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-023af57d90a3f8316a045c555f247e8d3a135184baaf4a754f14a998f340e797 | A chord of length $\ell$ divides the interior of a circle of radius $r$ into two regions, $D_1$ and $D_2$. A circle $S$ of maximal radius is inscribed in $D_2$. The area of the part of $D_1$ outside $S$ is $A$. If $r=1$, find the value of $\ell$ that maximizes $A$. | [
"2"
] | [
"A chord of length $\\ell$ divides the interior of a circle of radius $r$ into two regions, $D_1$ and $D_2$. A circle $S$ of maximal radius is inscribed in $D_2$. The area of the part of $D_1$ outside $S$ is $A$. If $r=1$, find the value of $\\ell$ that maximizes $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-a772fb7931a2ea620f5c02799b4c9ba33325abe31cd2d6826443d110d7c2a74e | A chord of length $\sqrt{3}$ divides a circle of unit radius into two regions. Find the rectangle of maximum area that can be inscribed in the smaller of those two regions. | [
"\\dfrac{(\\sqrt{33} - 3)\\sqrt{15 - \\sqrt{33}}}{16\\sqrt{2}}"
] | [
"A chord of length $\\sqrt{3}$ divides a circle of unit radius into two regions. Find the rectangle of maximum area that can be inscribed in the smaller of those two regions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-f725f5ba505246d0bdb07828658dd865d3a3f39620ce1793980bdb9a310e0998 | A chord perpendicular to the diameter divides it in the ratio $m : n$. Determine each of the arcs (arc measure) into which the circle is divided by the chord and diameter. | [
"The arc smaller than $\\frac{\\pi }{2}$ is equal to $\\arccos \\frac{m-n}{m+n}$ $(m>n)$; the arc larger than $\\frac{\\pi }{2}$ is equal to $\\pi - \\arccos \\frac{m-n}{m+n} = \\arccos \\frac{n-m}{m+n}$"
] | [
"A chord perpendicular to the diameter divides it in the ratio $m : n$. Determine each of the arcs (arc measure) into which the circle is divided by the chord and diameter."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03877227da6a56af3a6182681a2d1e3b09dcc373d5d46ec11e1b8bbd0539df68 | A chord, perpendicular to the circle's radius, bisects the radius. Find the ratio of the smaller area the chord splits the circle into to the larger area. | [
"\\(\\frac{4\\pi - 3\\sqrt{3}}{8\\pi + 3\\sqrt{3}}\\)"
] | [
"A chord, perpendicular to the circle's radius, bisects the radius. Find the ratio of the smaller area the chord splits the circle into to the larger area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7443dcbc81608b149ddc9a6e94251bfbdafb942f22d969141bc4170815152588 | A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form AB:CD without am or pm, such as 08:00) | [
"05\\!:\\!00"
] | [
"A church rings its bells every 15 minutes, the school rings its bells every 20 minutes and the day care center rings its bells every 25 minutes. If they all ring their bells at noon on the same day, at what time will they next all ring their bells together? (Answer in the form AB:CD without am or pm, such as 08:0... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d83aee7207bbc958943c5a9091d1df6ee2854ab4439b41f319011ad731f46ef | A circle $(I)$ is inscribed in a quadrilateral $ABCD$. Given that $IB = 5$, $IA = 3$, $IC = 7$, and $ID = 2$, calculate the value of $AB \cdot BC$. | [
"31"
] | [
"A circle $(I)$ is inscribed in a quadrilateral $ABCD$. Given that $IB = 5$, $IA = 3$, $IC = 7$, and $ID = 2$, calculate the value of $AB \\cdot BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b9b51aeac5d11c7dc2a908caf6b06fea7b231c8dbf66575994c19170621746f0 | A circle $(I)$ touches the sides $AB, BC, CD, DA$ of a quadrilateral $ABCD$ at points $E, F, G, H$ respectively. The lines $EG$ and $FH$ intersect at point $P$. The line passing through $P$ and perpendicular to $IP$ intersects $AD$ and $BC$ at points $K$ and $L$ respectively. Given that $EG \perp FH$, calculate the val... | [
"1"
] | [
"A circle $(I)$ touches the sides $AB, BC, CD, DA$ of a quadrilateral $ABCD$ at points $E, F, G, H$ respectively. The lines $EG$ and $FH$ intersect at point $P$. The line passing through $P$ and perpendicular to $IP$ intersects $AD$ and $BC$ at points $K$ and $L$ respectively. Given that $EG \\perp FH$, calculate t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bf5dd0863376efb0e84c3c68e0216b9e1f67e8e95258ecb99b5b3c8becc2199 | A circle $(I)$ touches the sides $AB, BC, CD, DA$ of a quadrilateral $ABCD$ at points $E, F, G, H$ respectively. The lines $EG$ and $FH$ intersect at point $P$. The line passing through $P$ and perpendicular to $IP$ intersects $AD$ and $BC$ at points $K$ and $L$ respectively. If $ABCD$ is a square with side length $10$... | [
"10"
] | [
"A circle $(I)$ touches the sides $AB, BC, CD, DA$ of a quadrilateral $ABCD$ at points $E, F, G, H$ respectively. The lines $EG$ and $FH$ intersect at point $P$. The line passing through $P$ and perpendicular to $IP$ intersects $AD$ and $BC$ at points $K$ and $L$ respectively. If $ABCD$ is a square with side length... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bce16293c4975b47d77a3e18e1d7a119a2920f2cef930c09a4693a0d50fc999 | A circle $(O')$ is inside circle $(O)$ and tangent to $(O)$ at $T$. A line $d$ is tangent to $(O')$ at $D$ and intersects $(O)$ at $B$ and $C$. From $B$, a tangent line to $(O)$ intersects the tangent line through $C$ at $A$. The lines $AT$ and $TD$ intersect $(O)$ at $N$ and $P$, respectively. The tangent from $T$ int... | [
"10"
] | [
"A circle $(O')$ is inside circle $(O)$ and tangent to $(O)$ at $T$. A line $d$ is tangent to $(O')$ at $D$ and intersects $(O)$ at $B$ and $C$. From $B$, a tangent line to $(O)$ intersects the tangent line through $C$ at $A$. The lines $AT$ and $TD$ intersect $(O)$ at $N$ and $P$, respectively. The tangent from $T... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-c47115003707a4f27ff9ad4780f9cda38f8aa94d6bc27304f1fc42dfefe97570 | A circle $(O)$ is tangent to the curves $y=\sqrt{e^{x}}$ and $y=2\ln x$ at points $P$ and $Q$ respectively. When the radius of the circle is minimized, calculate the length of $PQ$. | [
"2\\sqrt{2}(1 - \\ln 2)"
] | [
"A circle $(O)$ is tangent to the curves $y=\\sqrt{e^{x}}$ and $y=2\\ln x$ at points $P$ and $Q$ respectively. When the radius of the circle is minimized, calculate the length of $PQ$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d8d0a97cd9b762c164cc54d3a6e25e346dacbc30f0984b09a98116c147cecf7 | A circle $C$ with radius $\sqrt{2}$ touches the $Y$-axis at the origin $(0, 0)$. A ray of light $L$ parallel to the $X$-axis reflects off a point $P$ on the circumference of $C$ and the reflected ray $L'$ becomes parallel to the $Y$-axis. Find the distance between the ray $L$ and the $X$-axis. | [
"1"
] | [
"A circle $C$ with radius $\\sqrt{2}$ touches the $Y$-axis at the origin $(0, 0)$. A ray of light $L$ parallel to the $X$-axis reflects off a point $P$ on the circumference of $C$ and the reflected ray $L'$ becomes parallel to the $Y$-axis. Find the distance between the ray $L$ and the $X$-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-225540e10662825e967b74cf5f5808fe9a7a5f31de260d1c4b086523580eb8d3 | A circle $C$ with radius $\sqrt{2}$ touches the $Y$-axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects off a point $P$ on the circumference of $C$, and the reflected ray $L'$ becomes parallel to the $Y$-axis. Find the distance between the ray $L$ and the $X$-axis. | [
"1"
] | [
"A circle $C$ with radius $\\sqrt{2}$ touches the $Y$-axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects off a point $P$ on the circumference of $C$, and the reflected ray $L'$ becomes parallel to the $Y$-axis. Find the distance between the ray $L$ and the $X$-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a8d70bb247e1b8034e6fe6267278beb65909450d166c460f3e040e6b17b1422 | A circle $C_1$ has center $(h, k)$ with $h > k > 0$. It is tangent to the $x$-axis and to the line $L$, which has equation $y = mx$ for some positive constant $m$. A smaller circle $C_2$ is tangent to $C_1$, $L$, and the $x$-axis. If an infinite family $C_1$, $C_2$, $C_3$, ... of progressively smaller circles is constr... | [
"\\dfrac{\\pi k \\left( \\sqrt{h^2 + k^2} + k \\right)^2}{4 \\sqrt{h^2 + k^2}}"
] | [
"A circle $C_1$ has center $(h, k)$ with $h > k > 0$. It is tangent to the $x$-axis and to the line $L$, which has equation $y = mx$ for some positive constant $m$. A smaller circle $C_2$ is tangent to $C_1$, $L$, and the $x$-axis. If an infinite family $C_1$, $C_2$, $C_3$, ... of progressively smaller circles is c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-ecf921030f91f1b2478c5fd3aa41d2418bb5445f8787678bded3ba8b86938abd | A circle $K_1$ of radius $r_1 = \frac{1}{2}$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1$. A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n \geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$, and $AB$. Calculate the su... | [
"124"
] | [
"A circle $K_1$ of radius $r_1 = \\frac{1}{2}$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1$. A sequence of different circles $K_2, K_3, \\ldots$ with radii $r_2, r_3, \\ldots$ respectively are drawn so that for each $n \\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$, and $AB$. Calcula... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a2fee06e64347ec8cf217e2ebffaf1673abe2946dc1b9239a3c7ce27dead3c4 | A circle $O$ centered at the origin with radius $1$ intersects the x-axis at point $P$ and the graph of $y=x^2$ at point $Q$. Find $\sin\angle POQ$. | [
"\\(\\frac{-1 + \\sqrt{5}}{2}\\)"
] | [
"A circle $O$ centered at the origin with radius $1$ intersects the x-axis at point $P$ and the graph of $y=x^2$ at point $Q$. Find $\\sin\\angle POQ$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-af6cf7c6e06e17896329abe412bf85e3b80835d180c017420a531e9c6cadcd7b | A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} = \overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Calculate the... | [
"r\\sqrt{2}"
] | [
"A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\\frac{r}{\\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\\overrightarrow{PN} = \\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Calc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93fc66b7dfa4c5b3ca1f0a269e230c9decf535c958f570059a91a5dd9d0a3e69 | A circle $S$ of variable radius touches $BC$ at $M$. The tangents (other than $BC$) from $B$ and $C$ to $S$ intersect at $P$. Determine the locus of $P$ as the radius of $S$ varies. | [
"One branch of a hyperbola with foci at \\( B \\) and \\( C \\)."
] | [
"A circle $S$ of variable radius touches $BC$ at $M$. The tangents (other than $BC$) from $B$ and $C$ to $S$ intersect at $P$. Determine the locus of $P$ as the radius of $S$ varies."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d3bf62ebaf303418868127b34e2aae7d46631e332b2117526af30e086f0c4c5d | A circle $\Gamma$ has radius $1$. A line $l$ is such that the perpendicular distance from $l$ to the center of $\Gamma$ is strictly between $0$ and $2$. A frog starts at a point on $\Gamma$ whose perpendicular distance from $l$ is less than $1$. The frog performs a sequence of jumps where each jump has length $1$ and a... | [
"6"
] | [
"A circle $\\Gamma$ has radius $1$. A line $l$ is such that the perpendicular distance from $l$ to the center of $\\Gamma$ is strictly between $0$ and $2$. A frog starts at a point on $\\Gamma$ whose perpendicular distance from $l$ is less than $1$. The frog performs a sequence of jumps where each jump has length $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ecb3025f9815982be5b2601356d48ed969d235612023fa0ee4e8614c95876384 | A circle $\Gamma$ is completely inside a regular polygon $P$ with $1728$ sides. Given that $\Gamma$ is not the incircle of $P$, find the maximum number of sides of $P$ that $\Gamma$ can intersect. | [
"2"
] | [
"A circle $\\Gamma$ is completely inside a regular polygon $P$ with $1728$ sides. Given that $\\Gamma$ is not the incircle of $P$, find the maximum number of sides of $P$ that $\\Gamma$ can intersect."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-cef7478fb6974860998054798dbc915ecdb989c2e4147c7a2d914d4d7e0ffc4e | A circle $\Gamma$ is inscribed in the quadrilateral $ABCD$. The point $E$ is the point of intersection of $\Gamma$ with the diagonal $AC$, the closest to $A$. Tangent to $\Gamma$ at point $F$ intersects the lines $AB$ and $BC$ at points $A_1$ and $C_1$, respectively, and the lines $AD$ and $CD$ at points $A_2$ and $C_2... | [
"10"
] | [
"A circle $\\Gamma$ is inscribed in the quadrilateral $ABCD$. The point $E$ is the point of intersection of $\\Gamma$ with the diagonal $AC$, the closest to $A$. Tangent to $\\Gamma$ at point $F$ intersects the lines $AB$ and $BC$ at points $A_1$ and $C_1$, respectively, and the lines $AD$ and $CD$ at points $A_2$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e889309020cbc574e5ec1e1ff3c381dd23ef393f7562a5296bb4421098714f9 | A circle $\Gamma$ of radius $R$ is tangent to a line $t$ at a point $C$. Another circle $\omega$ of radius $r < R$ is tangent to $t$ at a point $B$ and intersects $\Gamma$ at points $A$ and $A'$, with $A$ being closest to $t$. What is the locus of the centers of the circles circumscribed to the triangles $ABC$ as the c... | [
"\\( x^{2}+y^{2}-2rx+r^{2}-rs=0 \\) (first and third quadrants)"
] | [
"A circle $\\Gamma$ of radius $R$ is tangent to a line $t$ at a point $C$. Another circle $\\omega$ of radius $r < R$ is tangent to $t$ at a point $B$ and intersects $\\Gamma$ at points $A$ and $A'$, with $A$ being closest to $t$. What is the locus of the centers of the circles circumscribed to the triangles $ABC$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-2b967e9ab19efb3994a438584b458e1f9d2da495e730561d45ed4eba8da8b4f0 | A circle $\Gamma$ with radius $R$ and center $\omega$ is drawn on a plane, such that the distance from $\omega$ to a line $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Let $P$ be the point in the plane from which all the segments $MN$ are visib... | [
"\\sqrt{D^2 - R^2}"
] | [
"A circle $\\Gamma$ with radius $R$ and center $\\omega$ is drawn on a plane, such that the distance from $\\omega$ to a line $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\\Gamma$. Let $P$ be the point in the plane from which all the segments $MN$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7f4475a45ecee224c5ddd10903a241e0964939db4c43ca83cae81a6235dd26c4 | A circle $\Gamma_2$ is internally tangent to the circumcircle $\Gamma_1$ of $\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection points of $\Gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\Gamma_1$ again at $G, H$ respectively. If $PF = 5$ and... | [
"no_valid_answer_found"
] | [
"A circle $\\Gamma_2$ is internally tangent to the circumcircle $\\Gamma_1$ of $\\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection points of $\\Gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\\Gamma_1$ again at $G, H$ respectively. If $P... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdf2bf84c1750d832d01e6686960ce85cec7f64d64e50ab833187b997d31ce6c | A circle $\Omega$ is internally tangent to two smaller circles $\omega_1$ and $\omega_2$, which do not intersect. The circumcircles of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. An internal common tangent of $\omega_1$ and $\omega_2$ intersects them at $X$ and $Y$ respectively. The circle with diameter... | [
"r_1 + r_2"
] | [
"A circle $\\Omega$ is internally tangent to two smaller circles $\\omega_1$ and $\\omega_2$, which do not intersect. The circumcircles of $\\omega_1$ and $\\omega_2$ are $O_1$ and $O_2$ respectively. An internal common tangent of $\\omega_1$ and $\\omega_2$ intersects them at $X$ and $Y$ respectively. The circle w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38ef95bd5f64bec25a12c665ff32759f639f69d7747f461ee0101776cc997a01 | A circle $\Omega$ passes through vertex $A$ of $\triangle ABC$ and is tangent to segment $BC$ at point $D$. Let $E$ be the projection of $D$ onto $AC$, and let $F$ be a point on $\Omega$ on the opposite side of $D$ with respect to line $EC$ such that $\angle EFC = 2 \angle ECD$. Suppose that $CF$ intersects $\Omega$ ag... | [
"\\dfrac{1}{6}"
] | [
"A circle $\\Omega$ passes through vertex $A$ of $\\triangle ABC$ and is tangent to segment $BC$ at point $D$. Let $E$ be the projection of $D$ onto $AC$, and let $F$ be a point on $\\Omega$ on the opposite side of $D$ with respect to line $EC$ such that $\\angle EFC = 2 \\angle ECD$. Suppose that $CF$ intersects $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34e6d7c5aeb2380ca221285d0e964a56ee9f2f3dbbb77f537ea1bd92b5503136 | A circle $\gamma_2$ is internally tangent to the circumcircle $\gamma_1$ of $\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection points of $\gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\gamma_1$ again at $G, H$ respectively. If $F, G, H$ ar... | [
"\\dfrac{625}{1024}"
] | [
"A circle $\\gamma_2$ is internally tangent to the circumcircle $\\gamma_1$ of $\\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection points of $\\gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\\gamma_1$ again at $G, H$ respectively. If $F... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-61059d50992ff3a85bd00468ae72630d0a28e21e811b9f0f0bed7600e5ab6e0c | A circle $\mathcal{C}$ with radius $1$ is centered at $O$. A circle $\mathcal{C}'$ of radius $r < 1$ passes through $O$. A point $O_1$ is selected on $\mathcal{C}$ such that line $OO_1$ is tangent to $\mathcal{C}'$. Point $O_2$ is the unique point on $\mathcal{C}$ such that $O_1O_2$ is tangent to $\mathcal{C}'$. Points... | [
"\\( r \\approx 0.2876211932940759513213193108382587113138 \\)"
] | [
"A circle $\\mathcal{C}$ with radius $1$ is centered at $O$. A circle $\\mathcal{C}'$ of radius $r < 1$ passes through $O$. A point $O_1$ is selected on $\\mathcal{C}$ such that line $OO_1$ is tangent to $\\mathcal{C}'$. Point $O_2$ is the unique point on $\\mathcal{C}$ such that $O_1O_2$ is tangent to $\\mathcal{C... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-555d7fdd383ada4bd90468a44d6a4edb6c8bde4b47cf98cbe85d462bd21bd43f | A circle $\omega$ and a straight line $\ell$ are drawn on the plane, not intersecting $\omega$. Pairs of parallels, tangent to $\omega$, intersect $\ell$ at points $A$ and $B$. Through $A$ and $B$, second tangents to $\omega$ are drawn, intersecting at the point $X$. If the radius of $\omega$ is $r$ and the distance fr... | [
"y = \\dfrac{d^2 + r^2}{2d}"
] | [
"A circle $\\omega$ and a straight line $\\ell$ are drawn on the plane, not intersecting $\\omega$. Pairs of parallels, tangent to $\\omega$, intersect $\\ell$ at points $A$ and $B$. Through $A$ and $B$, second tangents to $\\omega$ are drawn, intersecting at the point $X$. If the radius of $\\omega$ is $r$ and the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0351fbde5476db1118c7fa1bfbb207076ca9767ca673272516461c73cab2126a | A circle $\omega$ centered at $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of that arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ at points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilate... | [
"2"
] | [
"A circle $\\omega$ centered at $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of that arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\\omega$ at points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of qua... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-a882032504ec6fe718c315cb0f5b7a13ef1ac1b7cefb3d805780eeb68a76898b | A circle $\omega$ centered at $O$ with radius one intersects another circle $\omega'$ centered at $O'$. These two circles intersect at points $A$ and $B$ with $\angle OAO' = 90^\circ$. Point $E$ is the intersection of $\omega$ and segment $OO'$. Extending $AE$ and $BE$ to intersect $BO'$ and $AO'$ at points $Q$ and $P$... | [
"4"
] | [
"A circle $\\omega$ centered at $O$ with radius one intersects another circle $\\omega'$ centered at $O'$. These two circles intersect at points $A$ and $B$ with $\\angle OAO' = 90^\\circ$. Point $E$ is the intersection of $\\omega$ and segment $OO'$. Extending $AE$ and $BE$ to intersect $BO'$ and $AO'$ at points $... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-a29f91bf1fca3401adbba984b37ba8eab78c80d4d52c04bb379dbda248f136cc | A circle $\omega$ has a fixed chord $AB$. Let $C$ be a variable point on $\omega$. Segment $AB$ is rotated 45° about $C$, sending $AB$ to $A'B'$. If the points $A, B, A', B', C$ all lie on a parabola, find all possible locations of $C$. | [
"\\( C \\) can be at points \\( A \\) or \\( B \\)."
] | [
"A circle $\\omega$ has a fixed chord $AB$. Let $C$ be a variable point on $\\omega$. Segment $AB$ is rotated 45° about $C$, sending $AB$ to $A'B'$. If the points $A, B, A', B', C$ all lie on a parabola, find all possible locations of $C$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fa07b875fe34aa263549eb0a0ea263c3e9a94900e951294818839728cc2a526c | A circle $\omega$ intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. Define $A'$ as the circumcenter of triangle $A_1A_2P$, $B'$ as the circumcenter of triangle $B_1B_2P$, and $C'$ as the circumcen... | [
"36"
] | [
"A circle $\\omega$ intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, respectively. Let $P$ be the center of $\\omega$. Define $A'$ as the circumcenter of triangle $A_1A_2P$, $B'$ as the circumcenter of triangle $B_1B_2P$, and $C'$ as the ci... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c40c105d698cda824c2ac11a728af8208ab0e75d95201a82a3072f55d1c023c3 | A circle $\omega$ intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. Define $A'$ as the circumcenter of triangle $A_1A_2P$, $B'$ as the circumcenter of triangle $B_1B_2P$, and $C'$ as the circumcen... | [
"\\dfrac{13}{2}"
] | [
"A circle $\\omega$ intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, respectively. Let $P$ be the center of $\\omega$. Define $A'$ as the circumcenter of triangle $A_1A_2P$, $B'$ as the circumcenter of triangle $B_1B_2P$, and $C'$ as the ci... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88d006267466cf9185d48e3d57a577d190e477c0606857f303c092dc3656c2d1 | A circle $\omega$ is circumscribed around triangle $ABC$. The straight line tangent to $\omega$ at point $C$ intersects ray $BA$ at point $P$. On ray $PC$, behind point $C$, a point $Q$ is marked such that $PC = QC$. The segment $BQ$ intersects the circle $\omega$ at point $K$ for the second time. On the smaller arc $B... | [
"30^\\circ"
] | [
"A circle $\\omega$ is circumscribed around triangle $ABC$. The straight line tangent to $\\omega$ at point $C$ intersects ray $BA$ at point $P$. On ray $PC$, behind point $C$, a point $Q$ is marked such that $PC = QC$. The segment $BQ$ intersects the circle $\\omega$ at point $K$ for the second time. On the smalle... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-c0c708f324929d4bd92f72e6aaaa6f15731a52c4bf9ba4b226c139827c743f1d | A circle $\omega$ is inscribed in a convex quadrilateral $ABCD$. The incenters of triangles $\triangle ABC$ and $\triangle ACD$ are $I_1$ and $I_2$, respectively. Let $T$ and $U$ be the points on the arcs of $\omega$ lying within $\triangle ABC$ and $\triangle ACD$, respectively, such that the circles $ATC$ and $AUC$ a... | [
"1"
] | [
"A circle $\\omega$ is inscribed in a convex quadrilateral $ABCD$. The incenters of triangles $\\triangle ABC$ and $\\triangle ACD$ are $I_1$ and $I_2$, respectively. Let $T$ and $U$ be the points on the arcs of $\\omega$ lying within $\\triangle ABC$ and $\\triangle ACD$, respectively, such that the circles $ATC$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-61942081c67fece69d08f73f8afd6b146db779e566fd5cb3547fa13aefea6dca | A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Given that $AB = 12$, $BC = 16$, $BI = 5$, and $DI = 4$, calculate the value of $AI \cdot CI$. | [
"no_valid_answer_found"
] | [
"A circle $\\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\\omega$. Given that $AB = 12$, $BC = 16$, $BI = 5$, and $DI = 4$, calculate the value of $AI \\cdot CI$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67a1a97688a28a06f7157e75e7c68820b6425183b4031437b3afbc861613edc6 | A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Given that $BI^2+\cfrac{AI\cdot BI\cdot CI}{DI}=AB\cdot BC$, calculate the value of $BI^2$ if $AI = 6$, $CI = 8$, $DI = 4$, and $AB\cdot BC = 120$. | [
"192 - 24\\sqrt{39}"
] | [
"A circle $\\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\\omega$. Given that $BI^2+\\cfrac{AI\\cdot BI\\cdot CI}{DI}=AB\\cdot BC$, calculate the value of $BI^2$ if $AI = 6$, $CI = 8$, $DI = 4$, and $AB\\cdot BC = 120$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc6b340d914d6bd33bf0fa6d54b8352b30a2673fe0c904b6bdd4f15a023c0e25 | A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Given that \[(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2,\] find the value of \[\frac{AB + CD}{AD + BC}\] if $ABCD$ is an isosceles trapezoid. | [
"1"
] | [
"A circle $\\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\\omega$. Given that \\[(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2,\\] find the value of \\[\\frac{AB + CD}{AD + BC}\\] if $ABCD$ is an isosceles trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-820c98c31f7d10eed0664dc285fcecfeda1593e95c9ddc2a495c78ace2335304 | A circle $\omega$ is inscribed in triangle $ABC$, tangent to the side $BC$ at point $K$. Circle $\omega'$ is symmetrical to the circle $\omega$ with respect to point $A$. The point $A_0$ is chosen so that the segments $BA_0$ and $CA_0$ are tangent to $\omega'$. Let $M$ be the midpoint of side $BC$. If $AM$ intersects $... | [
"1"
] | [
"A circle $\\omega$ is inscribed in triangle $ABC$, tangent to the side $BC$ at point $K$. Circle $\\omega'$ is symmetrical to the circle $\\omega$ with respect to point $A$. The point $A_0$ is chosen so that the segments $BA_0$ and $CA_0$ are tangent to $\\omega'$. Let $M$ be the midpoint of side $BC$. If $AM$ int... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5e8c4d58f3013c4b61ffc2a2160bfa5c5fc05f58acaa6fed7826a3426167fb76 | A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ at points $A_1$ and $A_2$. Define $B_1, B_2, C_1, C_2$ similarly. Suppose five of the six points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle $\gamma$. If $A_1, A_2, B_1, B_2, C_1$ lie on $\gamma$, calculate the rati... | [
"2"
] | [
"A circle $\\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\\omega$ intersect $BC$ at points $A_1$ and $A_2$. Define $B_1, B_2, C_1, C_2$ similarly. Suppose five of the six points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle $\\gamma$. If $A_1, A_2, B_1, B_2, C_1$ lie on $\\gamma$, calculate... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abd36ce88ea38c4a5985470045a1ad1a1941ef702f3653cd2e713430a1a45505 | A circle $\omega$ is tangent to lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle intersects lines $a$ and $b$ at points $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Determine the locus of the projections of the ... | [
"The reflection of the circle \\(\\omega\\) about the line \\(AB\\)."
] | [
"A circle $\\omega$ is tangent to lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle intersects lines $a$ and $b$ at points $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Determine the locus of the projections o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27bdbdaae718dbb4140f0a242f3ba66635567ff7a8e231d73f1f2565f1835436 | A circle $\omega$ is tangent to the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $D$, $E$, and $F$ respectively. The segment $DP$ is the diameter of $\omega$. Line $AP$ intersects $\omega$ at points $P$ and $Q$. The point $M$ is the midpoint of side $BC$. Lines $MQ$ and $EF$ intersect at point $R$. Let $O$ be... | [
"\\dfrac{\\sqrt{3}}{2}"
] | [
"A circle $\\omega$ is tangent to the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $D$, $E$, and $F$ respectively. The segment $DP$ is the diameter of $\\omega$. Line $AP$ intersects $\\omega$ at points $P$ and $Q$. The point $M$ is the midpoint of side $BC$. Lines $MQ$ and $EF$ intersect at point $R$. Le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-684450aa3606326f4eaa7ab41af8fe3a92b3b39027917b3766c77677dbb70837 | A circle $\omega$ of diameter $AB$ is given. On this diameter, points $P$ and $M$ are chosen such that $M$ is the midpoint of segment $BP$. Parallel lines $r$ and $s$ are drawn through the points $M$ and $P$, respectively, which do not coincide with the line $AB$ and are not perpendicular to it. Point $H$ is the orthog... | [
"1"
] | [
"A circle $\\omega$ of diameter $AB$ is given. On this diameter, points $P$ and $M$ are chosen such that $M$ is the midpoint of segment $BP$. Parallel lines $r$ and $s$ are drawn through the points $M$ and $P$, respectively, which do not coincide with the line $AB$ and are not perpendicular to it. Point $H$ is the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02d9c375a80e89ff2cf108d9ad03cdd034236561a7be0205afeaf0464197c6ba | A circle $\omega$ of radius $r$ is inscribed in triangle $ABC$, touching side $AB$ at point $X$. Point $Y$ is diametrically opposite to point $X$ on the circle. Line $CY$ intersects side $AB$ at point $Z$. Find the area of triangle $ABC$ if it is known that $CA + AZ = 1$. | [
"r"
] | [
"A circle $\\omega$ of radius $r$ is inscribed in triangle $ABC$, touching side $AB$ at point $X$. Point $Y$ is diametrically opposite to point $X$ on the circle. Line $CY$ intersects side $AB$ at point $Z$. Find the area of triangle $ABC$ if it is known that $CA + AZ = 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-31a200995a095f1152327b18a4c2c9f0ac1ff4f14840c37974690047d95fb057 | A circle $\omega$ touches the sides $AB$ and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. Given that $AB = 9$ and $BC = 11$, what can be the difference $AK - CK$? | [
"2"
] | [
"A circle $\\omega$ touches the sides $AB$ and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. Given that $AB = 9$ and $BC = 11$, what can be the difference $AK - CK$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-f2bb1a083f29227ec296f5ca03368d369171347b5687569155d0d4b374fa9937 | A circle $\omega$ with center $I$ is inscribed in an isosceles acute-angled triangle $ABC$. Denote by $B_1$ and $C_1$ the projections of points $B$ and $C$ on the line $AE$, respectively. Points $X$ and $Y$ are chosen on the segment $BC$ such that $\angle B_1XC_1 = \angle B_1YC_1 = 90^\circ$. Let $O$ be the center of t... | [
"2"
] | [
"A circle $\\omega$ with center $I$ is inscribed in an isosceles acute-angled triangle $ABC$. Denote by $B_1$ and $C_1$ the projections of points $B$ and $C$ on the line $AE$, respectively. Points $X$ and $Y$ are chosen on the segment $BC$ such that $\\angle B_1XC_1 = \\angle B_1YC_1 = 90^\\circ$. Let $O$ be the ce... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42f22c94b64c630e1e58936e03c422d6925781f410801c0b68dcb0682eea501f | A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ at points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilate... | [
"2"
] | [
"A circle $\\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\\omega$ at points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of qua... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-dac7f727660fcafdd38114d9fd65122894671c95a0935cf91bf77e97a9bc792f | A circle $\omega$ with center $I$ is located inside the circle $\Omega$ with center $O$. Ray $IO$ intersects $\omega$ and $\Omega$ at $P_1$ and $P_2$ respectively. An arbitrary point $A \neq P_2$ is chosen on $\Omega$. The circumcircle of the triangle $P_1P_2A$ intersects $\omega$ for the second time at $X$. Line $AX$ ... | [
"0"
] | [
"A circle $\\omega$ with center $I$ is located inside the circle $\\Omega$ with center $O$. Ray $IO$ intersects $\\omega$ and $\\Omega$ at $P_1$ and $P_2$ respectively. An arbitrary point $A \\neq P_2$ is chosen on $\\Omega$. The circumcircle of the triangle $P_1P_2A$ intersects $\\omega$ for the second time at $X$... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-9a0e54bc1b801f6bcd613ff737e8eec026f844f4d643014c2dab18e2855ce48f | A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ at $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$ at $P$. The point $L$ lies on $\omega$ such that $PL$ is tangent to $\o... | [
"3"
] | [
"A circle $\\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\\omega$ at $A$ and $Q$. The tangent to $\\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$ at $P$. The point $L$ lies on $\\omega$ such that $PL$ is tange... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-207ddde58de5566e2cd95f038bc77827c23c4e79cbbf305c27344ce50afd371a | A circle $\omega_1$ of radius $6\sqrt{2}$ is internally tangent to a larger circle $\omega_2$ of radius $12\sqrt{2}$ such that the center of $\omega_2$ lies on $\omega_1$. A diameter $AB$ of $\omega_2$ is drawn tangent to $\omega_1$. A second line $\ell$ is drawn from $B$ tangent to $\omega_1$. Let the line tangent to ... | [
"\\(144\\)"
] | [
"A circle $\\omega_1$ of radius $6\\sqrt{2}$ is internally tangent to a larger circle $\\omega_2$ of radius $12\\sqrt{2}$ such that the center of $\\omega_2$ lies on $\\omega_1$. A diameter $AB$ of $\\omega_2$ is drawn tangent to $\\omega_1$. A second line $\\ell$ is drawn from $B$ tangent to $\\omega_1$. Let the l... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18f9deb00d52b8d7f51b7281c48a874b64bef8ff122d3ba43718b0719b1ac315 | A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. If the coordinates of $O_1$ are $(0,0)$ and $O_2$ are $(4,0)$, and the radius of $\omega_1$ i... | [
"-\\dfrac{3}{4}"
] | [
"A circle $\\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\\omega_2$. The tangent lines to $\\omega_2$ from a point $C$ on $\\omega_1$ intersect $\\omega_1$ again at points $A$ and $B$ respectively. If the coordinates of $O_1$ are $(0,0)$ and $O_2$ are $(4,0)$, and the radius of $\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-1b58b9ec5cab3bbe7bf4e049d0ee6b19cad549b8f4ece030a4773b34afb191d6 | A circle $k$ with radius $20$ contains a square $ABCD$ and a smaller circle $k_1$ with radius $5$. The circles $k$ and $k_1$ share a common point $P$, and points $A$ and $B$ lie on circle $k$. The line $CD$ intersects circle $k_1$ at point $Q$ such that $PQ$ is a diameter of $k_1$. Find the side length of the square $A... | [
"\\(8 + 4\\sqrt{19}\\)"
] | [
"A circle $k$ with radius $20$ contains a square $ABCD$ and a smaller circle $k_1$ with radius $5$. The circles $k$ and $k_1$ share a common point $P$, and points $A$ and $B$ lie on circle $k$. The line $CD$ intersects circle $k_1$ at point $Q$ such that $PQ$ is a diameter of $k_1$. Find the side length of the squa... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-74b6dd0d1666d038746ff22014a8265de9e43184638ea655311d2818bf08517b | A circle (C) intersects the parabola $(P): y=x^2$ at points $A(1,1)$ and three other points $B$, $C$, and $D$. The points $B$, $C$, and $D$ form an equilateral triangle. Calculate the radius of the circle (C). | [
"2"
] | [
"A circle (C) intersects the parabola $(P): y=x^2$ at points $A(1,1)$ and three other points $B$, $C$, and $D$. The points $B$, $C$, and $D$ form an equilateral triangle. Calculate the radius of the circle (C)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-1a5a50a10edbaa7e1320f4abf9555285ff55ab55817305940e4397861b08c068 | A circle \( C \) has two parallel tangents \( L' \) and \( L'' \). A circle \( C' \) touches \( L' \) at \( A \) and \( C \) at \( X \). A circle \( C'' \) touches \( L'' \) at \( B \), \( C \) at \( Y \), and \( C' \) at \( Z \). The lines \( AY \) and \( BX \) meet at \( Q \). If the radius of circle \( C \) is 10 an... | [
"\\dfrac{5\\sqrt{31}}{7}"
] | [
"A circle \\( C \\) has two parallel tangents \\( L' \\) and \\( L'' \\). A circle \\( C' \\) touches \\( L' \\) at \\( A \\) and \\( C \\) at \\( X \\). A circle \\( C'' \\) touches \\( L'' \\) at \\( B \\), \\( C \\) at \\( Y \\), and \\( C' \\) at \\( Z \\). The lines \\( AY \\) and \\( BX \\) meet at \\( Q \\).... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-ac41697e45a43e5f7981deb9c44f789d6fb7df42e39976c7683081b0e41d5492 | A circle \( C \) with center \( O \) and a line \( L \) which does not touch circle \( C \). \( OQ \) is perpendicular to \( L \), with \( Q \) on \( L \). \( P \) is on \( L \), and tangents \( L_1 \) and \( L_2 \) are drawn from \( P \) to circle \( C \). \( QA \) and \( QB \) are perpendicular to \( L_1 \) and \( L_... | [
"\\dfrac{119}{24}"
] | [
"A circle \\( C \\) with center \\( O \\) and a line \\( L \\) which does not touch circle \\( C \\). \\( OQ \\) is perpendicular to \\( L \\), with \\( Q \\) on \\( L \\). \\( P \\) is on \\( L \\), and tangents \\( L_1 \\) and \\( L_2 \\) are drawn from \\( P \\) to circle \\( C \\). \\( QA \\) and \\( QB \\) are... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8900a5ab2a9d3d4f77b9f77ab1a41f11284b38c1eadd03d432376d4caa0b259b | A circle \( C \) with radius \( 3 \) has an equilateral triangle inscribed in it. Let \( D \) be a circle lying outside the equilateral triangle, tangent to \( C \), and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of \( D \) can be written in the form \( \frac{m}{n} \), where \( ... | [
"7"
] | [
"A circle \\( C \\) with radius \\( 3 \\) has an equilateral triangle inscribed in it. Let \\( D \\) be a circle lying outside the equilateral triangle, tangent to \\( C \\), and tangent to the equilateral triangle at the midpoint of one of its sides. The radius of \\( D \\) can be written in the form \\( \\frac{m}... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-f76801dbac01632efa17950977a1f27fb27c766ad26ef22f2cb2f8eeac4610be | A circle \( C \) with radius \( \sqrt{2} \) touches the Y-axis at the origin \((0, 0)\). A ray of light \( L \), parallel to the X-axis, reflects off a point \( P \) on the circumference of \( C \), and the reflected ray \( L' \) becomes parallel to the Y-axis. Find the distance between the ray \( L \) and the X-axis. | [
"1"
] | [
"A circle \\( C \\) with radius \\( \\sqrt{2} \\) touches the Y-axis at the origin \\((0, 0)\\). A ray of light \\( L \\), parallel to the X-axis, reflects off a point \\( P \\) on the circumference of \\( C \\), and the reflected ray \\( L' \\) becomes parallel to the Y-axis. Find the distance between the ray \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7340ebab38a0417e909cca33a024c7236d8a4079dce79a75be004adaefc795ca | A circle \( C \) with radius \( \sqrt{2} \) touches the Y-axis at the origin \((0,0)\). A ray of light \( L \), parallel to the X-axis, reflects off a point \( P \) on the circumference of \( C \), and the reflected ray \( L' \) becomes parallel to the Y-axis. Find the distance between the ray \( L \) and the X-axis. | [
"1"
] | [
"A circle \\( C \\) with radius \\( \\sqrt{2} \\) touches the Y-axis at the origin \\((0,0)\\). A ray of light \\( L \\), parallel to the X-axis, reflects off a point \\( P \\) on the circumference of \\( C \\), and the reflected ray \\( L' \\) becomes parallel to the Y-axis. Find the distance between the ray \\( L... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87c35bfdc103f86343b3b9cb46f96d2f0aa9ac50e7aec73e2fd92939298a927e | A circle \( C1 \) has center \( Q \). A semicircle is drawn inside \( C1 \) sharing its diameter with the radius of \( C1 \). A second semicircle is drawn connected to the first semicircle, sharing its diameter with the radius of the first semicircle, on the side where it would be touching the edge of \( C1 \). A line ... | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"A circle \\( C1 \\) has center \\( Q \\). A semicircle is drawn inside \\( C1 \\) sharing its diameter with the radius of \\( C1 \\). A second semicircle is drawn connected to the first semicircle, sharing its diameter with the radius of the first semicircle, on the side where it would be touching the edge of \\( ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-09662b21f48f3f6bc6be2b61249b5a7d0e9aa9a353644727de6767f5a8e08d25 | A circle \( C_0 \) of radius 1 km is tangent to a line \( L \) at point \( Z \). A circle \( C_1 \) of radius 1 mm is drawn tangent to both \( C_0 \) and \( L \) on the right-hand side of \( C_0 \). A family of circles \( C_i \) is constructed outwardly to the right side such that each \( C_i \) is tangent to \( C_0 \)... | [
"1000"
] | [
"A circle \\( C_0 \\) of radius 1 km is tangent to a line \\( L \\) at point \\( Z \\). A circle \\( C_1 \\) of radius 1 mm is drawn tangent to both \\( C_0 \\) and \\( L \\) on the right-hand side of \\( C_0 \\). A family of circles \\( C_i \\) is constructed outwardly to the right side such that each \\( C_i \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 3,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-4890a45e236309e4346f5492e3fcfe66b89defe2f264d1aa17c51b1dc64c5e35 | A circle \( K \) is inscribed in a square \( C \). For any point \( M \) lying in \( C \) but not on \( K \), a tangent to \( K \) is drawn through \( M \) and intersects \( K \) at points \( A \) and \( B \). Let \( P \) be the center of the square with side \( AB \) that touches \( C \) only at \( M \). Find the locu... | [
"The locus of \\( P \\) is four line segments on \\( y = \\pm \\sqrt{2}r \\) and \\( x = \\pm \\sqrt{2}r \\) within the bounds \\( -\\frac{\\sqrt{2}}{2}r < x < \\frac{\\sqrt{2}}{2}r \\) and \\( -\\frac{\\sqrt{2}}{2}r < y < \\frac{\\sqrt{2}}{2}r \\). \n\nSince the problem asks for a specific numerical or mathematica... | [
"A circle \\( K \\) is inscribed in a square \\( C \\). For any point \\( M \\) lying in \\( C \\) but not on \\( K \\), a tangent to \\( K \\) is drawn through \\( M \\) and intersects \\( K \\) at points \\( A \\) and \\( B \\). Let \\( P \\) be the center of the square with side \\( AB \\) that touches \\( C \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e48ac89fb541e2f11723c831a0f2e201c108555ffc7cb4eca27a3e446b9cbd5 | A circle \( K \) is tangent to exactly three sides of a 6x4 rectangle. What is the radius of the largest circle that can fit inside the rectangle but is completely outside circle \( K \)? | [
"\\( 4(2-\\sqrt{3}) \\)"
] | [
"A circle \\( K \\) is tangent to exactly three sides of a 6x4 rectangle. What is the radius of the largest circle that can fit inside the rectangle but is completely outside circle \\( K \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-492e2896c308e6418aef32cd2b402fddd051e8f2adee470ce136369003ebfeb0 | A circle \( O \) is tangent to two parallel lines \( l_1 \) and \( l_2 \). A second circle \( O_1 \) is tangent to \( l_1 \) at \( A \) and to \( O \) externally at \( C \). A third circle \( O_2 \) is tangent to \( l_2 \) at \( B \) and to \( O \) externally at \( D \) and to \( O_1 \) externally at \( E \). Let \( Q ... | [
"\\dfrac{15\\sqrt{2}}{2}"
] | [
"A circle \\( O \\) is tangent to two parallel lines \\( l_1 \\) and \\( l_2 \\). A second circle \\( O_1 \\) is tangent to \\( l_1 \\) at \\( A \\) and to \\( O \\) externally at \\( C \\). A third circle \\( O_2 \\) is tangent to \\( l_2 \\) at \\( B \\) and to \\( O \\) externally at \\( D \\) and to \\( O_1 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-259cb1c36f848b61b57542b915da0188aabdbffe1cb7c3fe96b6d4e3c399ce16 | A circle \( P: (x - 3)^2 + (y - 3)^2 = 25 \) is rotated by an angle \( A \) where \( 0^\circ < A < 360^\circ \). If there is exactly one point \( L \) that lies on both the original circle \( P \) and the rotated circle \( P' \), find the value(s) of \( \cos(A) \). | [
"0"
] | [
"A circle \\( P: (x - 3)^2 + (y - 3)^2 = 25 \\) is rotated by an angle \\( A \\) where \\( 0^\\circ < A < 360^\\circ \\). If there is exactly one point \\( L \\) that lies on both the original circle \\( P \\) and the rotated circle \\( P' \\), find the value(s) of \\( \\cos(A) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38a59e5f0597a3e88ab48bb074da65e91b73a142d2fd624c0983cde8686da8b6 | A circle \( P: (x-3)^2 + (y-3)^2 = 25 \) is rotated by an angle \( A \) where \( 0^\circ < A < 360^\circ \). If there is only one point \( L \) that lies on both the original circle \( P \) and the rotated circle \( P' \), find the value of \( \cos(A) \). | [
"\\(\\frac{11}{36}\\)"
] | [
"A circle \\( P: (x-3)^2 + (y-3)^2 = 25 \\) is rotated by an angle \\( A \\) where \\( 0^\\circ < A < 360^\\circ \\). If there is only one point \\( L \\) that lies on both the original circle \\( P \\) and the rotated circle \\( P' \\), find the value of \\( \\cos(A) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce04b6194701fe38925bb660750a38ba78eeefdbb765c65a9cca3c67160aa16e | A circle \( S \) is inscribed in a quadrilateral \( ABCD \) with \( AB \) parallel to \( CD \). Let \( M \) and \( N \) be the points where the circle is tangent to \( AB \) and \( CD \) respectively. If \( X \) is the intersection of \( AN \) and \( DM \), and \( Y \) is the intersection of \( BN \) and \( MC \), find... | [
"\\text{Square}"
] | [
"A circle \\( S \\) is inscribed in a quadrilateral \\( ABCD \\) with \\( AB \\) parallel to \\( CD \\). Let \\( M \\) and \\( N \\) be the points where the circle is tangent to \\( AB \\) and \\( CD \\) respectively. If \\( X \\) is the intersection of \\( AN \\) and \\( DM \\), and \\( Y \\) is the intersection o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-55d9632687e9cba352e724d3f92f8e981a3e49e520697e2a18578522f09ea088 | A circle \( S \) of radius \( R \) has two parallel tangents \( t_1 \) and \( t_2 \). A circle \( S_1 \) of radius \( r_1 \) touches \( S \) and \( t_1 \), and a circle \( S_2 \) of radius \( r_2 \) touches \( S \) and \( t_2 \). Additionally, \( S_1 \) touches \( S_2 \) externally. Calculate \( R \) in terms of \( r_1... | [
"2\\sqrt{r_1 r_2}"
] | [
"A circle \\( S \\) of radius \\( R \\) has two parallel tangents \\( t_1 \\) and \\( t_2 \\). A circle \\( S_1 \\) of radius \\( r_1 \\) touches \\( S \\) and \\( t_1 \\), and a circle \\( S_2 \\) of radius \\( r_2 \\) touches \\( S \\) and \\( t_2 \\). Additionally, \\( S_1 \\) touches \\( S_2 \\) externally. Cal... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-c02f5d143c61a0f2ffc1ba1c027bfa0b81d92399541b1fcb8a56d374556b05a8 | A circle \( \Gamma \) is inscribed in a quadrilateral \( ABCD \). If \( \angle A = \angle B = 120^\circ \), \( \angle D = 90^\circ \), and \( BC = 1 \), find the length of \( AD \). | [
"\\(\\frac{\\sqrt{3} - 1}{2}\\)"
] | [
"A circle \\( \\Gamma \\) is inscribed in a quadrilateral \\( ABCD \\). If \\( \\angle A = \\angle B = 120^\\circ \\), \\( \\angle D = 90^\\circ \\), and \\( BC = 1 \\), find the length of \\( AD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 14
} | 53 | 0 | 0 |
omr-ws-48ea1d96cce31722453235c834cec0664c3542b954484cd4f4bbb6730302f74d | A circle \( \Gamma \) with radius \( R \) and center \( \omega \), and a line \( d \) are drawn on a plane, such that the distance of \( \omega \) from \( d \) is greater than \( R \). Two points \( M \) and \( N \) vary on \( d \) so that the circle with diameter \( MN \) is tangent to \( \Gamma \). Let \( P \) be the... | [
"\\sqrt{D^2 - R^2}"
] | [
"A circle \\( \\Gamma \\) with radius \\( R \\) and center \\( \\omega \\), and a line \\( d \\) are drawn on a plane, such that the distance of \\( \\omega \\) from \\( d \\) is greater than \\( R \\). Two points \\( M \\) and \\( N \\) vary on \\( d \\) so that the circle with diameter \\( MN \\) is tangent to \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f343e0e379ebb7fd68529150ae552790b32dbeba381937fca20c68541c637778 | A circle \( \Omega \) of unit radius is inscribed in quadrilateral \( ABCD \). Let circle \( \omega_A \) be the unique circle of radius \( r_A \) externally tangent to \( \Omega \) and also tangent to segments \( AB \) and \( DA \). Similarly define circles \( \omega_B \), \( \omega_C \), and \( \omega_D \) and radii \... | [
"3 - 2\\sqrt{2}"
] | [
"A circle \\( \\Omega \\) of unit radius is inscribed in quadrilateral \\( ABCD \\). Let circle \\( \\omega_A \\) be the unique circle of radius \\( r_A \\) externally tangent to \\( \\Omega \\) and also tangent to segments \\( AB \\) and \\( DA \\). Similarly define circles \\( \\omega_B \\), \\( \\omega_C \\), an... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-77b6bf2a47fed98b2c4983fc987c48a9c9a3cc213f2962677747cd2ad811a155 | A circle \( \Omega_1 \) is tangent outwardly to a circle \( \Omega_2 \) of a larger radius. Line \( t_1 \) is tangent to \( \Omega_1 \) at point \( A \) and to \( \Omega_2 \) at point \( D \). Line \( t_2 \), parallel to \( t_1 \), is tangent to \( \Omega_1 \) and intersects \( \Omega_2 \) at points \( E \) and \( F \)... | [
"12"
] | [
"A circle \\( \\Omega_1 \\) is tangent outwardly to a circle \\( \\Omega_2 \\) of a larger radius. Line \\( t_1 \\) is tangent to \\( \\Omega_1 \\) at point \\( A \\) and to \\( \\Omega_2 \\) at point \\( D \\). Line \\( t_2 \\), parallel to \\( t_1 \\), is tangent to \\( \\Omega_1 \\) and intersects \\( \\Omega_2 ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55346457e0696cbf245e4cade9a978f43d7b11ccf164597b246ae8a92e10340d | A circle \( \omega \) is circumscribed around an acute triangle \( ABC \). \( AN \) and \( CK \) are altitudes. The median \( BM \) intersects \( \omega \) at \( P \). Point \( Q \) is chosen on \( BM \) such that \( MQ = MP \). Calculate the value of \( \frac{BQ}{QM} \) if \( BK = 12 \) and \( BN = 16 \). | [
"\\dfrac{576 + 600\\sqrt{2}}{337}"
] | [
"A circle \\( \\omega \\) is circumscribed around an acute triangle \\( ABC \\). \\( AN \\) and \\( CK \\) are altitudes. The median \\( BM \\) intersects \\( \\omega \\) at \\( P \\). Point \\( Q \\) is chosen on \\( BM \\) such that \\( MQ = MP \\). Calculate the value of \\( \\frac{BQ}{QM} \\) if \\( BK = 12 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-831b792b802190403e1032ab2d30fdfd959493a32740f1bec0b23ef7c298b9a4 | A circle \( \omega \) is inscribed in quadrilateral \( ABCD \). The line segment \( AC \) intersects the circle \( \omega \) at points \( K \) and \( L \). Let \( M \) be the midpoint of \( KL \). If \( \angle BMA = \angle DMA = \theta \), find the value of \( \cos(2\theta) \) given that \( AB = 5 \), \( BC = 6 \), \( ... | [
"0"
] | [
"A circle \\( \\omega \\) is inscribed in quadrilateral \\( ABCD \\). The line segment \\( AC \\) intersects the circle \\( \\omega \\) at points \\( K \\) and \\( L \\). Let \\( M \\) be the midpoint of \\( KL \\). If \\( \\angle BMA = \\angle DMA = \\theta \\), find the value of \\( \\cos(2\\theta) \\) given that... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c85b79271f33f1d2265a52223fb1a5f19b2835019a36c797f8ffb412db328ff4 | A circle \( \omega \) with center \( O \) is tangent to the rays of an angle \( BAC \) at \( B \) and \( C \). Point \( Q \) is taken inside the angle \( BAC \). Assume that point \( P \) on the segment \( AQ \) is such that \( AQ \perp OP \). The line \( OP \) intersects the circumcircles \( \omega_{1} \) and \( \omeg... | [
"1"
] | [
"A circle \\( \\omega \\) with center \\( O \\) is tangent to the rays of an angle \\( BAC \\) at \\( B \\) and \\( C \\). Point \\( Q \\) is taken inside the angle \\( BAC \\). Assume that point \\( P \\) on the segment \\( AQ \\) is such that \\( AQ \\perp OP \\). The line \\( OP \\) intersects the circumcircles ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d8c1470a8b10fcea780d6df759c39eae0271d76badb6cac5642f0cd24177968 | A circle \( \omega_1 \) touches the sides of angle \( A \) at points \( B \) and \( C \). A straight line \( AD \) intersects \( \omega_1 \) at points \( D \) and \( Q \) with \( AD < AQ \). The circle \( \omega_2 \) with center \( A \) and radius \( AB \) intersects \( AQ \) at a point \( I \) and intersects some line... | [
"5"
] | [
"A circle \\( \\omega_1 \\) touches the sides of angle \\( A \\) at points \\( B \\) and \\( C \\). A straight line \\( AD \\) intersects \\( \\omega_1 \\) at points \\( D \\) and \\( Q \\) with \\( AD < AQ \\). The circle \\( \\omega_2 \\) with center \\( A \\) and radius \\( AB \\) intersects \\( AQ \\) at a poin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-403befbff3b58c063bd9532e58364f1272cd09f9e833e80a25e0f09358e7d127 | A circle \( k \) touches a larger circle \( K \) from inside at a point \( P \). Let \( Q \) be a point on \( k \) different from \( P \). The line tangent to \( k \) at \( Q \) intersects \( K \) at points \( A \) and \( B \). If \( \angle APB = 2\theta \), find the measure of \( \angle QPB \) in terms of \( \theta \)... | [
"\\theta"
] | [
"A circle \\( k \\) touches a larger circle \\( K \\) from inside at a point \\( P \\). Let \\( Q \\) be a point on \\( k \\) different from \\( P \\). The line tangent to \\( k \\) at \\( Q \\) intersects \\( K \\) at points \\( A \\) and \\( B \\). If \\( \\angle APB = 2\\theta \\), find the measure of \\( \\angl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-22499e80864c17e021920ec816a2e14aceee095af0a990d65e2737c42a11a9f7 | A circle \( k_1 \) lies within a second circle \( k_2 \) and touches it at point \( A \). A line through \( A \) intersects \( k_1 \) again in \( B \) and \( k_2 \) in \( C \). The tangent to \( k_1 \) through \( B \) intersects \( k_2 \) at points \( D \) and \( E \). The tangents to \( k_1 \) passing through \( C \) ... | [
"2\\sqrt{30}"
] | [
"A circle \\( k_1 \\) lies within a second circle \\( k_2 \\) and touches it at point \\( A \\). A line through \\( A \\) intersects \\( k_1 \\) again in \\( B \\) and \\( k_2 \\) in \\( C \\). The tangent to \\( k_1 \\) through \\( B \\) intersects \\( k_2 \\) at points \\( D \\) and \\( E \\). The tangents to \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9290c34adb0f7f2f24c9d3fa44225bfbbfc48fe05f812acf98182b9989cd564 | A circle \( w \) touches the sides \( AB \) and \( AC \) of an isosceles triangle \( ABC \) and intersects the side \( BC \) at points \( K \) and \( L \). The line \( AK \) intersects the circle \( w \) again at point \( M \). Points \( P \) and \( Q \) are the reflections of point \( K \) with respect to points \( B ... | [
"2"
] | [
"A circle \\( w \\) touches the sides \\( AB \\) and \\( AC \\) of an isosceles triangle \\( ABC \\) and intersects the side \\( BC \\) at points \\( K \\) and \\( L \\). The line \\( AK \\) intersects the circle \\( w \\) again at point \\( M \\). Points \\( P \\) and \\( Q \\) are the reflections of point \\( K \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-198087519b0e9b6faab03a53c3667b2a7fb19d3d2b0e1d65044d46bdbb2bc2aa | A circle \((\omega)\) is tangent to sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively and intersects \(BC\) at points \(K\) and \(L\) (with \(K\) on segment \(BL\)). Line \(AK\) intersects \((\omega)\) again at point \(M \neq K\). Points \(P\) and \(Q\) are on \(BK\) and \(CK\) respectively such that \(BK ... | [
"2"
] | [
"A circle \\((\\omega)\\) is tangent to sides \\(AB\\) and \\(AC\\) at points \\(D\\) and \\(E\\) respectively and intersects \\(BC\\) at points \\(K\\) and \\(L\\) (with \\(K\\) on segment \\(BL\\)). Line \\(AK\\) intersects \\((\\omega)\\) again at point \\(M \\neq K\\). Points \\(P\\) and \\(Q\\) are on \\(BK\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-9806c717323ea5b5ea0e801e220f3ab3f9a35e8fc2233dce94c3762522db0e79 | A circle \(\Gamma\) and a line \(\ell\) are given in a plane such that \(\ell\) does not intersect \(\Gamma\). Determine the intersection set of the circles with \([AB]\) as diameter for all pairs \(\{A, B\}\) (lying on \(\ell\)) that satisfy \(P, Q, R, S \in \Gamma\) such that \(PQ \cap RS = \{A\}\) and \(PS \cap QR =... | [
"\\text{The two points on the perpendicular from the center of } \\Gamma \\text{ to } \\ell \\text{ at distance } \\sqrt{d^2 - r^2} \\text{ from } \\ell"
] | [
"A circle \\(\\Gamma\\) and a line \\(\\ell\\) are given in a plane such that \\(\\ell\\) does not intersect \\(\\Gamma\\). Determine the intersection set of the circles with \\([AB]\\) as diameter for all pairs \\(\\{A, B\\}\\) (lying on \\(\\ell\\)) that satisfy \\(P, Q, R, S \\in \\Gamma\\) such that \\(PQ \\cap... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-f3a37ea7ed582f9797a1c52d196b7842aa6715b71c1c17a81b35d9dc261cd890 | A circle \(\omega\) and its chord \(AB\) are given. Find the locus of the midpoints of chords \(CD\) of the circle \(\omega\) for which the circle with diameter \(CD\) touches \(AB.\) | [
"\\text{An ellipse}"
] | [
"A circle \\(\\omega\\) and its chord \\(AB\\) are given. Find the locus of the midpoints of chords \\(CD\\) of the circle \\(\\omega\\) for which the circle with diameter \\(CD\\) touches \\(AB.\\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-a036827f933e0a8da54bdc2260d49c3e34b43e479129a83cd61424cb3988f64b | A circle \(\omega\) is tangent to two parallel lines \(l_{1}\) and \(l_{2}\). A second circle \(\omega_{1}\) is tangent to \(l_{1}\) at \(A\) and to \(\omega\) externally at \(C\). A third circle \(\omega_{2}\) is tangent to \(l_{2}\) at \(B\), to \(\omega\) externally at \(D\), and to \(\omega_{1}\) externally at \(E\... | [
"6"
] | [
"A circle \\(\\omega\\) is tangent to two parallel lines \\(l_{1}\\) and \\(l_{2}\\). A second circle \\(\\omega_{1}\\) is tangent to \\(l_{1}\\) at \\(A\\) and to \\(\\omega\\) externally at \\(C\\). A third circle \\(\\omega_{2}\\) is tangent to \\(l_{2}\\) at \\(B\\), to \\(\\omega\\) externally at \\(D\\), and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-552ca0c4827e981a98f81cafeb8edef4b0e8f36c398239342d6487ce79069841 | A circle \(w\) with radius \(r\) is inscribed in a triangle. Three additional circles are each tangent to \(w\) and to two sides of the triangle. Let \(r_1\), \(r_2\), and \(r_3\) be the radii of these three circles. Find the minimum possible value of \(r_1 + r_2 + r_3\). | [
"r"
] | [
"A circle \\(w\\) with radius \\(r\\) is inscribed in a triangle. Three additional circles are each tangent to \\(w\\) and to two sides of the triangle. Let \\(r_1\\), \\(r_2\\), and \\(r_3\\) be the radii of these three circles. Find the minimum possible value of \\(r_1 + r_2 + r_3\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db9ffd6bce3ecb439de0ba4be9142f3db5cd6c790c8052c3ba993141cbad6751 | A circle and a parabola are drawn in the $xy$-plane. The circle has its center at $(0, 5)$ with a radius of $4$, and the parabola has its vertex at $(0, 0)$. If the circle is tangent to the parabola at two points, give the equation of the parabola. | [
"\\( y = \\frac{1}{4}x^2 \\)"
] | [
"A circle and a parabola are drawn in the $xy$-plane. The circle has its center at $(0, 5)$ with a radius of $4$, and the parabola has its vertex at $(0, 0)$. If the circle is tangent to the parabola at two points, give the equation of the parabola."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-1c4e77e8f0dd71127cdeab2f0c367bac333e274316a73c4734255ce383798ff8 | A circle and a point \(A\) outside it are given in the plane. All possible triangles \(ABC\) circumscribed about a given circle are considered. Find the locus of the intersection points of the bisector of angle \(B\) with the midline (or its extension) of triangle \(ABC\) parallel to side \(BC\). | [
"\\text{The circle with diameter } OA"
] | [
"A circle and a point \\(A\\) outside it are given in the plane. All possible triangles \\(ABC\\) circumscribed about a given circle are considered. Find the locus of the intersection points of the bisector of angle \\(B\\) with the midline (or its extension) of triangle \\(ABC\\) parallel to side \\(BC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cdb5ddfc8a5e2a0348d598d9370cefc7312fa3bc8fa03a2cabdb4396216b5d5 | A circle and a square have the same center and equal areas. If the square has a side length of 2, what is the area of the intersection of the two shapes? | [
"\\( 4 - \\frac{16}{\\pi}\\cos^{-1}\\left(\\frac{\\sqrt{\\pi}}{2}\\right) + \\frac{4}{\\pi}\\sqrt{4-\\pi} \\)"
] | [
"A circle and a square have the same center and equal areas. If the square has a side length of 2, what is the area of the intersection of the two shapes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-394655e9a458104a17ba7a9d2990a845ead9b3dd73a23e9f18f880efad43e949 | A circle and an ellipse have the same center. The circle passes through the foci $F_1$ and $F_2$ of the ellipse, such that the two curves intersect at 4 points. Let $P$ be any one of their points of intersection. If the major axis of the ellipse is of length 17 and the area of the triangle $PF_1F_2$ is 30, then the dis... | [
"\\( P = 3 \\)"
] | [
"A circle and an ellipse have the same center. The circle passes through the foci $F_1$ and $F_2$ of the ellipse, such that the two curves intersect at 4 points. Let $P$ be any one of their points of intersection. If the major axis of the ellipse is of length 17 and the area of the triangle $PF_1F_2$ is 30, then th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6a911f7ecfd0c6cee28977ab8fcb431fe434150495d023ccbb9e3967ece48a5 | A circle can be inscribed in the quadrilateral $ABCD$. Extensions of side $AB$ beyond point $B$ and side $DC$ beyond point $C$ intersect at point $E$. Extensions of side $DA$ beyond point $A$ and side $CB$ beyond point $B$ intersect at point $F$. Let $I_1, I_2,$ and $I_3$ denote the centers of the circles inscribed in ... | [
"2"
] | [
"A circle can be inscribed in the quadrilateral $ABCD$. Extensions of side $AB$ beyond point $B$ and side $DC$ beyond point $C$ intersect at point $E$. Extensions of side $DA$ beyond point $A$ and side $CB$ beyond point $B$ intersect at point $F$. Let $I_1, I_2,$ and $I_3$ denote the centers of the circles inscribe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71380ae9a6ad8ec589c396605fe4346dd177faa1aeecee9961ff859e799d9930 | A circle can intersect the graph of $y = \sin x$ at at most how many points? | [
"6"
] | [
"A circle can intersect the graph of $y = \\sin x$ at at most how many points?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9357a79acc5e1661bb3b500e08262276d34ae13a30f5136a849d1114a5fe43d3 | A circle centered at $(0, 1)$ intersects the parabola $y = x^2$ at four points $A, B, C, D$. Find the largest possible value of the area of the quadrilateral $ABCD$. | [
"\\(\\frac{4\\sqrt{6}}{9}\\)"
] | [
"A circle centered at $(0, 1)$ intersects the parabola $y = x^2$ at four points $A, B, C, D$. Find the largest possible value of the area of the quadrilateral $ABCD$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-e7723e8fa73968d9c0d4e9ca85f6941bd0e7e99f2e67ade640db423affec9597 | A circle centered at $O$ has a radius of 25. Chord $AB$ with length 30 intersects chord $CD$ with length 14 at point $P$. If the distance between the midpoints of these chords is 12, then what is the length of $OP$? | [
"\\dfrac{45\\sqrt{14}}{7}"
] | [
"A circle centered at $O$ has a radius of 25. Chord $AB$ with length 30 intersects chord $CD$ with length 14 at point $P$. If the distance between the midpoints of these chords is 12, then what is the length of $OP$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc2e64973c782943686cedce1ad969a2652d848d4bae0db44e1b0aaf20061076 | A circle centered at $O$ intersects a circle centered at $O'$ at points $A$ and $B$. The line $TT'$ is tangent to the first circle at $T$ and to the second circle at $T'$. The perpendiculars from $T$ and $T'$ to the line $OO'$ meet at points $S$ and $S'$, respectively. The ray $AS$ intersects the first circle again at ... | [
"10"
] | [
"A circle centered at $O$ intersects a circle centered at $O'$ at points $A$ and $B$. The line $TT'$ is tangent to the first circle at $T$ and to the second circle at $T'$. The perpendiculars from $T$ and $T'$ to the line $OO'$ meet at points $S$ and $S'$, respectively. The ray $AS$ intersects the first circle agai... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b5e9f57aacecf7081a52d5422024ad4675892ce961ee8bd20201135afe04ab71 | A circle centered at $O$ is circumscribed about $\triangle ABC$ as follows: [asy] pair pA, pB, pC, pO; pO = (0, 0); pA = pO + dir(-20); pB = pO + dir(90); pC = pO + dir(190); draw(pA--pB--pC--pA); draw(pO--pA); draw(pO--pB); draw(pO--pC); label("$O$", pO, S); label("$110^\circ$", pO, NE); label("$100^\circ$", pO, NW); ... | [
"50^\\circ"
] | [
"A circle centered at $O$ is circumscribed about $\\triangle ABC$ as follows: [asy]\npair pA, pB, pC, pO;\npO = (0, 0);\npA = pO + dir(-20);\npB = pO + dir(90);\npC = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pO--pA);\ndraw(pO--pB);\ndraw(pO--pC);\nlabel(\"$O$\", pO, S);\nlabel(\"$110^\\circ$\", pO, NE);\nlabel(\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1edbdbd15cb6e1e7a1348779c9db8bd580c7b84f45992cb317227ceb80d3efaf | A circle centered at $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$, and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. If $DE = CF$, find the ratio $\frac{AB}{CD}$. | [
"2"
] | [
"A circle centered at $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$, and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. If $DE = CF$, find the ratio $\\frac{AB}{CD}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f459c1099a8cad8c76afccee22b34e6638d6c576cb2a02ac40f5201fc1fb7f25 | A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$. The rays $BA$ and $CD$ intersect at $K$, and the rays $AD$ and $BC$ intersect at $L$. Points $X$ and $Y$ are on the line segments $OA$ and $OC$, respectively. If $\angle XKY = \frac{1}{2} \angle AKC$, find the value of $\frac{\angle XL... | [
"\\dfrac{1}{2}"
] | [
"A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$. The rays $BA$ and $CD$ intersect at $K$, and the rays $AD$ and $BC$ intersect at $L$. Points $X$ and $Y$ are on the line segments $OA$ and $OC$, respectively. If $\\angle XKY = \\frac{1}{2} \\angle AKC$, find the value of $\\frac{... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80bd1897631bd1ed8d9ba0362e616b3fae7bee4d664dad549a4cea51d5029215 | A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$. The rays $BA$ and $CD$ intersect at $K$, the rays $AD$ and $BC$ intersect at $L$. Points $X$ and $Y$ are on the line segments $OA$ and $OC$, respectively. If $\angle XKY = \frac{1}{2} \angle AKC$, find the value of $\frac{\angle XLY}{\... | [
"\\dfrac{1}{2}"
] | [
"A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$. The rays $BA$ and $CD$ intersect at $K$, the rays $AD$ and $BC$ intersect at $L$. Points $X$ and $Y$ are on the line segments $OA$ and $OC$, respectively. If $\\angle XKY = \\frac{1}{2} \\angle AKC$, find the value of $\\frac{\\an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d872fa8f56a03a1074b2f3f4b261b7bce79ededc43e0dc31904c6950020bee86 | A circle centered at $O_1$ intersects a circle centered at $O_2$ at points $M$ and $M'$. The line $\gamma$, which is tangent to both circles, intersects $O_1O_2$ at point $P$. The line $PM$ intersects the circles $(O_1)$ and $(O_2)$ at points $M_1$ and $M_2$ respectively (with $M_1, M_2 \neq M$), and the line $PM'$ int... | [
"4"
] | [
"A circle centered at $O_1$ intersects a circle centered at $O_2$ at points $M$ and $M'$. The line $\\gamma$, which is tangent to both circles, intersects $O_1O_2$ at point $P$. The line $PM$ intersects the circles $(O_1)$ and $(O_2)$ at points $M_1$ and $M_2$ respectively (with $M_1, M_2 \\neq M$), and the line $P... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-4202f6684d8185b4409f834183b2013447835e78bfc86b013de0f157431d838a | A circle centered at (4, 0) is tangent to the curve \( y = x^3 + 1 \) at the point \((x_0, y_0)\). The sum of all possible values of \( x_0 \) is \( \frac{p}{q} \) where \( p \) and \( q \) are relatively prime positive integers. Find \( p + q \). | [
"1"
] | [
"A circle centered at (4, 0) is tangent to the curve \\( y = x^3 + 1 \\) at the point \\((x_0, y_0)\\). The sum of all possible values of \\( x_0 \\) is \\( \\frac{p}{q} \\) where \\( p \\) and \\( q \\) are relatively prime positive integers. Find \\( p + q \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-8ef1763138624a5daf1017b89bc86b2d1af47ec8b4406c313c67c1b21109fc62 | A circle centered at \( O \) is tangent to all sides of the convex quadrilateral \( ABCD \). The rays \( BA \) and \( CD \) intersect at \( K \), and the rays \( AD \) and \( BC \) intersect at \( L \). Points \( X \) and \( Y \) are on the line segments \( OA \) and \( OC \), respectively. If \( \angle XKY = \frac{1}{... | [
"60^\\circ"
] | [
"A circle centered at \\( O \\) is tangent to all sides of the convex quadrilateral \\( ABCD \\). The rays \\( BA \\) and \\( CD \\) intersect at \\( K \\), and the rays \\( AD \\) and \\( BC \\) intersect at \\( L \\). Points \\( X \\) and \\( Y \\) are on the line segments \\( OA \\) and \\( OC \\), respectively.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca66681c9b5a4590107a4632b99b9e14089896b33e3f5b6a9bce8cef2a070f36 | A circle centered at point $O$ is divided into $n$ equal parts by points $A_1, A_2, \ldots, A_n$ listed sequentially clockwise. The angle $A_2OA_3$ is divided into two equal angles, the angle $A_3OA_4$ into three equal angles, and so on, with the angle $A_nOA_1$ divided into $n$ equal angles. A point on each ray $OA_i$... | [
"L(n) = \\sum_{k=2}^{n} \\sin\\left(\\frac{2\\pi}{nk}\\right)"
] | [
"A circle centered at point $O$ is divided into $n$ equal parts by points $A_1, A_2, \\ldots, A_n$ listed sequentially clockwise. The angle $A_2OA_3$ is divided into two equal angles, the angle $A_3OA_4$ into three equal angles, and so on, with the angle $A_nOA_1$ divided into $n$ equal angles. A point on each ray ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
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