Olympiad problems

2011 NZMOC Camp Selection Problems, 2

In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?

2013 Dutch IMO TST, 2

Tags: ratio , circumcircle , geometry

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.

2023 Ukraine National Mathematical Olympiad, 9.6

Tags: geometry

A point $O$ lies inside $\triangle ABC$ so that $\angle BOC=90-\angle BAC$. Let $BO, CO$ meet $AC, AB$ at $K, L$. Points $K_1, L_1$ lie on the segments $CL, BK$ so that $K_1B=K_1K$ and $L_1C=L_1L$. If $M$ is the midpoint of $BC$, then prove that $\angle K_1ML_1=90^{o}$. [i]Proposed by Anton Trygub[/i]

1974 All Soviet Union Mathematical Olympiad, 202

Tags: convex polygon , geometry , area

Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.

2018 Auckland Mathematical Olympiad, 3

Tags: geometry , paper folding , folding , rectangle , area

A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]

2004 AIME Problems, 9

Tags: trapezoid , rectangle , number theory , relatively prime , geometry

Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is similar to $V_2$. The minimum value of the area of $U_1$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2017 Costa Rica - Final Round, G2

Tags: geometry , incenter , right triangle

Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$. Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$, and let $K$ and $ L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.

2005 Federal Competition For Advanced Students, Part 2, 3

Tags: 3d geometry , geometric transformation , reflection , geometry

Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.

OMMC POTM, 2023 8

Tags: geometry , geometric transformation , dilation

Find all polygons $P$ that can be covered completely by three (possibly overlapping) smaller dilated versions of itself. [i]Proposed by Evan Chang (squareman), USA[/i]

2014 Contests, 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.

2018 Pan African, 4

Tags: geometry , cyclic quadrilateral , angle chasing

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

2021 Saudi Arabia Training Tests, 20

Tags: geometry , bisects segment , circumcircle

Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$ ($D \in BC$), $M$ is the midpoint of $AD$ and $O$ is the circumcenter. Line $AO$ meets $BC$ at $K$ and circle of center $K$, radius $KA$ cuts $AB,AC$ at $E, F$ respectively. Prove that $AO$ bisects $EF$.

2012 South africa National Olympiad, 2

Tags: geometry , similar triangles

Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$. The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$, determine the length of $YZ$

2009 Tournament Of Towns, 5

Tags: tetrahedron , 3d geometry , geometry , concurrency

Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.

2015 Iran MO (2nd Round), 1

Tags: geometry

In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.

2001 Switzerland Team Selection Test, 3

Tags: ratio , pentagon , parallelogram , geometry

In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

2025 CMIMC Geometry, 9

Tags: geometry

Define the [i]ratio[/i] of an ellipse to be the length of the major axis divided by the length of its minor axis. Given a trapezoid $ABCD$ with $AB \parallel DC$ and that $\angle{ADC}$ is a right angle, with $AB=18, AD=33, CD=130,$ find the smallest ratio of any ellipse that goes through all vertices of $ABCD.$

2022-23 IOQM India, 9

Tags: geometry

Two sides of an integer sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer $y$ such that $18,x,y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have.

1989 All Soviet Union Mathematical Olympiad, 497

Tags: geometry , geometric inequality , equal ratio , area

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

1981 IMO Shortlist, 18

Tags: sphere , 3d geometry , area , geometry

Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

1968 All Soviet Union Mathematical Olympiad, 106

Tags: geometry , incircle , median

Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.

2012 China Northern MO, 1

Tags: geometry , perpendicular , right triangle

As shown in figure, given right $\vartriangle ABC$ with $\angle C=90^o$. $I$ is the incenter. The line $BI$ intersects segment $AC$ at the point $D$ . The line passing through $D$ parallel to $AI$ intersects $BC$ at point $E$. The line $EI$ intersects segment $AB$ at point $F$. Prove that $DF \perp AI$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/6fc94adb4ce12c3bf07948b8c57170ca01b256.png[/img]

2022 Tuymaada Olympiad, 7

Tags: geometry , angle

$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$ [i](K. Ivanov )[/i]

1999 Harvard-MIT Mathematics Tournament, 1

Tags: geometry

Two $10 \times 24$ rectangles are inscribed in a circle as shown. Find the shaded area. [img]https://cdn.artofproblemsolving.com/attachments/1/7/c97fb0e6f45a52fa751777da6ebc519839e379.png[/img]

2008 Greece Junior Math Olympiad, 4

Tags: geometry , trapezoid , isosceles , concurrency

Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that : i) triangle $BDZ$ is isosceles ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$ iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$