Olympiad problems

2014 Iran MO (3rd Round), 2

$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]

2015 Tournament of Towns, 2

Tags: geometry , geometric transformation , reflection , circumcircle

A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

2003 AIME Problems, 6

Tags: geometry , geometric transformation , rotation , ratio

In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

2007 Baltic Way, 12

Tags: circumcircle , geometric transformation , dilation , similar triangles , geometry

Let $M$ be a point on the arc $AB$ of the circumcircle of the triangle $ABC$ which does not contain $C$. Suppose that the projections of $M$ onto the lines $AB$ and $BC$ lie on the sides themselves, not on their extensions. Denote these projections by $X$ and $Y$, respectively. Let $K$ and $N$ be the midpoints of $AC$ and $XY$, respectively. Prove that $\angle MNK=90^{\circ}$ .

2010 Indonesia TST, 4

Tags: trigonometry , geometry , geometric transformation , reflection , circumcircle , angle bisector , similar triangles

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

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.

2024 Canadian Junior Mathematical Olympiad, 3

Tags: incenter , geometric transformation , reflection , angle bisector , geometry

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

2003 Tournament Of Towns, 4

Tags: circumcircle , geometric transformation , parallelogram , rectangle , geometry

In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

2013 Uzbekistan National Olympiad, 4

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

Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that : (a) $AA_2=BB_2=IO$; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$

1992 Dutch Mathematical Olympiad, 3

Tags: geometry , trigonometry , geometric transformation , reflection

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).

2013 Online Math Open Problems, 6

Tags: rotation , geometry , geometric transformation

Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$? [i]Ray Li[/i]

2010 Stanford Mathematics Tournament, 1

Tags: geometry , geometric transformation , reflection

Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.

2012 NIMO Problems, 8

Tags: geometry , parallelogram , geometric transformation , reflection , rotation , perimeter

A convex 2012-gon $A_1A_2A_3 \dots A_{2012}$ has the property that for every integer $1 \le i \le 1006$, $\overline{A_iA_{i+1006}}$ partitions the polygon into two congruent regions. Show that for every pair of integers $1 \le j < k \le 1006$, quadrilateral $A_jA_kA_{j+1006}A_{k+1006}$ is a parallelogram. [i]Proposed by Lewis Chen[/i]

2009 Tuymaada Olympiad, 1

Tags: geometry , geometric transformation , combinatorics

All squares of a $ 20\times 20$ table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i] [b]US Name Conversions: [/b] [i]Misha*: Naoki Sasha**: Richard[/i]

2006 Polish MO Finals, 3

Tags: geometry , geometric transformation , reflection , circumcircle , rhombus , congruent triangles , perpendicular bisector

Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

2010 Iran MO (3rd Round), 2

Tags: ratio , geometric transformation , geometry

in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$. a) in what ratio does $EF$ cut the digonals?(13 points) b) find $\frac{AF}{FD}$.(5 points)

2009 Sharygin Geometry Olympiad, 4

Tags: geometry , locus , parallel lines , geometric transformation , reflection , incenter

Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles. (C.Pohoata)

MBMT Team Rounds, 2020.20

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]

2014 Contests, 3

Tags: parallelogram , geometric transformation , homothety , geometry

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

2012 Math Prize for Girls Olympiad, 1

Tags: geometric transformation , vector , parallelogram , analytic geometry , geometry

Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that \[ \sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, . \]

2011 Today's Calculation Of Integral, 755

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

1978 Romania Team Selection Test, 2

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

Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.

2010 Bulgaria National Olympiad, 3

Tags: trapezoid , geometric transformation , incenter , inradius , geometry

Let $k$ be the circumference of the triangle $ABC.$ The point $D$ is an arbitrary point on the segment $AB.$ Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB,$ the segment $CD$ and the circle $k.$ We know that the points $A, B, I$ and $J$ are concyclic. The excircle of the triangle $ABC$ is tangent to the side $AB$ in the point $M.$ Prove that $M \equiv D.$

2020 Brazil Team Selection Test, 3

Tags: geometry , incenter , projective geometry , geometric transformation , inscribed quadrilateral

Let $ABCD$ be a quadrilateral with a incircle $\omega$. Let $I$ be the center of $\omega$, suppose that the lines $AD$ and $BC$ intersect at $Q$ and the lines $AB$ and $CD$ intersect at $P$ with $B$ is in the segment $AP$ and $D$ is in the segment $AQ$. Let $X$ and $Y$ the incenters of $\triangle PBD$ and $\triangle QBD$ respectively. Let $R$ be the intersection of $PY$ and $QX$. Prove that the line $IR$ is perpendicular to $BD$.

2008 IMAC Arhimede, 5

Tags: geometry , parallelogram , vector , trigonometry , geometric transformation , reflection , cyclic quadrilateral

The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$. $ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t. $ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.