Olympiad problems

1982 IMO Shortlist, 8

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

2011 ITAMO, 4

Tags: conic , ellipse , italy , geometry

$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

2012 Iran MO (3rd Round), 2

Tags: geometry , inequalities , combinatorics

Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$. [i]Proposed by Ali Khezeli[/i]

2010 Contests, 2

Tags: geometry , counting question , combinatorics

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P2

Tags: geometry , orthocenter , perpendicular bisector

Let $\triangle ABC$ be a scalene and acute triangle in which the angle at $A$ is second largest, $H$ is the orthocenter, and $k$ is the circumcircle with center $O$. Let the circumcircle of $\triangle AHO$ intersect the sides $AB$ and $AC$ again at $M$ and $N$, respectively, whereas the altitudes $CH$ and $BH$ intersect $k$ again at $K$ and $L$, respectively. Prove that the intersection of $KL$ and the perpendicular bisector of $AH$ is the orthocenter of $\triangle AMN$. Proposed by [i]Ilija Jovcevski[/i]

2018 ASDAN Math Tournament, 8

Tags: geometry

Aurick has a cup, a right cone with a circular base of radius $\frac12$, filled with milk tea. The slant height of the cup is $1$, and the tea fills the cup $\frac12$ of the way up the cup’s side. Suppose that Aurick tips the cup just to the point of spilling, as shown in the diagram. The new slant height EA and the tilted tea surface’s major axis $ET$ form $\angle T EA$. Compute $\cos(\angle T EA)$. [img]https://cdn.artofproblemsolving.com/attachments/e/e/76e12ee31ce4ba8a5daaf0f5538b98726a0d37.png[/img]

1957 Moscow Mathematical Olympiad, 349

Tags: rectangle table , table , sum , combinatorics , geometry

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

2008 Regional Olympiad of Mexico Center Zone, 6

Tags: geometry , perpendicular

In the quadrilateral $ABCD$, we have $AB = AD$ and $\angle B = \angle D = 90 ^ \circ $. The points $P$ and $Q $ lie on $BC$ and $CD$, respectively, so that $AQ$ is perpendicular on $DP$. Prove that $AP$ is perpendicular to $BQ$.

2019 Ramnicean Hope, 2

Tags: geometry , circumcircle , vector geometry , median

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

2002 Singapore MO Open, 1

Tags: geometry , circles

In the plane, $\Gamma$ is a circle with centre $O$ and radius $r, P$ and $Q$ are distinct points on $\Gamma , A$ is a point outside $\Gamma , M$ and $N$ are the midpoints of $PQ$ and $AO$ respectively. Suppose$ OA = 2a$ and $\angle PAQ$ is a right angle. Find the length of $MN$ in terms of $r$ and $a$. Express your answer in its simplest form, and justify your answer.

2014 Contests, 3

Tags: geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2018 Moscow Mathematical Olympiad, 11

Tags: geometry

Ivan want to paint ball. Ivan can put ball in the glass with some paint, and then one half of ball will be painted. Ivan use $5$ glasses to paint glass competely. Prove, that one glass was not needed, and Ivan can paint ball with $4$ glasses, putting ball in it by same way.

2012 Germany Team Selection Test, 2

Tags: trigonometry , circumcircle , geometry , geometric transformation

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

2006 Purple Comet Problems, 10

Tags: geometry , rectangle

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(2,2)); } } } [/asy]

2008 IMC, 2

Tags: geometry , trigonometry , search , ellipse , conic , analytic geometry , ratio

Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.

2014 AMC 8, 19

Tags: geometry , probability , 3d geometry

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? $\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$

2012 Brazil Team Selection Test, 4

Tags: parallelogram , circumcircle , power of a point , perpendicular bisector , geometry

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2009 Switzerland - Final Round, 1

Tags: geometric inequality , geometry , locus , hexagon

Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that: (a) $d_3$ takes the smallest possible value. (b) $d_4$ takes the smallest possible value.

1985 IMO Shortlist, 15

Tags: geometry , square , dissection

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

2022 USA TSTST, 7

Tags: parallelogram , angle chasing , trapezoid , geometry

Let $ABCD$ be a parallelogram. Point $E$ lies on segment $CD$ such that \[2\angle AEB=\angle ADB+\angle ACB,\] and point $F$ lies on segment $BC$ such that \[2\angle DFA=\angle DCA+\angle DBA.\] Let $K$ be the circumcenter of triangle $ABD$. Prove that $KE=KF$. [i]Merlijn Staps[/i]

2013 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2023 IFYM, Sozopol, 2

Tags: geometry

Given a triangle $ABC$, a line in its plane is called a [i]cool[/i] if it divides the triangle into two parts with equal areas and perimeters. a) Does there exist a triangle $ABC$ with at least seven [i]cool[/i] lines? b) Prove that all [i]cool[/i] lines intersect at a point $X$. If $\angle AXB = 126^\circ$, prove that $(8\sin^2 \angle ACB - 5)^2$ is an integer.

2020 Purple Comet Problems, 2

Tags: geometry

The diagram below shows a $18\times 35$ rectangle with eight points marked that divide each side into three equal parts. Four triangles are removed from each of the corners of the rectangle leaving the shaded region. Find the area of this shaded region. [img]https://cdn.artofproblemsolving.com/attachments/2/1/e0fd592d589a1f5d3324a637743d0e9d6e3480.png[/img]

1976 AMC 12/AHSME, 9

Tags: geometry , ratio

In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is $\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$

2021 Kosovo National Mathematical Olympiad, 4

Tags: geometry

Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.