Prove that in each polyhedron there exist two faces with the same number of edges.

For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.

Guess I should stop proposing problems at 2:00 AM, as this can lead to ones like this here: Let $a$, $b$, $c$ be three positive reals. Prove the inequality $\frac{a^2+2b^2}{b+c}+\frac{b^2+2c^2}{c+a}+\frac{c^2+2a^2}{a+b}\geq\frac32\left(a+b+c\right)$.

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either ``Karl" or ``Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say ``Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

Initially, the numbers $1,1,-1$ written on the board.At every step,Mikail chooses the two numbers $a,b$ and substitutes them with $2a+c$ and $\frac{b-c}{2}$ where $c$ is the unchosen number on the board. Prove that at least $1$ negative number must be remained on the board at any step.

In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle. $ CO$ is drawn and extended to $ D$. $ AO$ is drawn. Which of the following expresses the relationship between $ x$ and $ y$? [asy]size(200);defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0); draw(O--A--C--D); dot(A^^B^^C^^D^^O); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$O$", O, dir(285)); label("$x$", O+0.1*dir(172.5), dir(172.5)); label("$y$", C+0.4*dir(187.5), dir(187.5)); draw(Circle(O,1)); [/asy] $ \textbf{(A)}\ x\equal{}3y \\ \textbf{(B)}\ x\equal{}2y \\ \textbf{(C)}\ x\equal{}60^\circ \\ \textbf{(D)}\ \text{there is no special relationship between }x\text{ and }y \\ \textbf{(E)}\ x\equal{}2y \text{ or }x\equal{}3y\text{, depending upon the length of }AB$

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? $ \textbf{(A)}\ 5\sqrt{2}-7 \qquad \textbf{(B)}\ 7-4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9} $

Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.

In triangle $CAT$, we have $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$ [asy] pair A,C,T,R; C = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570); draw(C--A--T--cycle); draw(T--R); label("$A$",A,N); label("$T$",T,SE); label("$C$",C,SW); label("$R$",R,NW); [/asy] $\text{(A)}\ 36^\circ \qquad \text{(B)}\ 54^\circ \qquad \text{(C)}\ 72^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 108^\circ$

Let $ABCD$ be a rectangle with $P$ lies on the segment $AC$. Denote $Q$ as a point on minor arc $PB$ of $(PAB)$ such that $QB = QC$. Denote $R$ as a point on minor arc $PD$ of $(PAD)$ such that $RC = RD$. The lines $CB$, $CD$ meet $(CQR)$ again at $M, N$ respectively. Prove that $BM = DN$. by Tran Quang Hung

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

Point ${D}$ lies on the side ${BC}$ of $\vartriangle ABC$. The circumcenters of $\vartriangle ADC$ and $\vartriangle BAD$ are ${O_1}$ and ${O_2}$, respectively and ${O_1O_2\parallel AB}$. The orthocenter of $\vartriangle ADC$is ${H}$ and ${AH=O_1O_2}.$ Find the angles of $\vartriangle ABC$ if $2m\left( \angle C \right)=3m\left( \angle B \right).$

In an acute-angled triangle $ABC$, $AB>AC$. $M,N$ are distinct points on side $BC$ such that $\angle BAM=\angle CAN$. Let $O_1,O_2$ be the circumcentres of $\triangle ABC, \triangle AMN$, respectively. Prove that $O_1,O_2,A$ are collinear.

$I$ is incenter of triangle $ABC$. Incircle of $ABC$ touches $AB,AC$ at $X,Y$. $XI$ intersects incircle at $M$. Let $CM\cap AB=X'$. $L$ is a point on the segment $X'C$ that $X'L=CM$. Prove that $A,L,I$ are collinear iff $AB=AC$.

Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$ and $P$ its intersection of diagonals. Denote by $O_1$, $O_2$ the circumcenters of triangles $ABP$, $CDP$, respectively. Prove that $O_1BCO_2$ is a parallelogram. [i] (Patrik Bak)[/i]

A list of $8$ numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown: \[ \text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}} \] $ \text{(A)}\ \frac{1}{64}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ 1\qquad\text{(D)}\ 2\qquad\text{(E)}\ 4 $

Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions: (i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$; (ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.

Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_1B_1$ and $A_2B_2$ of these circles such that the segments $A_1A_2$ and $B_1B_2'$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_1A_2$ and $B_1B_2$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $ABC$ belongs to a fixed line that does not depend on the choice of diameters.

Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle EDC = 2\angle BCD = 2\angle CAB$. Find the angle $\angle BED$.

Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$. [i]Proposed by Andrew Gu.[/i]

Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$