Olympiad problems

2016 Belarus Team Selection Test, 1

Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively. Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$

Ukrainian TYM Qualifying - geometry, 2012.2

Tags: geometry , construction , angle bisector

The triangle $ABC$ is drawn on the board such that $AB + AC = 2BC$. The bisectors $AL_1, BL_2, CL_3$ were drawn in this triangle, after which everything except the points $L_1, L_2, L_3$ was erased. Use a compass and a ruler to reconstruct triangle $ABC$.

Ukrainian From Tasks to Tasks - geometry, 2013.13

Tags: angle , circumcenter , geometry

In the quadrilateral $ABCD$ it is known that $ABC + DBC = 180^o$ and $ADC + BDC = 180^o$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the diagonal $AC$.

2023 Sharygin Geometry Olympiad, 20

Tags: geometry , isogonal conjugate , tangent , symmedian , parallel lines

Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.

2005 APMO, 3

Tags: geometry , algorithm

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

1986 AIME Problems, 9

Tags: parallelogram , analytic geometry , symmetry , ratio , geometry

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

2008 IberoAmerican Olympiad For University Students, 2

Tags: geometry , algebra , polynomial , ratio

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

Ukraine Correspondence MO - geometry, 2017.11

Tags: parallelogram , geometry , tangent circle

Inside the parallelogram $ABCD$, choose a point $P$ such that $\angle APB+ \angle CPD= \angle BPC+ \angle APD$. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles $APB$, $BPC$, $CPD$ and $APD$.

1994 AIME Problems, 15

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection

Given a point $P$ on a triangular piece of paper $ABC,$ consider the creases that are formed in the paper when $A, B,$ and $C$ are folded onto $P.$ Let us call $P$ a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,$ and $\angle B=90^\circ.$ Then the area of the set of all fold points of $\triangle ABC$ can be written in the form $q\pi-r\sqrt{s},$ where $q, r,$ and $s$ are positive integers and $s$ is not divisible by the square of any prime. What is $q+r+s$?

2013 IMO Shortlist, G6

Tags: geometry , conditional geometry

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled. [i]Proposed by Alexander A. Polyansky, Russia[/i]

1987 Tournament Of Towns, (133) 2

Tags: geometry , altitude , parallel

In an acute angled triangle the feet of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .

2020 IMO Shortlist, G4

Tags: geometry , disks , geometric inequality , inequalities

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2023 Sharygin Geometry Olympiad, 7

Tags: geometry , tangent circle , euler circle

Let $A$ be a ﬁxed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a ﬁxed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some ﬁxed circle not depending on $BC$.

1979 AMC 12/AHSME, 16

Tags: geometry , arithmetic sequence

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is $\textbf{(A) }\frac{\sqrt{3}}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{2}{\sqrt{3}}\qquad\textbf{(D) }\frac{3}{2}\qquad\textbf{(E) }\sqrt{3}$

2007 India Regional Mathematical Olympiad, 1

Tags: angle bisector , geometry , trigonometry , rectangle

Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$. Show that $ \angle CED > 45^\circ .$ [b][weightage 17/100][/b]

2016 ASMT, T3

Tags: conic , geometry , ellipse , analytic geometry

An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.

2007 Romania Team Selection Test, 2

Tags: circumcircle , geometry

Let $ABC$ be a triangle, $E$ and $F$ the points where the incircle and $A$-excircle touch $AB$, and $D$ the point on $BC$ such that the triangles $ABD$ and $ACD$ have equal in-radii. The lines $DB$ and $DE$ intersect the circumcircle of triangle $ADF$ again in the points $X$ and $Y$. Prove that $XY\parallel AB$ if and only if $AB=AC$.

2019 OMMock - Mexico National Olympiad Mock Exam, 1

Tags: geometry , similarity , center

Let $C_1$ and $C_2$ be two circles with centers $O_1$ and $O_2$, respectively, intersecting at $A$ and $B$. Let $l_1$ be the line tangent to $C_1$ passing trough $A$, and $l_2$ the line tangent to $C_2$ passing through $B$. Suppose that $l_1$ and $l_2$ intersect at $P$ and $l_1$ intersects $C_2$ again at $Q$. Show that $PO_1B$ and $PO_2Q$ are similar triangles. [i]Proposed by Pablo Valeriano[/i]

2006 China Team Selection Test, 1

Tags: geometric transformation , geometry , power of a point , radical axis , circumcircle , reflection , projective geometry

The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

2009 Canada National Olympiad, 2

Tags: geometry , rotation

Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.

2017 BAMO, E/4

Tags: rectangle , geometry

Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon. As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):

2004 Argentina National Olympiad, 5

Tags: geometry , equal segments , pentagon , area

The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.

2012 Germany Team Selection Test, 2

Tags: geometry , circumcircle

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

2015 Peru IMO TST, 3

Tags: geometry

Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$

2024 Chile Junior Math Olympiad, 4

Tags: geometry

Consider a triangle with sides of length $ a $, $ b $, and $ c $ that satisfy the following conditions: \[ a + b = c + 3 \quad c^2 + 9 = 2ab \] Find the area of the triangle.