Olympiad problems

1998 Korea Junior Math Olympiad, 7

$O$ is a circumcircle of non-isosceles triangle $ABC$ and the angle bisector of $A$ meets $BC$ at $D$. If the line perpendicular to $BC$ passing through $D$ meets $AO$ at $E$, show that $ADE$ is an isosceles triangle.

2014 NIMO Problems, 9

Tags: rectangle , induction , geometry

Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win? (Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.) [i]Proposed by Lewis Chen[/i]

2014 Korea Junior Math Olympiad, 7

Tags: rhombus , parallelogram , incenter , geometry , projective geometry

In a parallelogram $\Box ABCD$ $(AB < BC)$ The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$. The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$. Let $S$ = $PQ$ $\cap$ $AD$ $U$ = $AR$ $\cap$ $CS$ $T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$ Prove that $AT, BU, PQ$ are concurrent

2009 Germany Team Selection Test, 3

Tags: parallelogram , geometry

Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms: (a) Show that \[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\] (b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?

2024 Princeton University Math Competition, A4 / B6

Tags: geometry

Let $\triangle ABC$ be such that $AB = 15, BC = 13, CA = 14.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $P$ is on the same side of $AB$ as $C$ and $AP = BP.$ Let $X$ be the foot of the perpendicular from $P$ to $AC.$ Then the length of $AX$ is $\tfrac{m}{n}$ for some relatively prime positive integers $m$ and $n.$ Find $m + n.$

2018 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle chasing , rhombus

The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal?

2024 Korea National Olympiad, 1

Tags: geometry

Let there be a circle with center $O$, and three distinct points $A, B, X$ on the circle, where $A, B, O$ are not collinear. Let $\Omega$ be the circumcircle of triangle $ABO$. Segments $AX, BX$ intersect $\Omega$ at points $C(\neq A), D(\neq B)$, respectively. Prove that $O$ is the orthocenter of triangle $CXD$.

2002 Austrian-Polish Competition, 2

Tags: geometry , convex polygon , convex

Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.

2002 District Olympiad, 3

Tags: geometry , midpoint , equilateral , centroid , distance

Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle. a) Show that $O$ is at equal distances from the midpoints of the three segments considered. b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.

2015 Bundeswettbewerb Mathematik Germany, 3

Tags: geometry , bwm , ray , angle , triangle , concurrency

Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$. Show that the rays $[AX$ and $[BY$ intersect on line $CM$.

2017 Poland - Second Round, 4

Tags: geometry , circumcircle

Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.

Ukrainian TYM Qualifying - geometry, 2019.10

Tags: orthocenter , altitude , geometry

At the altitude $AH_1$ of an acute non-isosceles triangle $ABC$ chose a point $X$ , from which draw the perpendiculars $XN$ and $XM$ on the sides $AB$ and $AC$ respectively. It turned out that $H_1A$ is the angle bisector $MH_1N$. Prove that $X$ is the point of intersection of the altitudes of the triangle $ABC$.

2018 IMO Shortlist, G6

Tags: geometry , isogonal conjugate , symmedian

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

2010 IMAR Test, 2

Tags: geometry

Given a triangle $ABC$, let $D$ be the point where the incircle of the triangle $ABC$ touches the side $BC$. A circle through the vertices $B$ and $C$ is tangent to the incircle of triangle $ABC$ at the point $E$. Show that the line $DE$ passes through the excentre of triangle $ABC$ corresponding to vertex $A$.

2002 AMC 10, 19

Tags: geometry

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? $ \text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi$

1972 USAMO, 4

Tags: inequalities , 3d geometry , geometry

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]

2024 Macedonian TST, Problem 3

Tags: geometry

Let $\triangle ABC$ be a triangle. On side $AB$ take points $K$ and $L$ such that $AK \;=\; LB \;<\;\tfrac12\,AB,$ on side $BC$ take points $M$ and $N$ such that $BM \;=\; NC \;<\;\tfrac12\,BC,$ and on side $CA$ take points $P$ and $Q$ such that $CP \;=\; QA \;<\;\tfrac12\,CA.$ Let $R \;=\; KN\;\cap\;MQ, \quad T \;=\; KN \cap LP, $ and $ D \;=\; NP \cap LM, \quad E \;=\; NP \cap KQ.$ Prove that the lines $DR, BE, CT$ are concurrent.

2020 Nigerian Senior MO Round 2, 2

Tags: geometry

Let $D$ be a point in the interior of $ABC$. Let $BD$ and $AC$ intersect at $E$ while $CD$ and $AB$ intersect at $F$. Let $EF$ intersect $BC$ at $G$. Let $H$ be an arbitrary point on $AD$. Let $HF$ and $BD$ intersect at $I$. Let $HE$ and $CD$ intersect at $J$ . prove that $G$,$I$ and $J$ are collinear

2009 Czech and Slovak Olympiad III A, 6

Tags: circumcircle , geometry , perpendicular bisector

Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.

2016 Tournament Of Towns, 6

Tags: geometry

Q. An automatic cleaner of the disc shape has passed along a plain floor. For each point of its circular boundary there exists a straight line that has contained this point all the time. Is it necessarily true that the center of the disc stayed on some straight line all the time? ($9$ marks)

2013 BAMO, 2

Tags: geometry

Let triangle $\triangle{ABC}$ have a right angle at $C$, and let $M$ be the midpoint of the hypotenuse $AB$. Choose a point $D$ on line $BC$ so that angle $\angle{CDM}$ measures $30$ degrees. Prove that the segments $AC$ and $MD$ have equal lengths.

1988 IberoAmerican, 1

Tags: trigonometry , arithmetic sequence , geometry

The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

2020 Balkan MO Shortlist, G1

Tags: geometry

Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$. [i]Proposed by Sam Bealing, United Kingdom[/i]

2014 Vietnam National Olympiad, 3

Tags: geometry , circumcircle , combinatorics

Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices. a) Find all possible values of pair $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

2010 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\tfrac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle. [asy] size(150); pair A = (-5,0); pair B = (5,0); pair C = (-3,4); pair D = (3,4); pair E = intersectionpoint(B--C,A--D); draw(A--B--D--cycle); draw(A--C); draw(C--E); draw(E--B,dashed); markscalefactor=0.06; draw(rightanglemark(A,C,B)); [/asy]