Let $I$ be the incenter of triangle $ABC$ and let $D$ be the midpoint of side $AB$. Prove that if the angle $\angle AOD$ is right, then $AB+BC=3AC$. [I]Proposed by S. Ivanov[/i]

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.

Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear. [i]Proposed by Dominik Burek, Poland[/i]

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.

Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.

We inscribed in triangle $ABC$ the rectangle $DEFG$ such that $D$ and $E$ fall on side $AB$, $F$ on side $BC$, and $G$ on side $AC$. We know that $AF$ bisects angle $\angle BAC$, and that $\frac{AD}{DE} = \frac12$. What is the measure of angle $\angle CAB$?

Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.

Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear. [i](Proposed by Karl Czakler)[/i]

Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

In an acute triangle $ABC$ the line $OI$ is parallel to side $BC$. Prove that the center of the nine-point circle of triangle $ABC$ lies on the line $MI$, where $M$ is the midpoint of $BC$.

Let a line, perpendicular to side $AD$ of parallelogram $ABCD$ passing through point $B$, intersect line $CD$ at point $M$, and a line, passing through point $B$ and perpendicular to side $CD$, intersect line $AD$ at point $N$. Prove that the line passing through point $B$ perpendicular to the diagonal $AC$, passes through the midpoint of the segment $MN$.

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.

Point $K$ lies on the circumcircle of a rectangle $ABCD$. Line $CK$ intersects line segment $AD$ at point $M$ so that $AM:MD=2$. $O$ is the center the rectangle. Prove that the centroid of triangle $OKD$ belongs to the circumcircle of triangle $COD$. (Author: V. Shmarov)

One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.) Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$. The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ . Show that there is a point $Z$ that lies on all of these lines $UV$.

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$? [asy] size(140); pair A, B, C, D, E, F, X, Y; real length = 12.5; real width = 10; A = origin; B = (length, 0); C = (length, width); D = (0, width); X = rotate(330, C)*B; E = extension(C, X, A, B); Y = rotate(30, C)*D; F = extension(C, Y, A, D); draw(E--C--F); label("$2$", A--F, dir(180)); label("$6$", E--B, dir(270)); draw(A--B--C--D--cycle); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$", A, dir(225)); label("$B$", B, dir(315)); label("$C$", C, dir(45)); label("$D$", D, dir(135)); label("$E$", E, dir(270)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$