Olympiad problems

2012 Argentina Cono Sur TST, 1

Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.

2024 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , circumradius , acute triangle , perpendicular

Let $ABC$ be an acute triangle with circumradius $R$. Let $D$ be the midpoint of $BC$ and $F$ the midpoint of $AB$. The perpendicular to $AC$ through $F$ and the perpendicular to $BC$ through $B$ intersect at $N$. Prove that $ND = R$.

2004 Federal Math Competition of S&M, 2

Tags: geometry

In a triangle $ABC$, points $D$ and $E$ are taken on rays $CB$ and $CA$ respectively so that $CD=CE = \frac{AC+BC}{2}$. Let $H$ be the orthocenter of the triangle, and $P$ be the midpoint of the arc $AB$ of the circumcircle of $ABC$ not containing $C$. Prove that the line $DE$ bisects the segment $HP$.

2021 LMT Fall, 9

Tags: geometry

Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .

2018 Balkan MO Shortlist, G1

Tags: geometry

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

1999 AMC 12/AHSME, 22

Tags: geometry , rectangle , analytic geometry

The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$. $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 18$

1959 IMO, 5

Tags: geometry , circumcircle , perpendicular bisector , angle bisector

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$. a) Prove that $N$ and $N'$ coincide; b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$; c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

2007 Kazakhstan National Olympiad, 1

Tags: rhombus , function , geometry

Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.

2011 Bundeswettbewerb Mathematik, 1

Tags: square , hexagon , tiling , combinatorial geometry , tiles , angle , geometry

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

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.

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: geometry , trapezoid , concurrency , concyclic

A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases of $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. Prove that the point $L$ lies on the diagonal $BD$.

2000 Tournament Of Towns, 2

Tags: equal segments , circumcenter , circumcircle , chord , geometry

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

2001 Czech And Slovak Olympiad IIIA, 5

Tags: geometry , trapezoid , isosceles trapezoid , area

A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.

2012-2013 SDML (Middle School), 6

Tags: geometry , perimeter , arithmetic sequence

How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.) $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$

1962 IMO, 3

Tags: geometry , 3d geometry , perimeter , vector , cube

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.

1997 Nordic, 2

Tags: diagonal , geometry , area of triangle , area , quadrilateral

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

2012 Oral Moscow Geometry Olympiad, 5

Tags: minimum , sum , distance , geometry , circles

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

2008 China Team Selection Test, 1

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , ratio , trigonometry

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

1999 IMO Shortlist, 2

Tags: geometry , point set , circles , combinatorial geometry

A circle is called a [b]separator[/b] for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

1993 All-Russian Olympiad, 2

Tags: geometry , rectangle , geometric transformation , rotation , symmetry , combinatorics

Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?

2016 Indonesia TST, 1

Tags: triangle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

2023 Sharygin Geometry Olympiad, 10.8

Tags: geometry

A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.

2020 JBMO TST of France, 1

Tags: geometry

Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

1967 IMO Shortlist, 4

Tags: trigonometric equation , trigonometry , algebra , geometry

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

2021 Polish Junior MO Finals, 4

Tags: geometry , angle bisector

On side $AB$ of a scalene triangle $ABC$ there are points $M$, $N$ such that $AN=AC$ and $BM=BC$. The line parallel to $BC$ through $M$ and the line parallel to $AC$ through $N$ intersect at $S$. Prove that $\measuredangle{CSM} = \measuredangle{CSN}$.