Let $x_0, x_1, \ldots, x_{n-1}, x_n=x_0$ be reals and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. The numbers $y_i$ for $i=0,1, \ldots, n-1$ are chosen such that $y_i$ is between $x_i$ and $x_{i+1}$. Prove that $\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i)$ can attain both positive and negative values, by varying the $y_i$.

Compute the largest prime factor of $3^{12}+3^9+3^5+1.$

Let $ABC$ be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$. Prove that the arc of the circle that is inside the triangle always has the same length.

A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$. [asy] unitsize(0.4 cm); for(int i = 0; i <= 11; ++i) { draw((i,0)--(i,7)); } for(int j = 0; j <= 7; ++j) { draw((0,j)--(11,j)); } label("$\underbrace{\hspace{4.4 cm}}$", (11/2,-0.5)); label("$\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}$", (11,7/2)); label("$m$ blocks", (11/2,-1.5)); label("$n$ blocks", (14,7/2)); [/asy]

Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.

Let $n$ be a positive integer. There exist $n$ ordered triples$$(x_1, y_1, z_1), (x_2, y_2, z_2), \dots, (x_n, y_n, z_n)$$where each coordinate is an integer between $1$ and $100$ (inclusive), satisfying the following condition: [center] [i]For every infinite sequence $(a_1, a_2, a_3, \dots)$ of integers between $1$ and $100$, there exist a positive integer $i$ and an index $j$ (with $1 \leqslant j \leqslant n$) such that $(a_i, a_{i+1}, a_{i+2}) = (x_j, y_j, z_j)$.[/i] [/center] Determine the minimum possible value of $n$.

Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .

Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f:\mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values and such that $$ f^{g(n)}(n)=f(n)+k$$ holds for every positive integer $n$. ([i]Remark.[/i] Here, $f^{i}$ denotes the function $f$ applied $i$ times i.e $f^{i}(j)=f(f(\dots f(j)\dots ))$.)

Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.

There are $1000$ balls and $500$ bins that can fit arbitrarily many balls. All of the balls are then placed independently and at random into the bins. Estimate how many bins, on average, are empty. (Estimate the expected number of empty bins. In other words, if this were done over and over again, how many bins would be empty on average?) Your estimate must be an integer, or you will receive a score of zero.

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: $\textbf{(A) }\dfrac2{n-1}\qquad \textbf{(B) }\dfrac{n-1}2\qquad \textbf{(C) }\dfrac2n\qquad \textbf{(D) }2n\qquad \textbf{(E) }2(n-1)$

Let be three real numbers $ u,v,t $ under the condition $ u+v+t=0. $ Prove that for any positive real number $ a\neq 1 $ the following inequality is true with equality only and only if $ u=v=t=0: $ $$ a^u/a^v+a^v/a^t+a^{v+t}\ge a^u+a^v+1 $$ [i]Ion Tecu[/i]

In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.

Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

Let $n$ be a positive integer. Elfie the Elf travels in $\mathbb{R}^3$. She starts at the origin: $(0,0,0)$. In each turn she can teleport to any point with integer coordinates which lies at distance exactly $\sqrt{n}$ from her current location. However, teleportation is a complicated procedure: Elfie starts off [i]normal[/i] but she turns [i]strange[/i] with her first teleportation. Next time she teleports she turns [i]normal[/i] again, then [i]strange [/i]again... etc. For which $n$ can Elfie travel to any point with integer coordinates and be [i]normal [/i]when she gets there?

Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties: [list] [*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points, [*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]

Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.

Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

Let $a, b$, and $c$ be positive real numbers not greater than $2$. Prove the inequality $\frac{abc}{a + b + c} \le \frac43$

For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.

Let $n \ge 6$ be an integer and $F$ be the system of the $3$-element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition: for every $1 \le i < j \le n$ there is at least $ \lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$. Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F $ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5 $ (Poland) PS. just in case my translation does not make sense, I leave the original in Slovak, in case someone understands something else

Anne and Bill decide to play a game together. At the beginning, they chose a positive integer $n$; then, starting from a positive integer $\mathcal{N}_0$, Anne subtracts to $\mathcal{N}_0$ an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1=\mathcal{N}_0-n^k$ is then passed to Bill, who repeats the same process starting from $\mathcal{N}_1$: he subtracts to $\mathcal{N}_1$ an integer $j$-th power of $n$ less than or equal to $\mathcal{N}_1$, and he then gives the resulting number $\mathcal{N}_2=\mathcal{N}_1-n^j$ to Anne. The game continues like that until one player gets $0$ as the result of his operation, winning the game. For each $1\leq n \leq 1000$, let $f(n)$ be the number of integers $1\leq \mathcal{N}_0\leq 5000$ such that Anne has a winning strategy starting from them. For how many values of $n$ we have that $f(n)\geq 2520$? [I]Proposed by [b]FedeX333X[/b][/I]

In a circle with center $O$, chords $AB$ and $AC$ are drawn, both equal to the radius. Points $A_1$, $B_1$ and $C_1$ are projections of points $A, B$ and $C$, respectively, onto an arbitrary diameter $XY$. Prove that one of the segments $XB_1$, $OA_1$ and $C_1Y$ is equal to the sum of the other two. (A. Shklover)

Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

for real number $a,b,c$ in interval $ (0,1]$ prove that: $\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$