Olympiad problems

1987 Balkan MO, 3

Tags: trigonometry , geometry

In the triangle $ABC$ the following equality holds: \[\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}\] Determine the value of $\frac{AC}{BC}$.

2024 Turkey EGMO TST, 1

Tags: geometry , circumcircle , incentre

Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.

2006 Romania National Olympiad, 3

Tags: geometry , 3d geometry

Let $ABCDA_1B_1C_1D_1$ be a cube and $P$ a variable point on the side $[AB]$. The perpendicular plane on $AB$ which passes through $P$ intersects the line $AC'$ in $Q$. Let $M$ and $N$ be the midpoints of the segments $A'P$ and $BQ$ respectively. a) Prove that the lines $MN$ and $BC'$ are perpendicular if and only if $P$ is the midpoint of $AB$. b) Find the minimal value of the angle between the lines $MN$ and $BC'$.

2007 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , perpendicular bisector , incenter , circumcircle

Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).

1976 IMO, 1

Tags: inequalities , geometry , trigonometry , area of a triangle , geometric inequality

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

2021 Sharygin Geometry Olympiad, 14

Tags: excircle , geometry

Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.

2011 JBMO Shortlist, 2

Tags: geometry

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

Cono Sur Shortlist - geometry, 2012.G6.6

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

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

2024 All-Russian Olympiad, 4

Tags: conditional geometry , geometry , cyclic quadrilateral , ratio

In cyclic quadrilateral $ABCD$, $\angle A+ \angle D=\frac{\pi}{2}$. $AC$ intersects $BD$ at ${E}$. A line ${l}$ cuts segment $AB, CD, AE, DE$ at $X, Y, Z, T$ respectively. If $AZ=CE$ and $BE=DT$, prove that the diameter of the circumcircle of $\triangle EZT$ equals $XY$.

2023 USA TSTST, 1

Tags: geometry

Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that \[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\] Prove that $\angle RAS+\angle BAC=\angle BGC$. [i]Merlijn Staps[/i]

2013 Sharygin Geometry Olympiad, 5

Tags: geometry , cyclic quadrilateral , rhombus , perpendicular bisector

Let ABCD is a cyclic quadrilateral inscribed in $(O)$. $E, F$ are the midpoints of arcs $AB$ and $CD$ not containing the other vertices of the quadrilateral. The line passing through $E, F$ and parallel to the diagonals of $ABCD$ meet at $E, F, K, L$. Prove that $KL$ passes through $O$.

2013 Romania Team Selection Test, 1

Tags: algebra , symmetry , geometry , 3d geometry , number theory , polynomial , modular arithmetic

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

1999 Baltic Way, 9

Tags: geometry , 3d geometry , combinatorics

A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?

1988 Irish Math Olympiad, 3

Tags: geometry

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.

2017 Iranian Geometry Olympiad, 5

Tags: circumcircle , geometry

Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B,Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$. [i]Proposed by Iman Maghsoudi[/i]

2025 Philippine MO, P7

Tags: midpoint , geometry , circumcircle , arbitrary point , diameter , perpendicular , parallel

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$. (a) Show that $AP$ and $BR$ are perpendicular. \\ (b) Show that $FM$ and $BM$ are perpendicular.

1966 Polish MO Finals, 6

Tags: geometry , combinatorics , combinatorial geometry

On the plane are chosen six points. Prove that the ratio of the longest distance between two points to the shortest is at least $\sqrt3$.

1974 Spain Mathematical Olympiad, 1

Tags: geometry , 3d geometry , dodecahedron , combinatorial geometry

It is known that a regular dodecahedron is a regular polyhedron with $12$ faces of equal pentagons and concurring $3$ edges in each vertex. It is requested to calculate, reasonably, a) the number of vertices, b) the number of edges, c) the number of diagonals of all faces, d) the number of line segments determined for every two vertices, d) the number of diagonals of the dodecahedron.

1998 IberoAmerican Olympiad For University Students, 2

Tags: conic , ellipse , geometry

In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$. Find both the locus of the centroid of all such triangles and its area.

2025 Belarusian National Olympiad, 8.8

Tags: parallelogram , geometry

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$. Point $M$ is the midpoint of $PE$. Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$. [i]Matsvei Zorka[/i]

1966 AMC 12/AHSME, 14

Tags: ratio , rectangle , geometry

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

2020 Princeton University Math Competition, B2

Tags: geometry

Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least $6$ feet apart from each other?

2015 Indonesia MO Shortlist, G6

Tags: circumcircle , geometry , perpendicular , equal segments

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2022 China Team Selection Test, 2

Tags: angle bisector , geometry , incenter

Let $ABCD$ be a convex quadrilateral, the incenters of $\triangle ABC$ and $\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\angle BAD$ and the outer angle bisector $\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.

1985 IMO Longlists, 30

Tags: geometry

A plane rectangular grid is given and a “rational point” is defined as a point $(x, y)$ where $x$ and $y$ are both rational numbers. Let $A,B,A',B'$ be four distinct rational points. Let $P$ be a point such that $\frac{A'B'}{AB}=\frac{B'P}{BP} = \frac{PA'}{PA}.$ In other words, the triangles $ABP, A'B'P$ are directly or oppositely similar. Prove that $P$ is in general a rational point and find the exceptional positions of $A'$ and $B'$ relative to $A$ and $B$ such that there exists a $P$ that is not a rational point.