Olympiad problems

2005 Postal Coaching, 16

The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC. Bomb [color=red][Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .][/color]

2016 Iran Team Selection Test, 5

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2021 Romanian Master of Mathematics Shortlist, G2

Tags: similarities , inversion , bashing , circumcircle , geometry , incenter

Let $ABC$ be a triangle with incenter $I$. The line through $I$, perpendicular to $AI$, intersects the circumcircle of $ABC$ at points $P$ and $Q$. It turns out there exists a point $T$ on the side $BC$ such that $AB + BT = AC + CT$ and $AT^2 = AB \cdot AC$. Determine all possible values of the ratio $IP/IQ$.

1998 May Olympiad, 2

Tags: combinatorial geometry , rectangle , geometry , combinatorics

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

2022 HMNT, 9

Tags: geometry

Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle $ABC$ and a point $D$ on segment $BC$ satisfying $BD=3$ and $CD=5$. Then, Alice chooses a point $P$ on line $AD$ and challenges Bob to mark a point $Q\neq P$ on line $AD$ such that $\frac{BQ}{QC}=\frac{BP}{PC}$. Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of $\frac{BP}{PC}$ for the $P$ she chose?

2022 Thailand TSTST, 3

Tags: geometry , circumcircle

An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.

2003 Federal Math Competition of S&M, Problem 4

Tags: locus , geometry

An acute angle with the vertex $O$ and the rays $Op_1$ and $Op_2$ is given in a plane. Let $k_1$ be a circle with the center on $Op_1$ which is tangent to $Op_2$. Let $k_2$ be the circle that is tangent to both rays $Op_1$ and $Op_2$ and to the circle $k_1$ from outside. Find the locus of tangency points of $k_1$ and $k_2$ when center of $k_1$ moves along the ray $Op_1$.

2017 Oral Moscow Geometry Olympiad, 3

Tags: construction , ruler , circumcircle , geometry

On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.

2020 Colombia National Olympiad, 5

Tags: geometry

Given an acute-angled triangle $ABC$ with $D$ is the foot of the altitude from $A.$ The perpendicular lines to $BC$ through $B$ and $C$ intersect the altitudes from $C$ and $B$ at points $M$ and $N$, respectively. Show that $AD$ $=$ $BC$ if and only if $A,M,N$ and $D$ lie on the same circle.

2002 Canada National Olympiad, 4

Tags: geometry , circumcircle , perpendicular bisector , angle bisector , trigonometry

Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally, let the line $CP$ meet $\Gamma$ again at $Q$. Prove that $PQ = r$.

1966 IMO Longlists, 22

Tags: geometry , parallelogram , vector

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

Ukrainian TYM Qualifying - geometry, V.8

Tags: equilateral , geometric inequality , trigonometry , geometry

Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that $$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$

1992 Taiwan National Olympiad, 1

Tags: trigonometry , quadratic , geometry

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

2023 LMT Spring, 5

Tags: geometry

How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct

2009 Mathcenter Contest, 2

Tags: sq , locus , geometry

Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$ [i](Anonymous314)[/i]

2013 Sharygin Geometry Olympiad, 23

Tags: symmetry , geometry

Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, b) $2013$ symmetry planes? c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

2024 Belarusian National Olympiad, 11.5

Tags: geometry

On the chord $AB$ of the circle $\omega$ points $C$ and $D$ are chosen such that $AC=BD$ and point $C$ lies between $A$ and $D$. On $\omega$ point $E$ and $F$ are marked, they lie on different sides with respect to line $AB$ and lines $EC$ and $FD$ are perpendicular to $AB$. The angle bisector of $AEB$ intersects line $DF$ at $R$ Prove that the circle with center $F$ and radius $FR$ is tangent to $\omega$ [i]V. Kamenetskii, D. Bariev[/i]

2014 India PRMO, 10

Tags: geometry , area , ratio

In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?

2014 Uzbekistan National Olympiad, 5

Tags: 3d geometry , pyramid , geometry , circumcircle

Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $. Find the volume of the pyramid.

Ukrainian TYM Qualifying - geometry, VIII.2

Tags: angle , 3d geometry , tetrahedron , geometry

Investigate the properties of the tetrahedron $ABCD$ for which there is equality $$\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}$$ where $\alpha, \beta, \gamma$ are the values of the dihedral angles at the edges $AD, BD$ and $CD$, respectively.

1973 IMO Shortlist, 13

Tags: 3d geometry , sphere , tetrahedron , geometry

Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to $1.$

2019 All-Russian Olympiad, 4

Tags: geometric transformation , circumcircle , reflection , geometry

Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.

2002 AMC 10, 7

Tags: geometry , ratio

If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is $ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$

2002 Germany Team Selection Test, 2

Tags: geometry , perimeter

Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have: \[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]

2007 IMAC Arhimede, 2

Tags: function , geometry , parallelogram

Let $ABCD$ be a parallelogram that is not rhombus. We draw the symmetrical half-line of $(DC$ with respect to line $BD$. Similarly we draw the symmetrical half- line of $(AB$ with respect to $AC$. These half- lines intersect each other in $P$. If $\frac{AP}{DP}= q$ find the value of $\frac{AC}{BD}$ in function of $q$.