Olympiad problems

1990 Federal Competition For Advanced Students, P2, 3

Tags: inequalities , geometry

In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that: $ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$

2013 Danube Mathematical Competition, 4

Tags: geometry , rectangle , perpendicular

Let $ABCD$ be a rectangle with $AB \ne BC$ and the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and $[AD]$ respectively. Prove that $FM \perp EN$ .

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , number theory

Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers.

1985 IMO Longlists, 52

Tags: geometry

In the triangle $ABC$, let $B_1$ be on $AC, E$ on $AB, G$ on $BC$, and let $EG$ be parallel to $AC$. Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$. Let $r, r_1$, and $r_2$ be the inradii of the triangles $ABC, ABB_1$, and $BFG$, respectively. Prove that $r = r_1 + r_2.$

2022 Iranian Geometry Olympiad, 3

Tags: geometry , similar triangles

Let $O$ be the circumcenter of triangle $ABC$. Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$, respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$, and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$. [i]Proposed by Medeubek Kungozhin, Kazakhstan[/i]

1975 Chisinau City MO, 111

Tags: square , geometry , concyclic

Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

2018 Hanoi Open Mathematics Competitions, 2

Tags: area , maximum , hexagon , equilateral , geometry

What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$? A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$

1991 Greece Junior Math Olympiad, 2

Tags: geometry , semicircle

Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.

2012 Tuymaada Olympiad, 2

Tags: rectangle , symmetry , angle bisector , geometry

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: midpoint , equal angles , angle , geometry

Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$

2013 Kosovo National Mathematical Olympiad, 5

Tags: geometry

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

2016 District Olympiad, 1

Tags: geometry , 3d geometry , pyramid

Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $

Geometry Mathley 2011-12, 11.2

Tags: equal segments , geometry

Let $ABC$ be a triangle inscribed in the circle $(O)$. Tangents at $B,C$ of the circles $(O)$ meet at $T$ . Let $M,N$ be the points on the rays $BT,CT$ respectively such that $BM = BC = CN$. The line through $M$ and $N$ intersects $CA,AB$ at $E, F$ respectively; $BE$ meets $CT$ at $P, CF$ intersects $BT$ at $Q$. Prove that $AP = AQ$. Trần Quang Hùng

2016 CMIMC, 7

Tags: geometry

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

2023 ISL, G3

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.

2013 Turkey Team Selection Test, 2

Tags: geometry , rectangle , induction , combinatorics

We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.

Oliforum Contest III 2012, 5

Tags: geometry , cyclic quadrilateral , orthogonal

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

1959 Poland - Second Round, 5

Tags: geometry , combinatorial geometry

In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.

2008 Mediterranean Mathematics Olympiad, 2

Tags: geometry

Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$, (i) $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$. (ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

1999 Croatia National Olympiad, Problem 1

Tags: geometry , 3d geometry , tetrahedron

For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.

2000 Brazil National Olympiad, 6

Tags: 3d geometry , geometry

Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?

2010 Saudi Arabia BMO TST, 2

Tags: equal area , ratio , geometry , cevian

Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.

Geometry Mathley 2011-12, 16.1

Tags: euler line , geometry , cyclic quadrilateral , concurrency

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

1902 Eotvos Mathematical Competition, 2

Tags: geometry , 3d geometry , sphere

Let $S$ be a given sphere with center $O$ and radius $r$. Let $P$ be any point outside then sphere $S$, and let $S'$ be the sphere with center $P$ and radius $PO$. Denote by $F$ the area of the surface of the part of $S'$ that lies inside $S$. Prove that $F$ is independent of the particular point $P$ chosen.

1976 IMO Longlists, 1

Tags: geometry , bisector , incircle

Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$