Olympiad problems

2010 Dutch IMO TST, 1

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

2013 Tournament of Towns, 4

Tags: geometry , isosceles , equal segments , angle

Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.

2015 Sharygin Geometry Olympiad, P13

Tags: midpoint , geometry , concurrency

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.

2023 Belarus Team Selection Test, 2.3

Tags: hyperbola , projective geometry , conic , geometry

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

2009 Greece National Olympiad, 2

Tags: circumcircle , geometric transformation , homothety , geometry

Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.

1962 IMO, 5

Tags: geometry , circumcircle , incenter , angle bisector

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

2013 Ukraine Team Selection Test, 1

Tags: midpoint , isosceles , circumcircle , geometry

Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.

Swiss NMO - geometry, 2004.9

Tags: geometry , cyclic quadrilateral , concurrency

Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.

2021 Greece Junior Math Olympiad, 4

Tags: geometry , equal circles

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

2010 VJIMC, Problem 4

Tags: function , geometry , rectangle

Let $f:[0,1]\to\mathbb R$ be a function satisfying $$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that $$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)

Russian TST 2022, P2

Tags: geometry , parallelogram , equal angles

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.

2024 Nordic, 2

Tags: geometry

There exists a quadrilateral $\mathcal{Q} _{1}$ such that the midpoints of its sides lie on a circle. Prove that there exists a cyclic quadrilateral $\mathcal{Q} _{2}$ with the same sides as $\mathcal{Q} _{1}$ with two of the same angles.

2022 Brazil National Olympiad, 2

Tags: geometry , angle chasing , geometric transformation , reflection

Let $ABC$ be an acute triangle, with $AB<AC$. Let $K$ be the midpoint of the arch $BC$ that does not contain $A$ and let $P$ be the midpoint of $BC$. Let $I_B,I_C$ be the $B$-excenter and $C$-excenter of $ABC$, respectively. Let $Q$ be the reflection of $K$ with respect to $A$. Prove that the points $P,Q,I_B,I_C$ are concyclic.

2025 Bangladesh Mathematical Olympiad, P9

Tags: altitude , geometry , similarity , parallel lines , congruent triangles

Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, respectively. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ and $N$, respectively, other than $D$. Prove that $BN = LC$.

2003 Mediterranean Mathematics Olympiad, 4

Tags: geometry , 3d geometry , sphere , combinatorics

Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

2012 Grigore Moisil Intercounty, 3

Tags: centroid , center of mass , vector geometry , geometry

Let $ M,N,P $ on the sides $ AB,BC,CA, $ respectively, of a triangle $ ABC $ such that $ AM=BN=CP $ and such that $$ AB\cdot \overrightarrow{AT} +BC\cdot \overrightarrow{BT} +CA\cdot \overrightarrow{CT} =0, $$ where $ T $ is the centroid of $ MNP. $ Prove that $ ABC $ is equilateral.

1993 All-Russian Olympiad, 2

Tags: parallelogram , geometry

A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.

2007 Nordic, 4

Tags: ratio , geometry

A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.

2006 Singapore Senior Math Olympiad, 3

Tags: angle bisector , tangent circle , geometry

Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Prove that the line TP bisects $\angle ATB$.

1970 AMC 12/AHSME, 2

Tags: geometry , ratio

A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is: $\textbf{(A) }\frac{4}{\pi}\qquad\textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad\textbf{(C) }\frac{4}{1}\qquad\textbf{(D) }\frac{\sqrt{2}}{\pi}\qquad \textbf{(E) }\frac{\pi}{4}$

1990 IMO Longlists, 44

Tags: binomial coefficient , pascal triangle , geometry , 3d geometry , pyramid , symmetry

Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.

2004 Argentina National Olympiad, 5

Tags: geometry , area , pentagon , equal segments

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.

1964 Bulgaria National Olympiad, Problem 3

Tags: geometry , locus

There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$. Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$. Prove that: (a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$. (b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$.

2004 Romania National Olympiad, 2

Tags: 3d geometry , tetrahedron , trigonometry , geometry

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles. Prove that it is regular.

2007 IMO Shortlist, 2

Tags: circumcircle , reflection , cyclic quadrilateral , mixtilinear incircle , geometry

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]