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

2004 Poland - Second Round, 2

Tags: geometry
In convex hexagon $ ABCDEF$ all sides have equal length and $ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$. Prove that the diagonals $ AD,BE,CF$ are concurrent.

2016 IMO, 3

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

2016 Japan Mathematical Olympiad Preliminary, 4

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$

1978 IMO Shortlist, 12

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

Bangladesh Mathematical Olympiad 2020 Final, #6

Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$[b]ABP =[/b] $\angle$[b]ADP[/b]. What is the radius of the circumcircle of $ABCD$?

2016 Bulgaria EGMO TST, 2

Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$. The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$, respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$. Determine the value of $\frac{CX \cdot CY}{AB^2}$.

2020 Taiwan TST Round 3, 1

Tags: excircle , geometry
Let $\Omega$ be the $A$-excircle of triangle $ABC$, and suppose that $\Omega$ is tangent to lines $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $M$ be the midpoint of segment $EF$. Two more points $P$ and $Q$ are on $\Omega$ such that $EP$ and $FQ$ are both parallel to $DM$. Let $BP$ meet $CQ$ at point $X$. Prove that the line $AM$ is the angle bisector of $\angle XAD$. [i]Proposed by Shuang-Yen Lee[/i]

2021 Switzerland - Final Round, 2

Tags: geometry
Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

2017 CHKMO, Q3

Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.

2010 Purple Comet Problems, 3

Tags: geometry
The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance two apart. Find the area of the irregularly shaped ten sided figure shown. [asy] import graph; size(5cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((-2,5)--(-3,4), linewidth(1.6)); draw((-3,4)--(-2,1), linewidth(1.6)); draw((-2,1)--(1,0), linewidth(1.6)); draw((1,0)--(2,1), linewidth(1.6)); draw((2,1)--(1,3), linewidth(1.6)); draw((1,3)--(1,4), linewidth(1.6)); draw((1,4)--(2,5), linewidth(1.6)); draw((2,5)--(0,5), linewidth(1.6)); draw((-2,5)--(-1,4), linewidth(1.6)); draw((-1,4)--(0,5), linewidth(1.6)); dot((-3,5),linewidth(6pt) + dotstyle); dot((-2,5),linewidth(6pt) + dotstyle); dot((-1,5),linewidth(6pt) + dotstyle); dot((0,5),linewidth(6pt) + dotstyle); dot((1,5),linewidth(6pt) + dotstyle); dot((2,5),linewidth(6pt) + dotstyle); dot((2,4),linewidth(6pt) + dotstyle); dot((2,3),linewidth(6pt) + dotstyle); dot((2,2),linewidth(6pt) + dotstyle); dot((2,1),linewidth(6pt) + dotstyle); dot((2,0),linewidth(6pt) + dotstyle); dot((-3,4),linewidth(6pt) + dotstyle); dot((-3,3),linewidth(6pt) + dotstyle); dot((-3,2),linewidth(6pt) + dotstyle); dot((-3,1),linewidth(6pt) + dotstyle); dot((-3,0),linewidth(6pt) + dotstyle); dot((-2,0),linewidth(6pt) + dotstyle); dot((-2,1),linewidth(6pt) + dotstyle); dot((-2,2),linewidth(6pt) + dotstyle); dot((-2,3),linewidth(6pt) + dotstyle); dot((-2,4),linewidth(6pt) + dotstyle); dot((-1,4),linewidth(6pt) + dotstyle); dot((0,4),linewidth(6pt) + dotstyle); dot((1,4),linewidth(6pt) + dotstyle); dot((1,3),linewidth(6pt) + dotstyle); dot((0,3),linewidth(6pt) + dotstyle); dot((-1,3),linewidth(6pt) + dotstyle); dot((-1,2),linewidth(6pt) + dotstyle); dot((-1,1),linewidth(6pt) + dotstyle); dot((-1,0),linewidth(6pt) + dotstyle); dot((0,0),linewidth(6pt) + dotstyle); dot((1,0),linewidth(6pt) + dotstyle); dot((1,1),linewidth(6pt) + dotstyle); dot((1,2),linewidth(6pt) + dotstyle); dot((0,2),linewidth(6pt) + dotstyle); dot((0,1),linewidth(6pt) + dotstyle); [/asy]

2013 Czech-Polish-Slovak Match, 1

Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.

1998 Harvard-MIT Mathematics Tournament, 8

Tags: geometry
It is not possible to construct a segment of length $\pi$ using a straightedge, compass, and a given segment of length $1$. The following construction, given in $1685$ by Adam Kochansky, yields a segment whose length agrees with $\pi$ to five decimal places: Construct a circle of radius $1$ and call its center $O$. Construct a diameter $AB$ of this circle and a line $\ell$ tangent to the circle at $A$. Next, draw a circle with radius $1$ centered at $A$, and call one of the intersections with the original circle $C$. Now from C draw an arc of radius $1$ intersecting the circle around $A$ at $D$, where $D$ lies outside of the circle centered at $O$. Draw $OD$ and let $E$ be its point of intersection with $\ell$ . Construct $H$ on $AE$ such that $A$ is between $H$ and $E$, and $HE=3$. The distance between $B$ and $H$ is then close to $\pi$; calculate its exact value.

2012 Kyoto University Entry Examination, 2

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

2018 Greece National Olympiad, 2

Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$.

2018 Peru MO (ONEM), 3

Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.

2016 Novosibirsk Oral Olympiad in Geometry, 6

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

1967 IMO Longlists, 45

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

2016 ASMT, T2

Tags: geometry
Let $ABCD$ be a square, and let $E$ be a point external to $ABCD$ such that $AE = CE = 9$ and $BE = 8$. Compute the side length of $ABCD$.

2003 AMC 10, 20

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

2017 Mid-Michigan MO, 10-12

[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends? [b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots? [b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$. [b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$. [b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions). [b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2020 JBMO TST of France, 1

Tags: geometry
Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

2015 Oral Moscow Geometry Olympiad, 2

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$. [img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]

2015 Oral Moscow Geometry Olympiad, 6

In the acute-angled non-isosceles triangle $ABC$, the height $AH$ is drawn. Points $B_1$ and $C_1$ are marked on the sides $AC$ and $AB$, respectively, so that $HA$ is the angle bisector of $B_1HC_1$ and quadrangle $BC_1B_1C$ is cyclic. Prove that $B_1$ and $C_1$ are feet of the altitudes of triangle $ABC$.

2005 AMC 10, 23

Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC \equal{} BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$ \textbf{(A)} \ \frac {1}{6} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {2}{3}$