Olympiad problems

2013 Online Math Open Problems, 26

Tags: trigonometry , geometry , geometric transformation , reflection , circumcircle , complex number

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

1997 Flanders Math Olympiad, 4

Tags: geometry , geometric transformation , reflection , algebra

Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)

2004 China Team Selection Test, 2

Tags: incenter , perimeter , geometric transformation , reflection , geometry

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

1993 Baltic Way, 20

Tags: geometry , 3d geometry , tetrahedron , geometric transformation , dilation

Let $ \mathcal Q$ be a unit cube. We say that a tetrahedron is [b]good[/b] if all its edges are equal and all of its vertices lie on the boundary of $ \mathcal Q$. Find all possible volumes of good tetrahedra.

2002 Moldova National Olympiad, 4

Tags: geometry , geometric transformation , reflection , symmetry

The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.

PEN R Problems, 3

Tags: analytic geometry , geometry , geometric transformation , rotation , trigonometry , group theory , abstract algebra

Prove no three lattice points in the plane form an equilateral triangle.

2020 MBMT, 20

Tags: geometry , geometric transformation , rotation

Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric? [i]Proposed by Gabriel Wu[/i]

2009 Princeton University Math Competition, 5

Tags: geometry , symmetry , rotation , geometric transformation , reflection

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

2010 Iran Team Selection Test, 5

Tags: geometry , circumcircle , geometric transformation , ratio , homothety , radical axis

Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$ \[\angle QXA=\angle QKP\]

2004 All-Russian Olympiad, 1

Tags: geometry , geometric transformation , rotation , analytic geometry , combinatorics , combinatorial geometry

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

1997 China Team Selection Test, 1

Tags: circumcircle , trigonometry , geometric transformation , homothety , rotation , geometry

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

2008 Bulgaria Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , homothety , trigonometry , power of a point , radical axis

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

2007 China Team Selection Test, 1

Tags: geometry , incenter , geometric transformation , homothety

Let $ ABC$ be a triangle. Circle $ \omega$ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of $ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.

2008 Romania National Olympiad, 1

Tags: circumcircle , geometric transformation , reflection , perpendicular bisector , geometry

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

2020 Cono Sur Olympiad, 4

Tags: geometry , geometric transformation , reflection

Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.

2009 Hong Kong TST, 4

Tags: geometric transformation , homothety , ratio , circumcircle , trapezoid , projective geometry , geometry

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

1993 Polish MO Finals, 2

Tags: inequalities , trapezoid , parallelogram , geometric transformation , homothety , geometry

A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.

2014 IMC, 5

Tags: geometric transformation , geometry

Let $A_{1}A_{2} \dots A_{3n}$ be a closed broken line consisting of $3n$ lines segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index $i=1,2,\dots,3n$, the triangle $A_{i}A_{i+1}A_{i+2}$ has counterclockwise orientation and $\angle A_{i}A_{i+1}A_{i+2} = 60º$, using the notation $A_{3n+1} = A_{1}$ and $A_{3n+2} = A_{2}$. Prove that the number of self-intersections of the broken line is at most $\frac{3}{2}n^{2} - 2n + 1$

2013 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , geometry , incenter , circumcircle , geometric transformation , reflection , analytic geometry

Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.

1998 Poland - First Round, 3

Tags: trigonometry , geometry , analytic geometry , geometric transformation , reflection , ratio , similar triangles

In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.

2007 QEDMO 4th, 5

Tags: geometry , geometric transformation , reflection , 3d geometry , prism , geometry theorems

Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.

2010 Contests, 2

Tags: circumcircle , geometric transformation , reflection , parallelogram , calculus , trigonometry , geometry

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

2007 Italy TST, 1

Tags: rectangle , incenter , geometric transformation , homothety , reflection , geometry

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

2004 District Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , geometric transformation , rotation

On the tetrahedron $ ABCD $ make the notation $ M,N,P,Q, $ for the midpoints of $ AB,CD,AC, $ respectively, $ BD. $ Additionally, we know that $ MN $ is the common perpendicular of $ AB,CD, $ and $ PQ $ is the common perpendicular of $ AC,BD. $ Show that $ AB=CD, BC=DA, AC=BD. $

2002 All-Russian Olympiad, 2

Tags: geometry , trapezoid , trigonometry , parallelogram , geometric transformation , reflection , trig identities

A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.