Olympiad problems

2004 German National Olympiad, 2

Tags: circles , line , tangent , concurrency , geometry

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

2022 Sharygin Geometry Olympiad, 9.4

Tags: geometry , angle bisector

Let $ABC$ be an isosceles triangle with $AB = AC$, $P$ be the midpoint of the minor arc $AB$ of its circumcircle, and $Q$ be the midpoint of $AC$. A circumcircle of triangle $APQ$ centered at $O$ meets $AB$ for the second time at point $K$. Prove that lines $PO$ and $KQ$ meet on the bisector of angle $ABC$.

1994 All-Russian Olympiad, 2

Tags: geometry , collinear , tangent circle

Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear. (A. Kalinin)

2016 Irish Math Olympiad, 6

Tags: centroid , right angle , geometry

Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.

2010 Contests, 3

Tags: geometry , rectangle , calculus , linear algebra , matrix , analytic geometry , number theory

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

2014 Albania Round 2, 5

Tags: geometry

Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$, Then $$\alpha < \beta$$

2017 Sharygin Geometry Olympiad, 4

Tags: geometry

Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.

2012 Kyiv Mathematical Festival, 1

Tags: combinatorial geometry , geometry , circles

Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?

1964 All Russian Mathematical Olympiad, 049

Tags: minimum , hexagon , geometry

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

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 $\angle DAE$.

2017 Novosibirsk Oral Olympiad in Geometry, 6

Tags: ratio , geometry , trapezoid

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

1940 Eotvos Mathematical Competition, 3

Tags: geometry , similar , median , triangle inequality

(a) Prove that for any triangle $H_1$, there exists a triangle $H_2$ whose side lengths are equal to the lengths of the medians of $H_1$. (b) If $H_3$ is the triangle whose side lengths are equal to the lengths of the medians of $H_2$, prove that $H_1$ and $H_3$ are similar.

2019 Hong Kong TST, 2

Tags: symmetry , angle chasing , miquel point , geometry

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

2017 Bosnia and Herzegovina Team Selection Test, Problem 1

Tags: geometry , circumcircle , inradius , incenter , ratio , trigonometry

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

2001 Romania National Olympiad, 2

Tags: trigonometry , inequalities , 3d geometry , tetrahedron , geometry

In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]

2010 All-Russian Olympiad Regional Round, 9.4

Tags: geometry , symmetry

In triangle $ABC$, $\angle A =60^o$. Let $BB_1$ and $CC_1$ be angle bisectors of this triangle. Prove that the point symmetrical to vertex $A$ with respect to line $B_1C_1$ lies on side $BC$.

1982 Vietnam National Olympiad, 3

Tags: geometry

Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that \[S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.\]

2007 Estonia Team Selection Test, 4

Tags: angle chasing , square , geometry , angle

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Kvant 2024, M2825

Tags: geometry

At the same time, three beetles with identical speeds began to crawl along the heights of an acute-angled non-isosceles triangle from its vertices. At some point, it turned out that the first and second beetles were on a circle inscribed in a triangle. Prove that at this moment the third beetle is also on this circle. [i]A. Kuznetsov[/i]

2019 Durer Math Competition Finals, 5

Tags: geometry , arc midpoint , concurrency

Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.

2002 India Regional Mathematical Olympiad, 1

Tags: geometry

In an acute triangle $ABC$ points $D,E,F$ are located on the sides $BC,CA, AB$ such that \[ \frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA} \] Prove that $AD,BE,CF$ are altitudes of triangle $ABC$.

2011 Indonesia MO, 8

Tags: circumcircle , geometry

Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.

2011 Saudi Arabia Pre-TST, 1.4

Tags: equal angles , geometry , equal segments

Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC = 40^o$. Points $S$ and $T$ lie on the sides $AB$ and $BC$, such that $\angle BAT = \angle BCS = 10^o$. Lines $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.

2003 JBMO Shortlist, 1

Tags: geometry

Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?

2021 CHKMO, 3

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma$ such that $AB=AD$. Let $E$ be a point on the segment $CD$ such that $BC=DE$. The line $AE$ intersect $\Gamma$ again at $F$. The chords $AC$ and $BF$ meet at $M$. Let $P$ be the symmetric point of $C$ about $M$. Prove that $PE$ and $BF$ are parallel.