Call a number $a-b\sqrt2$ with $a$ and $b$ both positive integers $tiny$ if it is closer to zero than any number $c-d\sqrt2$ such that $c$ and $d$ are positive integers with $c<a$ and $d<b$. Three numbers which are tiny are $1-\sqrt2$, $3-2\sqrt2$, and $7-5\sqrt2$. Without using any calculator or computer, prove whether or not each of the following is tiny: \[(a)\ 58-41\sqrt2,\qquad\qquad (b)\ 99-70\sqrt2.\]

Find all functions $f : \mathbb R \to \mathbb R$ such that for all $x, y,$ \[f(f(x) + y) = f(x^2 - y) + 4f(x)y.\]

Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.

On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid. Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.

Points $A$, $B$, $C$, $D$ lie on a circle (in that order) where $AB$ and $CD$ are not parallel. The length of arc $AB$ (which contains the points $D$ and $C$) is twice the length of arc $CD$ (which does not contain the points $A$ and $B$). Let $E$ be a point where $AC=AE$ and $BD=BE$. Prove that if the perpendicular line from point $E$ to the line $AB$ passes through the center of the arc $CD$ (which does not contain the points $A$ and $B$), then $\angle ACB = 108^\circ$.

A)Let $x,y$ be two complex numbers on the unit circle so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-x|+|z-y| \ge |zx-y|$ B)Let $x,y$ be two complex numbers so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$

Let $K$ be an equilateral triangle in the plane. Prove that for every $p>0$ there exists an $\varepsilon>0$ with the following property: If $n$ is a positive integer, and $T_1,\ldots,T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$ \sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon, $$ then $$ \sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p. $$

A tribe called Ababab uses only letters $A$ and $B$, and they create words according to the following rules: (1) $A$ is a word; (2) if $w$ is a word, then $ww$ and $w\overline{w}$ are also words, where $\overline{w}$ is obtained from $w$ by replacing all letters $A$ with $B$ and all letters $B$ with $A$ ( $xy$ denotes the concatenation of $x$ and $y$) (3) all words are created by rules (1) and (2). Prove that any two words with the same number of letters differ exactly in half of their letters.

A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1\times 4$ or $4 \times 1$ rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers $(i, j)$ of row and column, $1 \leq i, j \leq 20$, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by $1$ mod $20$.)

Among a set of $2N$ coins, all identical in appearance, $2N - 2$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times, if (a) $N = 16$, ( b ) $N = 11$ ? (A Shapovalov)

We have an acute triangle $ABC$. Consider the square $A_1A_2A_3A_4$ which has one vertex in $AB$, one vertex in $AC$ and two vertices ($A_1$ and $A_2$) in $BC$ and let $x_A=\angle A_1AA_2$. Analogously we define $x_B$ and $x_C$. Prove that $x_A+x_B+x_C=90$

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

Given a triangle $ABC$. Let $O$ be the circumcenter of this triangle $ABC$. Let $H$, $K$, $L$ be the feet of the altitudes of triangle $ABC$ from the vertices $A$, $B$, $C$, respectively. Denote by $A_{0}$, $B_{0}$, $C_{0}$ the midpoints of these altitudes $AH$, $BK$, $CL$, respectively. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$, respectively. Prove that the four lines $A_{0}D$, $B_{0}E$, $C_{0}F$ and $OI$ are concurrent. (When the point $O$ concides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$.)

Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear. (A. Voidelevich)

Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

The sequence of positive real numbers $(x_n)$ is defined recursively as follows: $x_1 = 1$, and for $n \geq 2$, \[ x_{n} = \begin{cases} nx_{n-1} & \quad \text{when }nx_{n-1}\leq 1, \\ \frac{x_{n-1}}{n} &\quad \text{otherwise}. \end{cases} \] Show that there is an integer $N > 1$ such that $|x_N - 1| < 0.00001$. Thus the elements of the sequence can get very close to 1 for large $N$; however, it is easy to see that they can never be 1 unless $N = 1$.