Olympiad problems

1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

A square $ ABCD$ is inscribed in a circle. Let $ \alpha \equal{} \angle DAB, \beta \equal{} \angle BDA,$ and $ \gamma \equal{} \angle CDB$. Then $ \angle DBC$ equals A. $ \alpha \minus{} \beta$ B. $ \alpha \minus{} \gamma$ C. $ 90^\circ \minus{} \alpha \plus{} \beta$ D. $ 90^\circ \minus{} \alpha \plus{} \gamma$ E. $ 180^\circ \minus{} \alpha \minus{} \gamma$

2024 Sharygin Geometry Olympiad, 10.4

Tags: geo , geometry

Let $I$ be the incenter of a triangle $ABC$. The lines passing through $A$ and parallel to $BI, CI$ meet the perpendicular bisector to $AI$ at points $S, T$ respectively. Let $Y$ be the common point of $BT$ and $CS$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $YA^*$ lies on the excircle of the triangle touching the side $BC$.

2013 District Olympiad, 4

Tags: square , equal segments , geometry

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

2001 JBMO ShortLists, 9

Tags: geometry

Consider a convex quadrilateral $ABCD$ with $AB=CD$ and $\angle BAC=30^{\circ}$. If $\angle ADC=150^{\circ}$, prove that $\angle BCA= \angle ACD$.

1995 IMO Shortlist, 3

Tags: harmonic division , geometry , incircle , power of a point , trigonometry

The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively. Show that $ E, F, Z, Y$ are concyclic.

2024 Assara - South Russian Girl's MO, 6

Tags: geometry

The points $A, B, C, D$ are marked on the straight line in this order. Circle $\omega_1$ passes through points $A$ and $C$, and the circle $\omega_2$ passes through points $B$ and $D$. On the circle $\omega_2$, the point $E$ is marked so that $AB = BE$, and on the circle $\omega_1$, the point $F$ is marked so that $CD = CF$. The line $AE$ intersects the circle $\omega_2$ a second time at point $X$, and the line $DF$ intersects the circle $\omega_1$ at point $Y$. Prove that the $XY$ lines and $AD$ is perpendicular. [i]A.D.Tereshin[/i]

2003 India IMO Training Camp, 4

Tags: euler , geometry

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

2001 Moldova National Olympiad, Problem 7

Tags: geometry

Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.

2004 Oral Moscow Geometry Olympiad, 4

Tags: projection , geometry , perpendicular

Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.

2015 Canadian Mathematical Olympiad Qualification, 4

Tags: cyclic quadrilateral , geometry

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.

2023 Taiwan Mathematics Olympiad, 3

Tags: geometry

Let $O$ be the center of circle $\Gamma$, and $A$, $B$ be two points on $\Gamma$ so that $O, A$ and $B$ are not collinear. Let $M$ be the midpoint of $AB$. Let $P$ and $Q$ be points on $OA$ and $OB$, respectively, so that $P \neq A$ and $P, M, Q$ are collinear. Let $X$ be the intersection of the line passing through $P$ and parallel to $AB$ and the line passing through $Q$ and parallel to $OM$. Let $Y$ be the intersection of the line passing through $X$ and parallel to $OA$ and the line passing through $B$ and orthogonal to $OX$. Prove that: if $X$ is on $\Gamma$, then $Y$ is also on $\Gamma$. [i] Proposed by usjl[/i]

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.

2010 Contests, 3

Tags: perimeter , geometry , inequalities

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

1969 IMO Longlists, 33

Tags: geometry , covering , circles

$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)

1975 Poland - Second Round, 2

Tags: tangential quadrilateral , tangential , geometry

In the convex quadrilateral $ ABCD $, the corresponding points $ M $ and $ N $ are chosen on the adjacent sides $ \overline{AB} $ and $ \overline{BC} $ and the intersection point of the segments $ AN $ and $ GM $ is marked by 0. Prove that if circles can be inscribed in the quadrilaterals $ AOCD $ and $ BMON $, then a circle can also be inscribed in the quadrilateral $ ABCD $.

2010 Gheorghe Vranceanu, 1

Tags: concurrency , median , geometry

Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.

1968 Vietnam National Olympiad, 2

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

$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$

2020-IMOC, G4

Tags: conic , incenter , geometry

Let $I$ be the incenter of triangle $ABC$. Let $BI$ and $AC$ intersect at $E$, and $CI$ and $AB$ intersect at $F$. Suppose that $R$ is another intersection of $\odot (ABC)$ and $\odot (AEF)$. Let $M$ be the midpoint of $BC$, and $P, Q$ are the intersections of $AI, MI$ and $EF$, respectively. Show that $A, P, Q, R$ are concyclic. (ltf0501).

2011 Germany Team Selection Test, 2

Tags: geometry , reflection , circumcircle , parallelogram

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2016 Romania National Olympiad, 3

Tags: geometric inequalities , geometric inequality , geometry

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

2015 IFYM, Sozopol, 7

Tags: equilateral triangle , geometry

In a square with side 1 are placed $n$ equilateral triangles (without having any parts outside the square) each with side greater than $\sqrt{\frac{2}{3}}$. Prove that all of the $n$ equilateral triangles have a common inner point.

2010 Contests, 1

Tags: geometry , circumcircle

Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.

2009 China Team Selection Test, 1

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

Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$

1998 Putnam, 6

Tags: geometry

Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

2000 Harvard-MIT Mathematics Tournament, 10

Tags: algebra , geometry

How many times per day do at least two of the three hands on a clock coincide?