Given is a triangle $ABC$ and on its sides the triangles $ABM, BCN$ and $CAP$ are build such that $\angle AMB = 150^o$, $AM = MB$, $\angle CAP = \angle CBN = 30^o$, $\angle ACP = \angle BCN = 45^o$. Prove that the triangle $MNP$ is an equilateral triangle.

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

A portion of a corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? [asy]/* AMC8 2002 #23 Problem */ fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey); fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey); fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey); fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey); draw((0,0)--(0,11)--(11,11)); for ( int x = 1; x &lt; 11; ++x ) { draw((x,11)--(x,0), linetype("4 4")); } for ( int y = 1; y &lt; 11; ++y ) { draw((0,y)--(11,y), linetype("4 4")); } clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy] $ \textbf{(A)}\ \frac13\qquad\textbf{(B)}\ \frac49\qquad\textbf{(C)}\ \frac12\qquad\textbf{(D)}\ \frac59\qquad\textbf{(E)}\ \frac58$

A cylindrical glass beaker is $8$ cm high and its circumference rim is $12$ cm wide . Inside, $3$ cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located $1$ cm from the base (or bottom) of the glass, there is a fly. What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path? [hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca. ¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?[/hide]

A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.

In a triangle $ABC$ it is given $\tan A,\tan B,\tan C$ are integers. Find their values.

$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.

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 $C.$

In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$, the triangle $MNP$ is right angled at $N$.

Let $ABC$ be a triangle and $P$ a point in its interior. Circle $\Gamma_A$ is considered such that it is tangent to rays $(PB$ and $(PC$. Define similarly $\Gamma_B$ and $\Gamma_C$. Let $\ell_A\neq PA$ be the other common internal tangent of $\Gamma_B$ and $\Gamma_C$. Prove that $\ell_A$, $\ell_B$ and $\ell_C$ meet at a point. [i]Proposed by Andrei Vila[/i]

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

The triangle below is divided into nine stripes of equal width each parallel to the base of the triangle. The darkened stripes have a total area of $135$. Find the total area of the light colored stripes. [img]https://cdn.artofproblemsolving.com/attachments/0/8/f34b86ccf50ef3944f5fbfd615a68607f4fadc.png[/img]

In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.

In a unit square $ABCD$, find the minimum of $\sqrt2 AP + BP + CP$ where $P$ is a point inside $ABCD$.

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.

5. Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25. Find the distance from T to D.

Today was realized the National Olimpiad in Cuba, this is the 3rd problem of the second day: Prof that we can color the lattice points in the plane with two color so that every rectangle with vertices in the lattice points and edges parallels to the co-ordinate axis that have area 2^n is not monocromatic [/img]

Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1,B_2,C_1,C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle $ABC$. Prove that the radical center of the circumcircle of the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ lies on the line $OI$.

Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.