Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.

Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$. [i]Andrew Carratu[/i]

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square? (C.Pohoata, A.Zaslavsky)

$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

$ABC$ is triangle. $l_1$- line passes through $A$ and parallel to $BC$, $l_2$ - line passes through $C$ and parallel to $AB$. Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$. $XY=AC$. What value can take $\angle A- \angle C$ ?

The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.

We are given a triangle $ABC$ in a plane $P$. To any line $D$, not parallel to any side of the triangle, we associate the barycenter $G_D$ of the set of intersection points of $D$ with the sides of $\triangle ABC$. The object of this problem is determining the set $\mathfrak F$ of points $G_D$ when $D$ varies. (a) If $D$ goes over all lines parallel to a given line $\delta$, prove that $G_D$ describes a line $\Delta_\delta$. (b) Assume $\triangle ABC$ is equilateral. Prove that all lines $\Delta_\delta$ are tangent to the same circle as $\delta$ varies, and describe the set $\mathfrak F$. (c) If $ABC$ is an arbitrary triangle, prove that one can find a plane $P$ and an equilateral triangle $A'B'C'$ whose orthogonal projection onto $P$ is $\triangle ABC$, and describe the set $\mathfrak F$ in the general case.

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order. [b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$. [b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$. [b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$. [i]Proposed by Lewis Chen[/i]

Two circles in the plane, $\gamma_1$ and $\gamma_2$, meet at points $M$ and $N$. Let $A$ be a point on $\gamma_1$, and let $D$ be a point on $\gamma_2$. The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$, respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$, respectively. Assume the order $M$, $N$, $F$, $A$, $E$ is circular around $\gamma_1$, and the segments $AB$ and $DE$ are congruent. Prove that the points $A$, $F$, $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above. [i]***[/i]

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$. Czech Republic

A rhombus has area $36$ and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus?

Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $AB$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$ Find all possible values of $\frac{AB+AC}{BC}$

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)

Let points $A_1$, $A_2$ and $A_3$ lie on the circle $\Gamma$ in a counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of rotation is equial to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))$, where the indices are taken modulo $3$ (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$). Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$. [i]Proposed by Anant Mudgal[/i]

Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$? $\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$

Let n be a fixed prime number >3. A triangle is said to be admissible if the measure of each of its angles is of the form $\frac{m\cdot 180^{\circ}}{n}$ for some positive integer m. We are given one admissible triangle. Every minute we cut one of the triangles we already have into two admissible triangles so that no two of the triangles we have after cutting are similar. After some time, it turns out that no more cuttings are possible. Prove that at this moment, the triangles we have contain all possible admissible triangles (we do not distinguish between triangles which have same sets of angles, i.e. similar triangles).

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear. (Patrik Bak, Slovakia)

Let $ABC$ be a triangle such that $\angle A=30^\circ$ and $\angle B=80^\circ$. Let $D$ and $E$ be points on sides $AC$ and $BC$ respectively so that $\angle ABD=\angle DBC$ and $DE\parallel AB$. Determine the measure of $\angle EAC$.