The points $D$ and $E$ lie on the side $(BC)$ of the triangle $ABC$ such that $D$ is between $B$ and $E.$ A point $R$ on the segment $(AE)$ is called [i]remarkable[/i] if the lines $PQ$ and $BC$ are parallel, where $\{P\}=DR \cap AC$ and $\{Q\}=CR \cap AB.$ A point $R'$ on the segment $(AD)$ is called [i]remarkable[/i] if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\}=BR' \cap AC$ and $\{Q'\}=ER' \cap AB.$ a) If there exists a remarkable point on the segment $(AE),$ prove that any point of the segment $(AE)$ is remarkable. b) If each of the segments $(AD)$ and $(AE)$ contains a remarkable point, prove that $BD=CE=\varphi \cdot DE,$ where $\varphi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.

Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius $r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches $k_n,k_0$, and $AB$. Prove that: (a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$. (b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.

Let $ABCD$ be a convex quadrilateral in which $\angle BAC = 50^{\circ}, \angle CAD = 60^{\circ}$and $\angle BDC = 25^{\circ}$. If $E$ is the point of intersection of $AC$ and $BD$, find $\angle AEB$.

Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$. As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$. [i]Proposed by Michael Kural[/i]

Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the side of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of the star. What is the difference between the maximum and minimum possible perimeter of $s$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(E)}\ \sqrt{5} $

Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]

Polygon $A_0A_1...A_{n-1}$ satisfies the following: $\bullet$ $A_0A_1 \le A_1A_2 \le ...\le A_{n-1}A_0$ and $\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 = ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles). Prove that this polygon is regular.

a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$. Point $M$ is the midpoint of $PE$. Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$. [i]Matsvei Zorka[/i]

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: [list] [*]All squares touching the border of the grid are coloured black. [*]No four squares forming a $2\times 2$ square are coloured in the same colour. [*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.[/list] Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?

Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$. A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega'$ denote the incircle of triangle $ADE$. The tangent from $D$ to $\omega$ (different from line $AB$) and the tangent from $E$ to $\omega$ (different from line $AC$) meet at point $P$. The tangent from $B$ to $\omega'$ (different from line $AB$) and the tangent from $C$ to $\omega'$ (different from line $AC$) meet at point $Q$. Prove that, independent of the choice of $\ell$, there is a fixed point that line $PQ$ always passes through.

Figure shows a semicircle with diameter $AD$. The chords $AC$ and $BD$ meet at $P$. $Q$ is the foot of the perpendicular from $P$ to $AD$. find $\angle BCQ$ in terms of $\theta$ and $\phi$ . [img]https://cdn.artofproblemsolving.com/attachments/a/2/2781050e842b2dd01b72d246187f4ed434ff69.png[/img]

Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that \[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\] Prove that $\angle RAS+\angle BAC=\angle BGC$. [i]Merlijn Staps[/i]

Find the sum of all positive integers $n$ such that \[x^3+y^3+z^3=nx^2y^2z^2\] is satisfied by at least one ordered triplet of positive integers $(x,y,z)$.

Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that: a) $QR = AD$, b) $QR \perp AD$.

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

Given a triangle $ABC$ with its incircle touching sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let the median from $A$ intersects $B_1C_1$ at $M$. Show that $A_1M\perp BC$.

The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.

It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$ $a)$ Prove that holds: $ a_1-a_4=a_2-a_5=a_3-a_6 $ $b)$ Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.

Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$. Denote by $M$ the midpoint of the side $AB$, and assume that $\measuredangle CMD=90^{\circ}$. Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$. The line $AK$ meets $BD$ at $P$, and the line $BK$ meets $AC$ at $Q$. Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$. [color=red][Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .][/color]

Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.

In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that $\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.