For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following. $$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$ ($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)

Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]

Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Determine the largest possible constant \( C \) such that for any positive real numbers \( x, y, z \), which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] [i]Proposed by Vadym Solomka[/i]

A circle $\mathcal C$ and five distinct points $M_1,M_2,M_3,M_4$ and $M$ on $\mathcal C$ are given in the plane. Prove that the product of the distances from $M$ to lines $M_1M_2$ and $M_3M_4$ is equal to the product of the distances from $M$ to the lines $M_1M_3$ and $M_2M_4$. What can one deduce for $2n+1$ distinct points $M_1,\ldots,M_{2n},M$ on $\mathcal C$?

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.

The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is $\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$

Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.

Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$: \[ f(xf(y)) \plus{} y \plus{} f(x) \equal{} f(x \plus{} f(y)) \plus{} yf(x)\]

Let $AXBY$ be a convex quadrilateral. The incircle of $\triangle AXY$ has center $I_A$ and touches $\overline{AX}$ and $\overline{AY}$ at $A_1$ and $A_2$ respectively. The incircle of $\triangle BXY$ has center $I_B$ and touches $\overline{BX}$ and $\overline{BY}$ at $B_1$ and $B_2$ respectively. Define $P = \overline{XI_A} \cap \overline{YI_B}$, $Q = \overline{XI_B} \cap \overline{YI_A}$, and $R = \overline{A_1B_1} \cap \overline{A_2B_2}$. [list=a] [*] Prove that if $\angle AXB = \angle AYB$, then $P$, $Q$, $R$ are collinear. [*] Prove that if there exists a circle tangent to all four sides of $AXBY$, then $P$, $Q$, $R$ are collinear. [/list]

Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.

Let $n \geq 1$ be an integer. A [b]path[/b] from $(0,0)$ to $(n,n)$ in the $xy$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$) or upwards (move denoted by $N$), all the moves being made inside the half-plane $x \geq y$. A [b]step[/b] in a path is the occurence of two consecutive moves of the form $EN$. Show that the number of paths from $(0,0)$ to $(n,n)$ that contain exactly $s$ steps $(n \geq s \geq 1)$ is \[\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.\]

Find all the functions $f : R\to R$ such that $f(x^2 + f(y)) = y - x^2$ for all $x, y$ reals.

Let $S$ denote the sum of all rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive divisors of $1300$. If $S$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find $m+n$. [i]Proposed by Ephram Chun[/i]

Let $x_1, x_2, \ldots, x_{11}$ be nonnegative reals such that their sum is $1$. For $i = 1,2, \ldots, 11$, let \[ y_i = \begin{cases} x_{i} + x_{i + 1}, & i \, \, \textup{odd} ,\\ x_{i} + x_{i + 1} + x_{i + 2}, & i \, \, \textup{even} ,\end{cases} \] where $x_{12} = x_{1}$. And let $F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}$. Prove that $x_6 < x_8$ when $F$ achieves its maximum.

Show that there is no integer $ a$ such that $ a^2 \minus{} 3a \minus{} 19$ is divisible by $ 289$.

Let $ABC$ be a triangle with $AB = 13$, $AC = 14$, and $BC = 15$, and let $\Gamma$ be its incircle with incenter $ I$. Let $D$ and $E$ be the points of tangency between $\Gamma$ and $BC$ and $AC$ respectively, and let $\omega$ be the circle inscribed in $CDIE$. If $Q$ is the intersection point between $\Gamma$ and $\omega$ and $P$ is the intersection point between $CQ$ and $\omega$, compute the length of $P Q$.

Prove that $1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0$ for every $x \ge - 1$ .

Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16 $

Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.

While eating out, Mike and Joe each tipped their server $ 2$ dollars. Mike tipped $ 10\%$ of his bill and Joe tipped $ 20\%$ of his bill. What was the difference, in dollars between their bills? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 20$