Olympiad problems

2020 Yasinsky Geometry Olympiad, 2

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

1999 Romania Team Selection Test, 2

Tags: incenter , geometry

Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.

2021 Final Mathematical Cup, 2

Tags: geometry , tangent

The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .

2010 Sharygin Geometry Olympiad, 4

Tags: construction , incircle , geometry

In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.

1995 Tournament Of Towns, (469) 3

Tags: geometry , angle bisector , angle

Let $AK$, $BL$ and $CM$ be the angle bisectors of a triangle $ABC$, with $K$ on $BC$. Let $P$ and $Q$ be the points on the lines $BL$ and $CM$ respectively such that $AP = PK$ and $AQ = QK$. Prove that $\angle PAQ = 90^o -\frac12 \angle B AC.$ (I Sharygin)

2018 Romania National Olympiad, 2

Tags: geometry , rhombus , area , square , symmetry

In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that: a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus. b) The area of the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.

2015 CHMMC (Fall), 1

Tags: combinatorics , geometry

$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?

2022 Kosovo & Albania Mathematical Olympiad, 3

Tags: geometry

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $X$ and $Y$ be points on the segments $AB$ and $CD$, respectively. Prove that $\angle XMY = 90^\circ$ if and only if $BX + CY = XY$. [i]Note: In the competition, students were only asked to prove the 'only if' direction.[/i]

2011 Tuymaada Olympiad, 3

Tags: circumcircle , geometry

An excircle of triangle $ABC$ touches the side $AB$ at $P$ and the extensions of sides $AC$ and $BC$ at $Q$ and $R$, respectively. Prove that if the midpoint of $PQ$ lies on the circumcircle of $ABC$, then the midpoint of $PR$ also lies on that circumcircle.

2019 Oral Moscow Geometry Olympiad, 4

Tags: excircle , geometry , angle , right triangle

Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

2020 LMT Fall, B2

Tags: geometry

The area of a square is $144$. An equilateral triangle has the same perimeter as the square. The area of a regular hexagon is $6$ times the area of the equilateral triangle. What is the perimeter of the hexagon?

2006 MOP Homework, 3

Tags: circumcircle , geometry

In triangle $ ABC$,$ \angle BAC \equal{} 120^o$. Let the angle bisectors of angles $ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively. Prove that the circle on diameter $ EF$ passes through $ D.$

2018 USA TSTST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$. Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$. [i]Evan Chen and Yannick Yao[/i]

2013 CHMMC (Fall), 2

Tags: geometry

Two circles of radii $7$ and $17$ have a distance of $25$ between their centers. What is the difference between the lengths of their common internal and external tangents (positive difference)?

2019 Kosovo National Mathematical Olympiad, 3

Tags: geometry , incenter , angle bisector

Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$. Let points $D,E$ be on sides $AC,AB$, respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$, respectively. Show that $ID=IE$.

2021 Brazil National Olympiad, 6

Tags: geometry , bicentric polygon , convex polygon , inscribed circle , circumscribed , tangent circle

Let $n \geq 5$ be integer. The convex polygon $P = A_{1} A_{2} \ldots A_{n}$ is bicentric, that is, it has an inscribed and circumscribed circle. Set $A_{i+n}=A_{i}$ to every integer $i$ (that is, all indices are taken modulo $n$). Suppose that for all $i, 1 \leq i \leq n$, the rays $A_{i-1} A_{i}$ and $A_{i+2} A_{i+1}$ meet at the point $B_{i}$. Let $\omega_{i}$ be the circumcircle of $B_{i} A_{i} A_{i+1}$. Prove that there is a circle tangent to all $n$ circles $\omega_{i}$, $1 \leq i \leq n$.

2021 CMIMC, 2.4

Tags: geometry

A $2\sqrt5$ by $4\sqrt5$ rectangle is rotated by an angle $\theta$ about one of its diagonals. If the total volume swept out by the rotating rectangle is $62\pi$, find the measure of $\theta$ in degrees. [i]Proposed by Connor Gordon[/i]

2007 Nordic, 2

Tags: rectangle , geometry

Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.

2016 Postal Coaching, 3

Tags: geometry , combinatorial geometry

Given a convex polygon, show that it has three consecutive vertices such that the circle through them contains the polygon.

2016 Iranian Geometry Olympiad, 2

Tags: geometry

In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili

2015 NIMO Problems, 8

Tags: geometry , incenter

Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of \[IM_a^2+IM_b^2+IM_c^2.\][i]Proposed by David Altizio[/i]

2008 District Olympiad, 4

Tags: geometry , parallelogram , trigonometry , gauss , geometric transformation , reflection , ratio

Let $ ABCD$ be a cyclic quadrilater. Denote $ P\equal{}AD\cap BC$ and $ Q\equal{}AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F\equal{}CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

1988 Tournament Of Towns, (187) 4

Tags: coloring , combinatorics , cube , 3d geometry , geometry

Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.

2008 China Team Selection Test, 1

Tags: algorithm , induction , geometry

Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)

2009 Sharygin Geometry Olympiad, 1

Tags: geometry

Points $ B_1$ and $ B_2$ lie on ray $ AM$, and points $ C_1$ and $ C_2$ lie on ray $ AK$. The circle with center $ O$ is inscribed into triangles $ AB_1C_1$ and $ AB_2C_2$. Prove that the angles $ B_1OB_2$ and $ C_1OC_2$ are equal.