Olympiad problems

2010 Tournament Of Towns, 2

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$. $(a)$ Prove that $ABCD$ has an incircle. $(b)$ Prove that $ABCD$ is symmetric about one of its diagonals.

2001 China Team Selection Test, 1

Tags: geometry

In $ \triangle ABC $ with $ AB > BC $, a tangent to the circumcircle of $ \triangle ABC $ at point $ B $ intersects the extension of $ AC $ at point $ D $. $ E $ is the midpoint of $ BD $, and $ AE $ intersects the circumcircle of $ \triangle ABC $ at $ F $. Prove that $ \angle CBF = \angle BDF $.

1998 Swedish Mathematical Competition, 4

Tags: area , angle , geometry

$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?

1995 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry

Consider $2n$ points in the plane. Two players $A$ and $B$ alternately choose a point on each move. After $2n$ moves, there are no points left to choose from and the game ends. Add up all the distances between the points chosen by $A$ and add up all the distances between the points chosen by $B$. The one with the highest sum wins. If $A$ starts the game, describe the winner's strategy. Clarification: Consider that all the partial sums of distances between points give different numbers.

2005 Bulgaria Team Selection Test, 1

Tags: geometry , trigonometry

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

Estonia Open Senior - geometry, 2018.1.1

Tags: equilateral , lattice points , geometry

Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

2005 AMC 12/AHSME, 12

Tags: graphing lines , slope , rectangle , greatest common divisor , analytic geometry , number theory , geometry

A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

2018 Iran Team Selection Test, 1

Tags: geometry

Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$ at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$. [i]Proposed by Ali Zamani [/i]

1991 Tournament Of Towns, (318) 5

Tags: geometric transformation , centroid , equilateral , rotation , geometry

Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide. (Bykovsky, Khabarovsksk)

2013 Sharygin Geometry Olympiad, 6

Tags: geometry

Dear Mathlinkers, 1. A, B the end points of an arch circle 2. (O) a circle tangent to AB intersecting the arch in question 3. T the point of contact of (O) and AB 4. C, D the points of intersection of (O) with the arch in the order A, D, C, B 5. E, F the points of intersection of AC and DT, BD and CT. Prove : EF is parallel to AB. Sincerely Jean-Louis

2005 MOP Homework, 1

Tags: pigeonhole principle , rectangle , rotation , combinatorics , geometry

Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board. (a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields. (b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.

2009 JBMO TST - Macedonia, 4

Tags: combinatorics , algorithm , rectangle , induction , geometry

In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?

2008 India Regional Mathematical Olympiad, 6

Tags: triangle inequality , combinatorics , perimeter , trigonometry , geometry , inequalities

Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$. [16 points out of 100 for the 6 problems]

Ukraine Correspondence MO - geometry, 2006.7

Tags: right triangle , geometric inequality , geometry

Let $D$ and $E$ be the midpoints of the sides $BC$ and $AC$ of a right triangle $ABC$. Prove that if $\angle CAD=\angle ABE$, then $$\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.$$

1963 Polish MO Finals, 3

Tags: geometric inequalities , rectangle , geometry

From a given triangle, cut out the rectangle with the largest area.

1998 Italy TST, 2

Tags: equilateral , median , altitude , angle bisector , geometry

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

1999 AIME Problems, 8

Tags: inequalities , analytic geometry , geometry , parallelogram

Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2012 India National Olympiad, 1

Tags: inequalities , topology , circumcircle , geometry , trigonometry , function

Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.

2004 Germany Team Selection Test, 2

Tags: menelaus , angle bisector , geometry

In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2009 CentroAmerican, 5

Tags: geometric transformation , circumcircle , homothety , projective geometry , geometry

Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.

2014 Middle European Mathematical Olympiad, 6

Tags: geometric transformation , reflection , geometry , homothety , perpendicular bisector

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B, C, N,$ and $L$ are concyclic.

2023 Novosibirsk Oral Olympiad in Geometry, 6

Tags: construction , geometric construction , area , geometry

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

2024 Saint Petersburg Mathematical Olympiad, 6

Tags: circumcircle , geometry

Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.

2001 National High School Mathematics League, 9

Tags: 3d geometry , geometry

The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.

2018 Germany Team Selection Test, 2

Tags: inscribed circle , pentagon , geometry

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.