A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively. (a) Prove that $AP/AD=BQ/BC$. (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

Points $E$ and $F$ are located on square $ABCD$ so that $\Delta BEF$ is equilateral. What is the ratio of the area of $\Delta DEF$ to that of $\Delta ABE$? [asy] pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=B+2*dir(165), E=intersectionpoint(B--X, A--D), Y=B+2*dir(105), F=intersectionpoint(B--Y, D--C); draw(B--C--D--A--B--F--E--B); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy] $\textbf{(A)}\; \frac43\qquad \textbf{(B)}\; \frac32\qquad \textbf{(C)}\; \sqrt3\qquad \textbf{(D)}\; 2\qquad \textbf{(E)}\; 1+\sqrt3\qquad$

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles, and similar to the large rectangle, so that the sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine what is the minimum possible value of the area of the large rectangle.

Sides of a triangle are equal to $3$, $4$ and $5$. Each side is extended until it intersects the bisector of the external angle to the angle opposite to it. Three such points are obtained in all. Prove that one of the three points we get is the midpoint of the segment joining the other two points. (V. Prasolov)

the edges of triangle $ABC$ are $a,b,c$ respectively,$b<c$,$AD$ is the bisector of $\angle A$,point $D$ is on segment $BC$. (1)find the property $\angle A$,$\angle B$,$\angle C$ have,so that there exists point $E,F$ on $AB,AC$ satisfy $BE=CF$,and $\angle NDE=\angle CDF$ (2)when such $E,F$ exist,express $BE$ with $a,b,c$

The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

Let $O$ be the centre of a regular $n$-gon whose vertices are labelled $A_1$,$...$, $A_n$. Let $a_1>a_2>...>a_n>0$. Prove that the vector $$a_1\overrightarrow{OA_1}+a_2\overrightarrow{OA_2}+...+a_n\overrightarrow{OA_n}$$ is not equal to the zero vector. (D. Fomin, Alexey Kirichenko)

Jimmy’s garden has right angled triangle shape that lies on island of circular shape in such a way that the corners of the triangle are on the shore of the island. When he made fences along the garden he realized that the length of the shortest side is $36$ meter shorter than the longest side, and third side required $48$ meter long fence. In the middle of the garden he built a house of circular shape that has the largest possible size. Jimmy measured the distance between the center of his house and the center of the island. What is the square of this distance?

Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that (a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and (b) the Euclidean norm of every point $v$ in $A$ satisfies \[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\] Prove that the cardinality of $A$ is at most $2d+4$.

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that $$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.

We are given an acute-angled triangle $ABC$ with $AB < AC < BC$. Points $K$ and $L$ are chosen on segments $AC$ and $BC$, respectively, so that $AB = CK = CL$. Perpendicular bisectors of segments $AK$ and $BL$ intersect the line $AB$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at point $M$. Prove that $AK + KM = BL + LM$. Poland

Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$, and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$. Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

Let $ABC$ be a triangle. Consider three circles, centered at $A, B, C$, with respective radii $$\sqrt{AB \cdot AC},\sqrt{BC \cdot BA},\sqrt{CA \cdot CB}.$$ Given that there are six distinct pairwise intersections between these three circles, show that they lie on two concentric circles. [i](Two circles are concentric if they have the same center.)[/i]