This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 25757

2017-IMOC, C7
Tags: combinatorics , geometry
There are $12$ monsters in a plane. Each monster is capable of spraying fire in a $30$-degree cone. Prove that monsters can destroy the plane.

Kvant 2021, M2678
Tags: geometry
The triangle $ABC$ is given. Let $A', B'$ and $C'$ be the midpoints of the sides $BC, CA$ and $AB$ and $O_a,O_b$ and $O_c$ be the circumcenters of the triangles $CAC', ABA'$ and $BCB'$ respectively. Prove that the triangles $ABC$ and $O_aO_bO_c$ are similar. [i]Proposed by Don Luu (Vietnam)[/i]

1989 Turkey Team Selection Test, 6
Tags: geometry , homothety , geometric transformation , circumcircle
The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .

2014-2015 SDML (High School), 13
Tags: geometry
Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices. $\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$

2018 Caucasus Mathematical Olympiad, 6
Tags: geometric inequality , geometry , inequalities
Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leqslant AD+BD$.

2013 IMO Shortlist, G1
Tags: geometry
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear. [i]Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand[/i]

2014 IMO Shortlist, G6
Tags: humpty point , geometry
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}$ [i]Proposed by Ali Zamani, Iran[/i]

2001 USAMO, 6
Tags: geometry , reflection
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.

2004 AMC 10, 9
Tags: geometry
A square has sides of length $ 10$, and a circle centered at one of its vertices has radius $ 10$. What is the area of the union of the regions enclosed by the square and the circle? $ \textbf{(A)}\ 200 \plus{} 25\pi\qquad \textbf{(B)}\ 100 \plus{} 75\pi\qquad \textbf{(C)}\ 75 \plus{} 100\pi\qquad \textbf{(D)}\ 100 \plus{} 100\pi$ $ \textbf{(E)}\ 100 \plus{} 125\pi$

1999 Irish Math Olympiad, 4
Tags: rectangle , combinatorics , geometry
A $ 100 \times 100$ square floor consisting of $ 10000$ squares is to be tiled by rectangular $ 1 \times 3$ tiles, fitting exactly over three squares of the floor. $ (a)$ If a $ 2 \times 2$ square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles. $ (b)$ If, instead, a $ 2 \times 2$ square is removed from the corner, prove that such a tiling is not possble.

2014 South East Mathematical Olympiad, 7
Tags: exterior angle , geometry
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.

2010 Chile National Olympiad, 3
Tags: incenter , geometry , midpoint
The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.

1991 ITAMO, 1
Tags: circumcircle , geometry , angle bisector
For every triangle $ABC$ inscribed in a circle $\Gamma$ , let $A',B',C'$ be the intersections of the bisectors of the angles at $A,B,C$ with $\Gamma$ . Consider the triangle $A'B'C'$ . (a) Do triangles $A'B'C'$ go over all possible triangles inscribed in $\Gamma$ as $\vartriangle ABC$ varies? If not, what are the constraints? (b) Prove that the angle bisectors of $\vartriangle ABC$ are the altitudes of $\vartriangle A',B',C'$ .

2010 Stanford Mathematics Tournament, 17
Tags: geometry , perpendicular bisector
An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle

2003 Manhattan Mathematical Olympiad, 1
Tags: geometry
There are 2003 points chosen randomly in the plane in such a way that no three of them lie on a straight line. Prove that there exists a circle which contains at least three of the given points on its circumference, and no other given points inside.

2009 Indonesia TST, 2
Tags: geometry , perpendicular bisector
Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.

2020 USOJMO, 4
Tags: geometry , dennisthepro
Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$. [i]Milan Haiman[/i]

2005 Iran Team Selection Test, 2
Tags: reflection , geometry , symmetry , circumcircle , ratio , geometric transformation , parallelogram
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

2009 Tournament Of Towns, 5
Tags: geometry , rhombus , circumcenter
In rhombus $ABCD$, angle $A$ equals $120^o$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ so that angle $NAM$ equals $30^o$. Prove that the circumcenter of triangle $NAM$ lies on a diagonal of of the rhombus.

2017 China Northern MO, 6
Tags: geometry
Find all integers \(n\) such that there exists a concave pentagon which can be dissected into \(n\) congruent triangles.

2010 China Team Selection Test, 1
Tags: reflection , circumcircle , geometry , geometric transformation
Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

2024 AIME, 8
Tags: geometry
Eight circles of radius $34$ can be placed tangent to side $\overline{BC}$ of $\triangle ABC$ such that the first circle is tangent to $\overline{AB}$, subsequent circles are externally tangent to each other, and the last is tangent to $\overline{AC}$. Similarly, $2024$ circles of radius $1$ can also be placed along $\overline{BC}$ in this manner. The inradius of $\triangle ABC$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2023 BMT, 2
Tags: geometry
Consider an equilateral triangle with side length $9$. Each side is divided into $3$ equal segments by $2$ points, for a total of $6$ points. Compute the area of the circle passing through these$ 6$ points. [img]https://cdn.artofproblemsolving.com/attachments/7/b/1860a3ff86a0e4b93a4891861300dcb09adad4.png[/img]

1960 AMC 12/AHSME, 24
Tags: integration , geometry , calculus , logarithm , 3d geometry , irrational number , function
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

1999 Korea Junior Math Olympiad, 5
Tags: circumcircle , geometry
$O$ is a circumcircle of $ABC$ and $CO$ meets $AB$ at $P$, and $BO$ meets $AC$ at $Q$. Show that $BP=PQ=QC$ if and only if $\angle A=60^{\circ}$.