Let $(v_1, ..., v_{2^n})$ be the vertices of an $n$-dimensional hypercube. Label each vertex $v_i$ with a real number $x_i$. Label each edge of the hypercube with the product of labels of the two vertices it connects. Let $S$ be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$ in terms of $ n$. Note: an $n$ dimensional hypercube is a graph on $2^n$ vertices labeled labeled with the binary strings of length $n$, where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices $100$ and $101$ on the $3$ dimensional hypercube are connected, but the vertices $100$ and $111$ are not.

Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? [img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.

Let a circle through $B$ and $C$ of a triangle $ABC$ intersect $AB$ and $AC$ in $Y$ and $Z$ , respectively. Let $P$ be the intersection of $BZ$ and $CY$ , and let $X$ be the intersection of $AP$ and $BC$ . Let $M$ be the point that is distinct from $X$ and on the intersection of the circumcircle of the triangle $XYZ$ with $BC$. Prove that $M$ is the midpoint of $BC$

Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.

Let $M$ be an arbitrary positive real number greater than $1$, and let $a_1,a_2,...$ be an infinite sequence of real numbers with $a_n\in [1,M]$ for any $n\in \mathbb{N}$. Show that for any $\epsilon\ge 0$, there exists a positive integer $n$ such that $$\frac{a_n}{a_{n+1}}+\frac{a_{n+1}}{a_{n+2}}+\cdots+\frac{a_{n+t-1}}{a_{n+t}}\ge t-\epsilon$$ holds for any positive integer $t$.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.

Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.

Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square. [b]Generalization:[/b] Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.

In the diagram below, how many different routes are there from point $M$ to point $P$ using only the line segments shown? A route is not allowed to intersect itself, not even at a single point. [asy] unitsize(40); draw( (1, 0) -- (2, 0) -- (2, 1) -- (3, 1) -- (3, 2) -- (2, 2) -- (2, 3) -- (1, 3) -- (1, 2) -- (0, 2) -- (0, 1) -- (1, 1) -- cycle); draw( (1, 1) -- (2, 1) -- (2, 2) -- (1, 2) -- cycle); draw( (1, 0) -- (2, 1)); draw((1, 1) -- (2, 2)); draw((1, 2) -- (2, 3)); label( "$M$", (1, 0), SW); label("$P$", (1, 3), NW); label("$F$", (2, 3), NE); label("$G$", (2, 0), SE); [/asy]

Let $n$ be a fixed positive integer. Ana and Banana are playing a game. First, Ana picks a subset $S$ of $\{1,2,\ldots,n\}$. Then for each $k=1,2,\ldots,n$, she tells Banana how many numbers from $k-1$ to $k+1$ she has picked (i.e. $\lvert S \cap \{k-1,k,k+1\}\rvert$). Then Banana guesses $S$; she wins if her guess is correct and she loses otherwise. (a) Determine all $n$ for which Banana will win regardless of what Ana chooses. (b) For the values of $n$ for which Ana can win, determine the number of sets $S$ she can choose so as to do so.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$

Each white point in the figure below has to be completed with one of the integers $1, 2, ..., 9$, without repetitions, such that the sum of the three numbers in the external circle is equal to the sum of the four numbers in each internal circle that don't belong to the external circle. $(a)$ Show a solution. $(b)$ Prove that, in any solution, the number $9$ must belong to the external circle.

What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$? $ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$

Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.

Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$? [i]Proposed by Evan Chen[/i]

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Given a triangle $ABC$, point $D$ on $[AB], E$ on $[AC]$, $|AD| = |DE| = |AC| , |BD| = |AE| , DE$ is parallel to $BC$. Prove that the length $|BD|$ equals to the side of a regular decagon inscribed in a circle with the radius $R=|AC|$.

Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ \quad f(1)=1$ $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$ $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken $f$.

Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.