Olympiad problems

2023 Pan-African, 1

Tags: geometry

In a triangle $ABC$ with $AB<AC$, $D$ is a point on segment $AC$ such that $BD = CD$. A line parallel to $BD$ meets segment $BC$ at $E$ and line $AB$ at $F$. Point $G$ is the intersection of $AE$ and $BD$. Show that $\angle BCG = \angle BCF$. [i](Côte d’Ivoire)[/i]

Oliforum Contest III 2012, 3

Tags: hexagon , equal angles , geometry

Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.

2009 Dutch IMO TST, 2

Tags: geometry , equal segments , midpoint

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

2016 BMT Spring, 1

Tags: geometry

A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas?

1986 AMC 8, 18

Tags: geometry

A rectangular grazing area is to be fenced off on three sides using part of a $ 100$ meter rock wall as the fourth side. Fence posts are to be placed every $ 12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $ 36$ m by $ 60$ m? \[ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16 \]

1988 All Soviet Union Mathematical Olympiad, 467

Tags: ratio , fixed , geometry , cyclic , midpoint

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

2020 Czech-Austrian-Polish-Slovak Match, 6

Tags: equal angles , fixed , fixed point , geometry

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point. (Dominik Burek, Poland)

2024 Canadian Mathematical Olympiad Qualification, 7b

Tags: geometry , equal segments

In triangle $ABC$, let $I$ be the incentre, $O$ be the circumcentre, and $H$ be the orthocentre. It is given that $IO = IH$. Show that one of the angles of triangle $ABC$ must be equal to $60$ degrees.

2021 Peru PAGMO TST, P2

Tags: geometry , rectangle

The bisector of the diagonal $BD$ of a rectangle $ABCD$ (with $AB < BC$) intersects the lines $BC$ and $BA$ at points $E$ and $F$, respectively. The line passing through point $F$ and parallel to segment $AC$ intersects line $CD$ at point $G$. Prove that lines $EG$ and $AC$ are perpendicular

2017 Germany Team Selection Test, 3

Tags: geometry , incenter , reflection , inversion , cyclic quadrilateral

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2007 Paraguay Mathematical Olympiad, 2

Tags: geometry

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

2010 Stars Of Mathematics, 2

Tags: geometric transformation , reflection , perpendicular bisector , geometry

Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$. 1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$; 2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$. (Dan Schwarz)

2006 Regional Competition For Advanced Students, 3

Tags: circumcircle , angle bisector , exterior angle , geometry

In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$. Given are the points $ A$ and $ B$. Determine the set of points $ P\equal{}P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.

1998 APMO, 4

Tags: geometry , circumcircle , rectangle

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let $E$ and $F$ lie on a line passing through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

V Soros Olympiad 1998 - 99 (Russia), 10.10

Tags: geometry , tangent circle

A circle inscribed in triangle $ABC$ touches $BC$ at point $K$, $M$ is the midpoint of the altitude drawn on $BC$. The straight line $KM$ intersects the circle inscribed in $ABC$ for the second time at point $P$. Prove that the circle passing through $B$, $C$ and $P$ touches the circle inscribed in triangle $ABC$.

2012 Kosovo National Mathematical Olympiad, 2

Tags: geometry , inequality

If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that $a+b\leq c\sqrt 2$

2010 239 Open Mathematical Olympiad, 2

Tags: geometry

The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.

1971 All Soviet Union Mathematical Olympiad, 150

Tags: geometry , projection , circles

The projections of the body on two planes are circles. Prove that they have the same radius.

2003 Federal Math Competition of S&M, Problem 2

Tags: circumcircle , geometry

Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.

2017 Hong Kong TST, 1

Tags: geometry , combinatorics

a) Do there exist 5 circles in the plane such that each circle passes through exactly 3 centers of other circles? b) Do there exist 6 circles in the plane such that each circle passes through exactly 3 centers of other circles?

2007 APMO, 2

Tags: geometry , incenter

Let $ABC$ be an acute angled triangle with $\angle{BAC}=60^\circ$ and $AB > AC$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $ABC$ . Prove that $2\angle{AHI}= 3\angle{ABC}$.

2005 Thailand Mathematical Olympiad, 4

Tags: incenter , geometry

Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.

2010 Switzerland - Final Round, 10

Tags: circumcircle , induction , combinatorics , geometry

Let $ n\geqslant 3$ and $ P$ a convex $ n$-gon. Show that $ P$ can be, by $ n \minus{} 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $ P$. Under which conditions is there exactly one such triangulation?

2005 Croatia National Olympiad, 2

Tags: circumcircle , incenter , geometric transformation , homothety , geometry

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.

2005 Cono Sur Olympiad, 2

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.