FI\cap \Gamma$. Let $\ell$ be the line through $G$ perpendicular to $OH$. If $T$ is the point of tang... | [
"r"
] | [
"Triangle $ABC$ has $AB=AC$, circumcircle $\\Gamma$ with center $O$, and incircle $\\omega$ with center $I$. Let $EF$ be a variable chord of $\\Gamma$ parallel to $BC$. We construct $D=AI\\cap \\Gamma$, $G=DE\\cap BC$, and $H=FI\\cap \\Gamma$. Let $\\ell$ be the line through $G$ perpendicular to $OH$. If $T$ is the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94b41809f0971377e9004b2a8c7366df624ddd67a488c6b25bd77e6a09df6f1e | Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ f... | [
"072"
] | [
"Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overline{CL}$, and let $M$ be the point on l... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-1e0de2904628422e3272a2368815af00b7bf581341746808d58022febdcb7e1c | Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ABC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $\frac{r_a}{r_b}$? | [
"\\(\\frac{4+\\sqrt{2}}{4}\\)"
] | [
"Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ABC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $\\frac{r_a}{r_b}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1d436c1d8bb0fb4aa1494d9e89aadee680f509d79c03177b98a044e5670069c8 | Triangle $ABC$ has $BC<AC$ and circumradius $8$. Let $O$ be the circumcenter of $\triangle ABC$, $M$ be the midpoint of minor arc $AB$, and $C'$ be the reflection of $C$ across $OM$. If $AB$ bisects $\angle OAM$, and $\angle COC' = 120^\circ$, find the square of the area of the convex pentagon $CC'AMB$. | [
"19200"
] | [
"Triangle $ABC$ has $BC<AC$ and circumradius $8$. Let $O$ be the circumcenter of $\\triangle ABC$, $M$ be the midpoint of minor arc $AB$, and $C'$ be the reflection of $C$ across $OM$. If $AB$ bisects $\\angle OAM$, and $\\angle COC' = 120^\\circ$, find the square of the area of the convex pentagon $CC'AMB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-77159f3ce75b144ae09a5e66e664adbaf8f775fcfdaf2f3671b5aa6d801b86ce | Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\angle A$? | [
"\\(30^\\circ\\)"
] | [
"Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\\angle A$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-25b5ec1511dea655b28bda7194a5137174c588ea5dc82ea46fd107c9239bde8a | Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\angle{A}$? | [
"\\(30^\\circ\\)"
] | [
"Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\\angle{A}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4c6434eb30f5fcffb35948cc0d4f51e8808c517ae6f511234cfc03eb98d3e3ba | Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. Compute the area of $ABC$. | [
"100\\sqrt{3}"
] | [
"Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. Compute the area of $ABC$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-c3909b80b091b82f5bcf0169ccbcd356bc238657cd16bce70de33a98c0abd803 | Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$. | [
"38"
] | [
"Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-1bef7c36fcbff06e9e2f3929421fde15582bc692d1d8a70b9d3f3e860b48420b | Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$ | [
"038",
"38"
] | [
"Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$",
"Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects th... | [
"MATH_training_set",
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"cot",
"genselect"
] | {
"additional_problems": 1,
"cot": 1,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-8fd4ff130ee616a4a6b1bc40008f237c98515b42bacea76123175ffd4215afa2 | Triangle $ABC$ has $\angle A = 67^\circ$ and $\angle B = 79^\circ$. Let $P$ be on side $AB$, $Q$ on side $BC$, and $R$ on side $CA$ such that the angles $\angle APR = \angle BPQ$, $\angle BQP = \angle CQR$, and $\angle CRQ = \angle ARP$. Calculate the angles of triangle $PQR$. | [
"22^\\circ"
] | [
"Triangle $ABC$ has $\\angle A = 67^\\circ$ and $\\angle B = 79^\\circ$. Let $P$ be on side $AB$, $Q$ on side $BC$, and $R$ on side $CA$ such that the angles $\\angle APR = \\angle BPQ$, $\\angle BQP = \\angle CQR$, and $\\angle CRQ = \\angle ARP$. Calculate the angles of triangle $PQR$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-570e44ee14d7c19858c265a3799acd764afc68b63457d4090e1bb9f03e48d8b2 | Triangle $ABC$ has $\angle A = 73^\circ$, and $\angle C = 38^\circ$. Let $M$ be the midpoint of $AC$, and let $X$ be the reflection of $C$ over $BM$. If $E$ and $F$ are the feet of the altitudes from $A$ and $C$, respectively, then what is the measure, in degrees, of $\angle EXF$? | [
"21"
] | [
"Triangle $ABC$ has $\\angle A = 73^\\circ$, and $\\angle C = 38^\\circ$. Let $M$ be the midpoint of $AC$, and let $X$ be the reflection of $C$ over $BM$. If $E$ and $F$ are the feet of the altitudes from $A$ and $C$, respectively, then what is the measure, in degrees, of $\\angle EXF$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-f36116d3b0425dd44b5636b6aca306a0baf2f82af109829da2c041d8ba9fc749 | Triangle $ABC$ has $\angle A=90^\circ$, $\angle B=60^\circ$, and $AB=8$. A point $P$ is chosen inside the triangle. The interior angle bisectors $\ell_A$, $\ell_B$, and $\ell_C$ of respective angles $PAB$, $PBC$, and $PCA$ intersect pairwise at $X=\ell_A\cap\ell_B$, $Y=\ell_B\cap\ell_C$, and $Z=\ell_C\cap\ell_A$. If tr... | [
"85"
] | [
"Triangle $ABC$ has $\\angle A=90^\\circ$, $\\angle B=60^\\circ$, and $AB=8$. A point $P$ is chosen inside the triangle. The interior angle bisectors $\\ell_A$, $\\ell_B$, and $\\ell_C$ of respective angles $PAB$, $PBC$, and $PCA$ intersect pairwise at $X=\\ell_A\\cap\\ell_B$, $Y=\\ell_B\\cap\\ell_C$, and $Z=\\ell_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-8bfc810199810a6c5ab8cd185ef327207f0daa765434a093cb54cb0cfe667936 | Triangle $ABC$ has $\angle ABC = 75^\circ$ and $\angle BAC = 45^\circ$. $\overline{AB} = 10$. Point $D$ is on $\overline{AB}$, and point $E$ is on $\overline{BC}$ such that $DE \parallel AC$. If $\overline{BD} = 6$, find the ratio of the area of triangle $DBE$ to the area of triangle $ABC$. | [
"$\\frac{9}{25}$"
] | [
"Triangle $ABC$ has $\\angle ABC = 75^\\circ$ and $\\angle BAC = 45^\\circ$. $\\overline{AB} = 10$. Point $D$ is on $\\overline{AB}$, and point $E$ is on $\\overline{BC}$ such that $DE \\parallel AC$. If $\\overline{BD} = 6$, find the ratio of the area of triangle $DBE$ to the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6f4232fce0ab27a2c692c5f37e2255aecb1085d5f5f5c17a0594421567c5ec3a | Triangle $ABC$ has $\angle ABC = 90^{\circ}$ and $2BC = AC$. Let $P$ be a point in $\triangle ABC$ such that $\angle BPA = 120^{\circ}$ and $\frac{PB}{PA} = \frac{1}{27}$. If the value of $\frac{PC^2}{PB^2}$ can be written as $\frac{m}{n}$ with $m$ and $n$ coprime positive integers, find $m+n$. | [
"682"
] | [
"Triangle $ABC$ has $\\angle ABC = 90^{\\circ}$ and $2BC = AC$. Let $P$ be a point in $\\triangle ABC$ such that $\\angle BPA = 120^{\\circ}$ and $\\frac{PB}{PA} = \\frac{1}{27}$. If the value of $\\frac{PC^2}{PB^2}$ can be written as $\\frac{m}{n}$ with $m$ and $n$ coprime positive integers, find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-3763034c89cf37d6a84630e2d09f5a87b307d1b3e341ba4ac462c663161adcc0 | Triangle $ABC$ has $\angle BAC = 30^\circ$ and $AB = AC = 2$. A semicircle with diameter $BC$ is drawn. Find the probability that a randomly chosen point inside the triangle is in the semicircle. | [
"\\(\\left(2 - \\sqrt{3}\\right)\\left(\\frac{\\pi}{3} + \\frac{1}{2}\\right)\\)"
] | [
"Triangle $ABC$ has $\\angle BAC = 30^\\circ$ and $AB = AC = 2$. A semicircle with diameter $BC$ is drawn. Find the probability that a randomly chosen point inside the triangle is in the semicircle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-da3255f03c942b8d6efc96e855315cbf87dfbe0b31695db4a250ac1e0eb379d8 | Triangle $ABC$ has $\angle BAC = 60^\circ$. Let $P$ be the intersection point of the bisector of $\angle ABC$ with side $AC$, and let $Q$ be the intersection point of the bisector of $\angle ACB$ with side $AB$. Let $R$ be the symmetric point to $A$ with respect to the straight line $PQ$. If $R$ lies on the straight li... | [
"\\dfrac{2(\\sqrt{7} - 1)}{9}"
] | [
"Triangle $ABC$ has $\\angle BAC = 60^\\circ$. Let $P$ be the intersection point of the bisector of $\\angle ABC$ with side $AC$, and let $Q$ be the intersection point of the bisector of $\\angle ACB$ with side $AB$. Let $R$ be the symmetric point to $A$ with respect to the straight line $PQ$. If $R$ lies on the st... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8277b2515b2469b27b6f09aa1d31ca5a1aeea94e0c7b655186552780eb61a1a | Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? | [
"$80^\\circ$",
"90^\\circ"
] | [
"Triangle $ABC$ has $\\angle BAC = 60^{\\circ}$, $\\angle CBA \\leq 90^{\\circ}$, $BC=1$, and $AC \\geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\\angle CBA$?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-177081288eabfd699018db65ab8f33d63a27ddcd546499c298dbf66f8c843ab9 | Triangle $ABC$ has $\angle BAC = 80^\circ$ and $\angle ACB = 40^\circ$. Point $D$ is on the ray $BC$ beyond $C$ such that $CD = AB + BC + CA$. Find the measure of $\angle ADB$. | [
"\\(10^\\circ\\)"
] | [
"Triangle $ABC$ has $\\angle BAC = 80^\\circ$ and $\\angle ACB = 40^\\circ$. Point $D$ is on the ray $BC$ beyond $C$ such that $CD = AB + BC + CA$. Find the measure of $\\angle ADB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-b03d2e5bf3fb7ea2f3fb21333c9c91c6209de2d2d55d004fb713011a77fa4e62 | Triangle $ABC$ has a circumcircle $O$. Construct circle $O'$ such that $O'$ is internally tangent to $O$ at $A$ and $O'$ is tangent to $BC$. Determine the radius of $O'$ given that $AB = 6$, $BC = 10$, $AC = 8$. | [
"\\(\\frac{120}{49}\\)"
] | [
"Triangle $ABC$ has a circumcircle $O$. Construct circle $O'$ such that $O'$ is internally tangent to $O$ at $A$ and $O'$ is tangent to $BC$. Determine the radius of $O'$ given that $AB = 6$, $BC = 10$, $AC = 8$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 27
} | 74 | 0 | 0 |
omr-ws-507509bf79534d2ca5ec11007b1aecaa52212be1f127f67b8a69c9d11de5563c | Triangle $ABC$ has a circumcircle $O$. Construct circle $O'$ such that $O'$ is internally tangent to $O$ at $A$ and $O'$ is tangent to $BC$. Determine the radius of $O'$ given that $AB=6$, $BC=10$, $AC=8$. Express your answer as an improper fraction reduced to lowest terms. | [
"\\dfrac{120}{49}"
] | [
"Triangle $ABC$ has a circumcircle $O$. Construct circle $O'$ such that $O'$ is internally tangent to $O$ at $A$ and $O'$ is tangent to $BC$. Determine the radius of $O'$ given that $AB=6$, $BC=10$, $AC=8$. Express your answer as an improper fraction reduced to lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f58b5c9ad638d4e5f7260791ac32494fbf94368e56f6b1b9f61d3130b88b837e | Triangle $ABC$ has a circumcircle of radius $1$. If the two medians of triangle $ABC$ each have a length of $1$, then the perimeter of the triangle $ABC$ is ... | [
"\\dfrac{2\\sqrt{2} + \\sqrt{7}}{2}"
] | [
"Triangle $ABC$ has a circumcircle of radius $1$. If the two medians of triangle $ABC$ each have a length of $1$, then the perimeter of the triangle $ABC$ is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de46412b055a01acd8b0180dcdb1891e5128d74e47d88d95e58da4eb4b4f823d | Triangle $ABC$ has a circumradius of $201$ and an inradius of $6$. Three circles, each with radius $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, meet at exactly one point and are each tangent to two distinct sides of triangle $ABC$. Find $p + q$. | [
"467"
] | [
"Triangle $ABC$ has a circumradius of $201$ and an inradius of $6$. Three circles, each with radius $\\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, meet at exactly one point and are each tangent to two distinct sides of triangle $ABC$. Find $p + q$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0fc231d051ebe07c1f27065a7f3b8bd57e8bdaddc0635c7594678a92e9f20bc7 | Triangle $ABC$ has a perimeter of $4$. Points $X$ and $Y$ lie on rays $AB$ and $AC$, respectively, such that $AX = AY = 1$. Segments $BC$ and $XY$ intersect at point $M$. Let $P_1$ be the perimeter of $\triangle ABM$ and $P_2$ be the perimeter of $\triangle ACM$. Calculate the value of $P_1 + P_2$. | [
"6"
] | [
"Triangle $ABC$ has a perimeter of $4$. Points $X$ and $Y$ lie on rays $AB$ and $AC$, respectively, such that $AX = AY = 1$. Segments $BC$ and $XY$ intersect at point $M$. Let $P_1$ be the perimeter of $\\triangle ABM$ and $P_2$ be the perimeter of $\\triangle ACM$. Calculate the value of $P_1 + P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-18ec4a643cef793a9db59dc0edf2d3716bbd36f56e8ea8c2c166371355792545 | Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \leq AC$. What is the smallest possible value of $BC - AB$? | [
"1"
] | [
"Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \\leq AC$. What is the smallest possible value of $BC - AB$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0aee476589e6dbffa2e2587fb821cdbb9751c40fca656693dfa4a5c325d1d40b | Triangle $ABC$ has a right angle at $A$. Point $P$ lies inside triangle $ABC$, and perpendiculars $PA'$, $PB'$, and $PC'$ are dropped to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ respectively. If $PA'=3$, $PB'=4$, and $PC'=5$, what is the maximum possible area of $A'B'C'$? | [
"\\dfrac{20 + 3\\sqrt{41}}{2}"
] | [
"Triangle $ABC$ has a right angle at $A$. Point $P$ lies inside triangle $ABC$, and perpendiculars $PA'$, $PB'$, and $PC'$ are dropped to $\\overline{BC}$, $\\overline{AC}$, and $\\overline{AB}$ respectively. If $PA'=3$, $PB'=4$, and $PC'=5$, what is the maximum possible area of $A'B'C'$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-677866ee79d108672c9af8be1993f9cc5ec792daef63fa467c0540f49b5468d6 | Triangle $ABC$ has a right angle at $B$ and contains a point $P$ such that $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find the length of $PC$. | [
"33"
] | [
"Triangle $ABC$ has a right angle at $B$ and contains a point $P$ such that $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find the length of $PC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 1
} | 33 | 0 | 0 |
omr-ws-a4058ab8b41f04d18566a034ba8b2fe2b5aaae866e9d9c5d9aa14f90dfb33a3c | Triangle $ABC$ has a right angle at $B$ with $AB=3$ and $BC=4$. A point $P$ exists inside the triangle such that $AP=x+2$, $BP=x+1$, and $CP=x+3$. Find $x$. | [
"\\(\\frac{6}{23}\\)"
] | [
"Triangle $ABC$ has a right angle at $B$ with $AB=3$ and $BC=4$. A point $P$ exists inside the triangle such that $AP=x+2$, $BP=x+1$, and $CP=x+3$. Find $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0cb7ef0d4362aae789ae307a90e4829f42072da9e36aea3ddecd463dccb2cd42 | Triangle $ABC$ has a right angle at $B$. Points $D$ and $E$ are chosen on $\overline{AC}$ and $\overline{BC}$, respectively, such that $AB = BE = ED = DC = 2$. Find the area of $\triangle CDE$. | [
"\\(\\sqrt{5} - 1\\)"
] | [
"Triangle $ABC$ has a right angle at $B$. Points $D$ and $E$ are chosen on $\\overline{AC}$ and $\\overline{BC}$, respectively, such that $AB = BE = ED = DC = 2$. Find the area of $\\triangle CDE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-f4dec3871169d7535af0c78b899f3031a78c6800a5d4faa03bf97e89d59258d3 | Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then what is $\tan B$? | [
"\\(\\frac{\\sqrt{5}}{2}\\)"
] | [
"Triangle $ABC$ has a right angle at $C$. If $\\sin A = \\frac{2}{3}$, then what is $\\tan B$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8f39c7f6abb4227eff1459b053bd351e79e67cc39e5baba6d439baf43da2ff41 | Triangle $ABC$ has a right angle at $C$. Point $P$ lies on side $AB$, and segment $CP$ intersects the incircle of triangle $ABC$ at points $X$ and $Y$, with $X$ closer to $C$ than $Y$. If $CX = XY = YP = 1$, compute the area of triangle $ABC$. | [
"\\(\\frac{126+48\\sqrt{7}}{7}\\)"
] | [
"Triangle $ABC$ has a right angle at $C$. Point $P$ lies on side $AB$, and segment $CP$ intersects the incircle of triangle $ABC$ at points $X$ and $Y$, with $X$ closer to $C$ than $Y$. If $CX = XY = YP = 1$, compute the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-ae91000fad6c1e9c1fb3738deb2235c7e766a1ed2ddef7b29a41455942f01b8e | Triangle $ABC$ has a right angle at $C$. Sides $AC$ and $BC$ have lengths $2-\sqrt{3}+x$ and $1-2x+x\sqrt{3}$ respectively. Find the angle measure of $B$ in terms of $x$. | [
"\\( \\frac{\\pi}{12} + \\arctan x \\)"
] | [
"Triangle $ABC$ has a right angle at $C$. Sides $AC$ and $BC$ have lengths $2-\\sqrt{3}+x$ and $1-2x+x\\sqrt{3}$ respectively. Find the angle measure of $B$ in terms of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a716e0f6d02c206d5530f41d389d79bf4f5e0603c84afc771cf9bd14be1e5736 | Triangle $ABC$ has a right angle at $C$. The internal bisectors of angles $BAC$ and $ABC$ meet $BC$ and $CA$ at $P$ and $Q$ respectively. The points $M$ and $N$ are the feet of the perpendiculars from $P$ and $Q$ to $AB$. Find $\angle MCN$. | [
"45^\\circ"
] | [
"Triangle $ABC$ has a right angle at $C$. The internal bisectors of angles $BAC$ and $ABC$ meet $BC$ and $CA$ at $P$ and $Q$ respectively. The points $M$ and $N$ are the feet of the perpendiculars from $P$ and $Q$ to $AB$. Find $\\angle MCN$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d31557b9c4ceb88dc915c6da67b05c92d434c36eda13a603340f6e75db7c5e8c | Triangle $ABC$ has a right angle at point $C$. Let $P$ be the point on $AB$ such that $2AP=BP$. If $CP=1$, maximize the area of $\triangle ABC$. | [
"\\dfrac{9}{8}"
] | [
"Triangle $ABC$ has a right angle at point $C$. Let $P$ be the point on $AB$ such that $2AP=BP$. If $CP=1$, maximize the area of $\\triangle ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c3a892b720ce83d07c763f410365ec8bba032d396d89b65563a6d238d0065e71 | Triangle $ABC$ has a right-angle at $C$, and the point $M$ on $AB$ is strictly between $A$ and $B$. Let $S, S_1$ and $S_2$ denote the circumcentres of $\triangle ABC, \triangle AMC$ and $\triangle MBC$ respectively. For which position of $M$ does the circle passing through $M, C, S, S_1$ and $S_2$ have the least radius... | [
"\\( M \\) is the midpoint of \\( AB \\)."
] | [
"Triangle $ABC$ has a right-angle at $C$, and the point $M$ on $AB$ is strictly between $A$ and $B$. Let $S, S_1$ and $S_2$ denote the circumcentres of $\\triangle ABC, \\triangle AMC$ and $\\triangle MBC$ respectively. For which position of $M$ does the circle passing through $M, C, S, S_1$ and $S_2$ have the leas... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 6
} | 14 | 0 | 0 |
omr-ws-69305ae6bf3b8405bab1387ee23614b1513c905467aa8e7f070acf32cc0b691a | Triangle $ABC$ has altitudes $\overline{BB'}$ and $\overline{CC'}$ that intersect at $H$. Additionally, $AC = 24$ and $BH = 10$. Circle $\Omega$ with center $O$ is circumscribed around $\triangle ABC$. Calculate the radius of $\Omega$. | [
"13"
] | [
"Triangle $ABC$ has altitudes $\\overline{BB'}$ and $\\overline{CC'}$ that intersect at $H$. Additionally, $AC = 24$ and $BH = 10$. Circle $\\Omega$ with center $O$ is circumscribed around $\\triangle ABC$. Calculate the radius of $\\Omega$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29683a50a759af935df262d4be7fdca1102ae7390c6063504eb2769b6fa80dc9 | Triangle $ABC$ has altitudes of length $6$, $9.6$, and $9.6$. Find the area of triangle $ABC$. | [
"48"
] | [
"Triangle $ABC$ has altitudes of length $6$, $9.6$, and $9.6$. Find the area of triangle $ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-541becf057e58670f13420464f63517bcc42e2947e8b41810927c2c5cc95d2ae | Triangle $ABC$ has an area of $10$. Points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, with $AD=2$ and $DB=3$. If triangle $ABE$ and quadrilateral $DBEF$ have equal areas, what is that area? | [
"6"
] | [
"Triangle $ABC$ has an area of $10$. Points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, with $AD=2$ and $DB=3$. If triangle $ABE$ and quadrilateral $DBEF$ have equal areas, what is that area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-f30944ae469c1b3d457ebe1503f695486339677ff9d171f3ec79ee6991d03e51 | Triangle $ABC$ has an area of $12$ cm$^2$. The sides of the triangle are extended to points $P, Q, R, S, T,$ and $U$ such that $PA = AB = BS$, $QA = AC = CT$, and $RB = BC = CU$. What is the area of hexagon $PQRSTU$ in cm$^2$? | [
"276"
] | [
"Triangle $ABC$ has an area of $12$ cm$^2$. The sides of the triangle are extended to points $P, Q, R, S, T,$ and $U$ such that $PA = AB = BS$, $QA = AC = CT$, and $RB = BC = CU$. What is the area of hexagon $PQRSTU$ in cm$^2$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3743255febe157ee6cc8d2f2af4915093310c8456b53bb80dc19398bff423a68 | Triangle $ABC$ has an area of $48$. Let $P$ be the midpoint of the median $AM$ and let $N$ be the midpoint of side $AB$. If $G$ is the intersection of $MN$ and $BP$, find the area of triangle $MPG$. | [
"4"
] | [
"Triangle $ABC$ has an area of $48$. Let $P$ be the midpoint of the median $AM$ and let $N$ be the midpoint of side $AB$. If $G$ is the intersection of $MN$ and $BP$, find the area of triangle $MPG$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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