Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n\geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, what is $a_{2013}$? $ \textbf{(A)}\ 6025 \qquad\textbf{(B)}\ 5555 \qquad\textbf{(C)}\ 4025 \qquad\textbf{(D)}\ 3456 \qquad\textbf{(E)}\ 2013 $

Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$ What is the minimum possible value of $h+k$? $ \textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad $

Suppose that $X$ and $Y$ are metric spaces and $f:X \longrightarrow Y$ is a continious function. Also $f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R$ with equation $f_1(x,t)=(f(x),t)$ for all $x\in X$ and $t\in \mathbb R$ is a closed function. Prove that for every compact set $K\subseteq Y$, its pre-image $f^{pre}(K)$ is a compact set in $X$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

Draw the smaller number of line segments connecting points of the figure such that the new figure obtained to have exactly: [img]https://cdn.artofproblemsolving.com/attachments/d/1/098e03714904573a1eacd2d3dc28b4e8c42c7c.png[/img] i) one axis of symmetry ii) two axes of symmetry iii) four axes of symmetry Draw a new figure, at each case.

If ${a, b}$ and $c$ are positive real numbers, prove that \begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*} [i](Montenegro).[/i]

A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$. Prove that infinitely many multiples of $n$ are palindromes.

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6, 8^8,$ and $k$?

Find the sum of all distinct possible values of $x^2-4x+100$, where $x$ is an integer between 1 and 100, inclusive. [i]Proposed by Robin Park[/i]

In the triangle $ABC$ on the side $BC$, the points$ D$ and $E$ are chosen so that the angle $BAD$ is equal to the angle $EAC$. Let $I$ and $J$ be the centers of the inscribed circles of triangles $ABD$ and $AEC$ respectively, $F$ be the point of intersection of $BI$ and $EJ$, $G$ be the point of intersection of $DI$ and $CJ$. Prove that the points $I, J, F, G$ lie on one circle, the center of which belongs to the line $I_bI_c$, where $I_b$ and $I_c$ are the centers of the exscribed circles of the triangle $ABC$, which touch respectively sides $AC$ and $AB$.

For every nonnegative integer $n{}$ let $f(n)$ be the smallest number of digits $1$ which can represent the number $n{}$ using the symbols $"+", "-", "\times", "(", ")"$. For example, $80=(1+1+1+1+1)\times(1+1+1+1)\times(1+1+1+1)$ and $f(80)\leq 13$. Prove that $2\log_3 n \leq f(n) < 5\log_3 n$ for every $n>1$.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $

Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that $\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,\ldots, P_n$ are congruent, $\bullet$ all of the $n$ line segments $P_2P_3,\ldots, P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Evaluate \[ \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]

Prove that every positive rational number can be represented in the form \[\frac{a^{3}+b^{3}}{c^{3}+d^{3}}\] for some positive integers $a, b, c$, and $d$.

Find all pairs of positive integers $(a,b)$ such that: \[\dfrac{a^3+4b}{a+2b^2+2a^2b}\] is a positive integer.

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.

Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: $\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad \textbf{(B)}\ y = \dfrac{3}{4}x\qquad \textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$ $ \textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad \textbf{(E)}\ y = \dfrac{3}{4}x + 2$