Olympiad problems

1994 Balkan MO, 1

Tags: geometry , parallelogram , analytic geometry , trigonometry , parameterization , parabola , conic

An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

2007 Ukraine Team Selection Test, 6

Tags: geometry , 3d geometry , number theory

Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes). E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$ only.

2004 Germany Team Selection Test, 3

Tags: geometry , incenter , circumcircle , triangle

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

Ukraine Correspondence MO - geometry, 2009.11

Tags: geometry , angle bisector , angle

In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.

2020 Serbian Mathematical Olympiad, Problem 3

Tags: circumcircle , triangle , angle , geometry

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2005 All-Russian Olympiad, 2

Tags: analytic geometry , pigeonhole principle , geometry

Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?

1969 IMO Longlists, 20

Tags: geometry , polygon , lattice points , area

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

2015 Romania Team Selection Tests, 2

Tags: circles , circumcircle , euler circle , geometry

Let $ABC$ be a triangle . Let $A'$ be the center of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$ , respectively ; the points $B'$ and $C'$ are defined similarly . Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.

Brazil L2 Finals (OBM) - geometry, 2010.5

Tags: diagonal , geometry , circumcircle , cyclic quadrilateral , cyclic

The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.

2024 Israel TST, P3

Tags: geometry , parallelogram , concentric circles , perpendicular bisector , circumcircle

Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$. Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.

2021 Indonesia TST, G

Tags: geometry , parallel , circles , kharkiv masters

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

2018 Costa Rica - Final Round, G1

Tags: geometry , equal segments , equal circles

Let $O$ be the center of the circle circumscribed to $\vartriangle ABC$, and let $ P$ be any point on $BC$ ($P \ne B$ and $P \ne C$). Suppose that the circle circumscribed to $\vartriangle BPO$ intersects $AB$ at $R$ ($R \ne A$ and $R \ne B$) and that the circle circumscribed to $\vartriangle COP$ intersects $CA$ at point $Q$ ($Q \ne C$ and $Q \ne A$). 1) Show that $\vartriangle PQR \sim \vartriangle ABC$ and that$ O$ is orthocenter of $\vartriangle PQR$. 2) Show that the circles circumscribed to the triangles $\vartriangle BPO$, $\vartriangle COP$, and $\vartriangle PQR$ all have the same radius.

2012 Regional Olympiad of Mexico Center Zone, 3

Tags: geometry , parallelogram , congruent triangles

In the parallelogram $ABCD$, $\angle BAD =60 ^ \circ$. Let $E $ be the intersection point of the diagonals. The circle circumscribed to the triangle $ACD$ intersects the line $AB$ at the point $K$ (different from $A$), the line $BD$ at the point $P$ (different from $D$), and to the line $BC$ in $L$ (different from $C$). The line $EP$ intersects the circumscribed circle of the triangle $CEL$ at the points $E$ and $M$. Show that the triangles $KLM$ and $CAP$ are congruent.

1989 Romania Team Selection Test, 3

Tags: rhombus , parallelogram , concyclic , geometry

Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.

2017 Tuymaada Olympiad, 2

Tags: geometry , inequalities

$ABCD $ is a cyclic quadrilateral such that the diagonals $AC $ and $BD $ are perpendicular and their intersection is $P $. Point $Q $ on the segment $CP$ is such that $CQ=AP $. Prove that the perimeter of triangle $BDQ $ is at least $2AC $. Tuymaada 2017 Q2 Juniors

1972 IMO Longlists, 10

Tags: geometry

Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.

Indonesia Regional MO OSP SMA - geometry, 2013.5

Tags: geometry , geometric inequality , angle

Given an acute triangle $ABC$. The longest line of altitude is the one from vertex $A$ perpendicular to $BC$, and it's length is equal to the length of the median of vertex $B$. Prove that $\angle ABC \le 60^o$

2001 Tournament Of Towns, 7

Tags: analytic geometry , geometric transformation , absolute value , geometry

The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle. (a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$? (b) What is the maximum area of this triangle?

2002 Spain Mathematical Olympiad, Problem 2

Tags: geometry

In the triangle $ABC$, $A'$ is the foot of the altitude to $A$, and $H$ is the orthocenter. $a)$ Given a positive real number $k = \frac{AA'}{HA'}$ , find the relationship between the angles $B$ and $C$, as a function of $k$. $b)$ If $B$ and $C$ are fixed, find the locus of the vertice $A$ for any value of $k$.

2013 India IMO Training Camp, 2

Tags: trigonometry , geometry , geometric transformation , reflection , trig identities , law of sines

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

2015 China Team Selection Test, 3

Tags: circumcircle , tangent , geometry

Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$ Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC . $ Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $ Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $ Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $ Prove that $ \odot (AMN) $ is tangent to $ \omega $

2024 India IMOTC, 5

Tags: geometry

Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\] Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$. [i]Proposed by Atul Shatavart Nadig and Shantanu Nene[/i]

2015 British Mathematical Olympiad Round 1, 5

Tags: geometry

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to $BC, CA$ and $AB$ respectively. Let $P, Q, R$ and $S$ be the feet of the perpendiculars from $D$ to $BA, BE, CF$ and $CA$ respectively. Prove that $P, Q, R$ and $S$ are collinear.

2021 CHMMC Winter (2021-22), 6

Tags: geometry

Let $ABC$ be an acute triangle with orthocenter $H$. A point $L \ne A$ lies on the plane of $ABC$ such that $\overline{HL} \perp \overline{AL}$ and $LB : LC = AB : AC$. Suppose $M_1 \ne B$ lies on $\overline{BL}$ such that $\overline{HM_1} \perp \overline{BM_1}$ and $M_2 \ne C$ lies on $\overline{CL}$ such that $\overline{HM_2} \perp \overline{CM_2}$. Prove that $\overline{M_1M_2}$ bisects $\overline{AL}$.

2023 Oral Moscow Geometry Olympiad, 5

Tags: geometry

In an acute-angled triangle $ABC$ with orthocenter $H$, the line $AH$ cuts $BC$ at point $A_1$. Let $\Gamma$ be a circle centered on side $AB$ tangent to $AA_1$ at point $H$. Prove that $\Gamma$ is tangent to the circumscribed circle of triangle $AMA_1$, where $M$ is the midpoint of $AC$.