Olympiad problems

2024 Regional Olympiad of Mexico West, 2

Tags: geometry , collinear

Let $\triangle ABC$ be a triangle and $H$ its orthocenter. We draw the circumference $\mathcal{C}_1$ that passes through $H$ and its tangent to $BC$ at $B$ and the circumference $\mathcal{C}_2$ that passes through $H$ and its tangent to $BC$ at $C$. If $\mathcal{C}_1$ cuts line $AB$ again at $X$ and $\mathcal{C}_2$ cuts line $AC$ again at $Y$. Prove that $X,Y$ and $H$ are collinear.

2014 PUMaC Geometry A, 4

Tags: geometric transformation , reflection , trigonometry , cyclic quadrilateral , perpendicular bisector , trig identities , geometry

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

2007 National Olympiad First Round, 21

Tags: geometry

Let $ABCD$ be a quadrilateral such that $m(\widehat{A}) = m(\widehat{D}) = 90^\circ$. Let $M$ be the midpoint of $[DC]$. If $AC\perp BM$, $|DC|=12$, and $|AB|=9$, then what is $|AD|$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the above} $

2010 Stanford Mathematics Tournament, 17

Tags: geometry , perpendicular bisector

An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle

2022 Thailand TST, 3

Tags: geometry

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

2020 ASDAN Math Tournament, 1

Tags: geometry , team test

Consider triangle $\vartriangle ABC$ with $\angle C = 90^o$. Let $P$ be the midpoint of $\overline{AC}$ so that $AP = PC = 1$, and suppose $\angle BAC = \angle CBP$. Compute $AB^2$.

2021 Tuymaada Olympiad, 2

Tags: geometry

The bisector of angle $B$ of a parallelogram $ABCD$ meets its diagonal $AC$ at $E$, and the external bisector of angle $B$ meets line $AD$ at $F$. $M$ is the midpoint of $BE$. Prove that $CM // EF$.

2018 Iranian Geometry Olympiad, 1

Tags: geometry , circles , equal segments

Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $ \omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $ \omega_1$ for the second time at $X'$. Prove that $PX = PX'$. Proposed by Morteza Saghafian

2024 Indonesia TST, G

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2012 Sharygin Geometry Olympiad, 5

Tags: geometry , perimeter , convex polygon

Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter? (A.Akopyan)

1997 All-Russian Olympiad Regional Round, 10.4

Tags: combinatorics , combinatorial geometry , geometry , 3d geometry

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?

1999 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , combinatorial geometry , 3d geometry , geometry

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

2011 Oral Moscow Geometry Olympiad, 2

Tags: geometry , isosceles , equal segments , right angle , angle

In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.

2016 Uzbekistan National Olympiad, 1

Tags: geometry , circumcircle

$\omega$ is circumcircle of triangle $ABC$ and $BB_1, CC_1$ are bisectors of ABC. $I$ is center incirle . $B_1 C1$ and $\omega$ intersects at $M$ and $N$ . Find the ratio of circumradius of $ABC$ to circumradius $MIN$.

2012 IFYM, Sozopol, 8

Tags: geometry , polygon

The lengths of the sides of a convex decagon are no greater than 1. Prove that for each inner point $M$ of the decagon there is at least one vertex $A$, for which $MA\leq \frac{\sqrt{5}+1}{2}$.

2010 Romania National Olympiad, 2

Tags: rectangle , ratio , angle bisector , geometry

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

2024 Indonesia TST, 1

Tags: geometry

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

2022 AMC 12/AHSME, 10

Tags: geometry , perimeter , trigonometry

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$? $ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4\sqrt5 \qquad \textbf{(D)}\ 4\sqrt7 \qquad \textbf{(E)}\ 12$

1945 Moscow Mathematical Olympiad, 101

Tags: parallelogram , geometry

The side $AD$ of a parallelogram $ABCD$ is divided into $n$ equal segments. The nearest to $A$ division point $P$ is connected with $B$. Prove that line $BP$ intersects the diagonal $AC$ at point $Q$ such that $AQ = \frac{AC}{n + 1}$

2022 Malaysia IMONST 2, 4

Tags: geometry , angle chasing , polygon

Given a pentagon $ABCDE$ with all its interior angles less than $180^\circ$. Prove that if $\angle ABC = \angle ADE$ and $\angle ADB = \angle AEC$, then $\angle BAC = \angle DAE$.

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , incircle , inradius , square

A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones. (L.Steingarts)

1997 Czech And Slovak Olympiad IIIA, 6

Tags: diagonal , cyclic , parallelogram , locus , geometry

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

2018 Taiwan APMO Preliminary, 1

Tags: geometry

Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

2021 Baltic Way, 13

Tags: geometry

Let $D$ be the foot of the $A$-altitude of an acute triangle $ABC$. The internal bisector of the angle $DAC$ intersects $BC$ at $K$. Let $L$ be the projection of $K$ onto $AC$. Let $M$ be the intersection point of $BL$ and $AD$. Let $P$ be the intersection point of $MC$ and $DL$. Prove that $PK \perp AB$.

2016 IFYM, Sozopol, 4

Tags: geometry , area of a triangle

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that: $\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.