In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

Acute triangle $ABC$ is inscribed in a circle with center $O$. The reflections of $O$ across the three altitudes of the triangle are called $U$, $V$, $W$: $U$ over the altitude from $A$, $V$ over the altitude from $B$, and $W$ over the altitude from $C$. Let $\ell_A$ be a line through $A$ parallel to $VW$, and define $\ell_B$, $\ell_C$ similarly. Prove that the three lines $\ell_A$, $\ell_B$, $\ell_C$ are concurrent.

Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

Let $ABC$ be a triangle with incenter $I$. The line $AI$ intersects $BC$ and the circumcircle of $ABC$ at the points $T$ and $S$, respectively. Let $K$ and $L$ be the incenters of $SBT$ and $SCT$, respectively, $M$ be the midpoint of $BC$ and $P$ be the reflection of $I$ with respect to $KL$. a) Prove that $M$, $T$, $K$ and $L$ are concyclic. b) Determine the measure of $\angle BPC$.

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$

An isosceles triangle $ ABC$ with $ AB \equal{} AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$. Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$, respectively. Find the locus of the orthocenter of triangle $ AMN$.

Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$. a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$. b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$. Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.

Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.

Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is \[ \displaystyle \prod_{t \in T} f(t) \text{?} \] [asy] size((120)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); label("$\circ$", (0,0)); label("$S$", (-.7,2.5)); [/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$

Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$. Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.

Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .

The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$

Let $ABC$ be a triangle contained in one of the halfplanes determined by a line $r$. Points $A',B',C'$ are the reflections of $A,B,C$ in $r,$ respectively. Consider the line through $A'$ parallel to $BC$, the line through $B'$ parallel to $AC$ and the line through $C'$ parallel to $AB$. Show that these three lines have a common point.

Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and ${H_{c}^{'} }$ are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.

Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$. [i](Russia) Fedor Ivlev[/i]

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.