Let $ f:[0,1]\longrightarrow\mathbb{R}_+^* $ be a Riemann-integrable function. Calculate $ \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) . $

Let $f: \mathbb{N} \to \mathbb{N}$ be a function that satisfies: \[ f(1) = 2008, \] \[ f(4n^2) = 4f(n^2), \] \[ f(4n^2 + 2) = 4f(n^2) + 3, \] \[ f(4n(n+1)) = 4f(n(n+1)) + 1, \] \[ f(4n(n+1) + 3) = 4f(n(n+1)) + 4. \] Determine whether there exists a natural number $m$ such that: \[ 1^2 + 2^2 + \dots + m^2 + f(1^2) + \dots + f(m^2) = 2008m + 251. \]

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $$f(x)=f \left( \frac{x}{2} \right) + \frac{x}{2} f'(x), ~\forall x \in \mathbb{R}.$$ Prove that $f$ is a polynomial function of degree at most one. [hide=Note]The problem was posted quite a few times before: [url]https://artofproblemsolving.com/community/c7h100225p566080[/url] [url]https://artofproblemsolving.com/community/q11h564540p3300032[/url] [url]https://artofproblemsolving.com/community/c7h2605212p22490699[/url] [url]https://artofproblemsolving.com/community/c7h198927p1093788[/url] I'm reposting it just to have a more suitable statement for the [url=https://artofproblemsolving.com/community/c13_contests]Contest Collections[/url]. [/hide]

Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1. \] [i]Proposed by Dusan Djukic, Serbia[/i]

Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by $20$ and increases the larger number by $23,$ only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.

Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$. Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$. Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$´s for eigenvalues.

Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that $$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.

For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them. Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.

Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

Find the number of ways to place a digit in each cell of a $2 \times 3$ grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991 = 999$ and $9+9+81 = 99$. [asy] unitsize(0.7cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); label("$9$", (0.5,0.5)); label("$9$", (1.5,0.5)); label("$1$", (2.5,0.5)); label("$0$", (0.5,1.5)); label("$0$", (1.5,1.5)); label("$8$", (2.5,1.5)); [/asy]

In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)

Let $ABC$ be an acute triangle with orthocenter $H$. A point $L \ne A$ lies on the plane of $ABC$ such that $\overline{HL} \perp \overline{AL}$ and $LB : LC = AB : AC$. Suppose $M_1 \ne B$ lies on $\overline{BL}$ such that $\overline{HM_1} \perp \overline{BM_1}$ and $M_2 \ne C$ lies on $\overline{CL}$ such that $\overline{HM_2} \perp \overline{CM_2}$. Prove that $\overline{M_1M_2}$ bisects $\overline{AL}$.

Find all positive integer $n$ and nonnegative integer $a_1,a_2,\dots,a_n$ satisfying: $i$ divides exactly $a_i$ numbers among $a_1,a_2,\dots,a_n$, for each $i=1,2,\dots,n$. ($0$ is divisible by all integers.)

The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$. [i]Mihai Baluna[/i]

One of the factors of $ x^4\plus{}4$ is: $ \textbf{(A)}\ x^2\plus{}2 \qquad\textbf{(B)}\ x\plus{}1 \qquad\textbf{(C)}\ x^2\minus{}2x\plus{}2 \qquad\textbf{(D)}\ x^2\minus{}4\\ \textbf{(E)}\ \text{none of these}$