Olympiad problems

2003 IberoAmerican, 2

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

In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.

2013 Romania Team Selection Test, 2

Tags: incenter , circumcircle , geometric transformation , homothety , power of a point , geometry

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

2012 ELMO Shortlist, 5

Tags: parallelogram , trigonometry , ellipse , geometric transformation , reflection , conic , geometry

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

2006 Australia National Olympiad, 4

Tags: geometry , geometric transformation , reflection , absolute value , combinatorics

There are $n$ points on a circle, such that each line segment connecting two points is either red or blue. $P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$. (a) For which values of $n$ is this possible? (b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.

1993 China Team Selection Test, 3

Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , ring theory

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2014 Taiwan TST Round 3, 3

Tags: geometry , circumcircle , incenter , geometric transformation , homothety , ratio , parallelogram

Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

1988 Romania Team Selection Test, 1

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

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

2013 Lusophon Mathematical Olympiad, 6

Tags: geometry , geometric transformation , homothety

Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.

2001 Pan African, 2

Tags: geometry , 3d geometry , geometric transformation , reflection , floor function

Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?

2013 Today's Calculation Of Integral, 865

Tags: calculus , integration , geometric transformation , rotation , analytic geometry , rectangle , geometry

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

2008 All-Russian Olympiad, 7

Tags: geometry , geometric transformation , dilation , angle bisector

In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.

2012 Grigore Moisil Intercounty, 3

Tags: circumcircle , geometric transformation , rotation , geometry

Let $ \Delta ABC$ be a triangle, with $ m(\angle A)=90^{\circ}$ and $ m(\angle B)=30^{\circ}.$ If $M$ is the middle of $[AB],$ $N$ is the middle of $[BC],$ and $P\in[BC],\ Q\in[MN],$ such that \[\frac{PB}{PC}=4\cdot\frac{QM}{QN}+3,\] prove that $ \Delta APQ$ is an equilateral triangle. [b]Author: MARIN BANCOȘ[/b] [b]Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 24.03.2012, 7th grade[/b]

1999 Tuymaada Olympiad, 1

Tags: geometry , geometric transformation , combinatorics

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

2006 Iran MO (3rd Round), 2

Tags: vector , geometric transformation , rotation , algebra , polynomial , group theory , geometry

$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$

1991 AIME Problems, 2

Tags: geometry , rectangle , geometric transformation

Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.

1991 Arnold's Trivium, 95

Tags: algebra , polynomial , invariant , geometry , geometric transformation , rotation , quadratic

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

2007 International Zhautykov Olympiad, 3

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

Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle. Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.

2013 Today's Calculation Of Integral, 880

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

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2013 Romanian Masters In Mathematics, 3

Tags: geometry , geometric transformation , reflection , combinatorics

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

2007 Kyiv Mathematical Festival, 2

Tags: geometric transformation , reflection , trapezoid , geometry

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

2012 NIMO Problems, 7

Tags: rectangle , geometric transformation , reflection , geometry

Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$. [i]Proposed by Aaron Lin[/i]

2010 Greece Team Selection Test, 3

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

Cono Sur Shortlist - geometry, 2020.G1.4

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 $DE$.

2007 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometric transformation , reflection , trigonometry , angle bisector , power of a point , radical axis , geometry

Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.

2012 Germany Team Selection Test, 2

Tags: circumcircle , trigonometry , geometric transformation , geometry

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]