Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: for all real numbers $a_1,\ldots,a_d$ satisfying $a_1+\cdots+a_d=n$ and $0\leq a_i\leq1$, it is possible to group the numbers into $k$ packets (possibly empty) so that every packet has sum at most 1.
Let $ABC$ be a triangle. Construct equilateral triangles $ABE$ and $ACF$ externally to $ABC$. Let $G$ be the centroid of $ABE$ and $K$ the midpoint of $EF$. Determine the angles of triangle $KCG$.
Six teams will take part in a volleyball tournament. Each pair of the teams should play one match. All the matches will be realized in five rounds, each involving three simultaneous matches on the courts numbered 1, 2 and 3. Find the number of all possible draws for such a tournament. By a draw we mean a table $5 \times 3$ in which an unordered pair of teams is written on the field $(i, j)$, where $i \in \{1, 2, 3, 4, 5\}$ and $j \in \{1, 2, 3\}$, if these two teams will meet each other in the $i$-th round on the court $j$. You are allowed to write down the resulting number as a product of prime factors (instead of writing its decimal expansion).
Let $ABC$ be a non-isosceles triangle inscribed in a circle $\Gamma$ of radius $R$. The circle through $A$ tangent to $BC$ at $C$ meets again at $D$ the circle through $B$ tangent to $AC$ at $C$.
a) Prove that $CD\leq R$.
b) Prove that as $C$ moves on $\Gamma$, the line $CD$ passes through a fixed point.
Let $ABC$ be a triangle with $\angle A=40^\circ$, $\angle B=60^\circ$. Choose $D\in AC$ and $E\in AB$ such that $\angle CBD=40^\circ$ and $\angle ECB=70^\circ$. Let $F=BD\cap CE$. Prove $AF\perp BC$.
Choose 1008 distinct integers between 1 and 2014 inclusive.
a. Prove that three of them $a,b,c$ can be found such that $\gcd(a,b)$ divides $c$ (the greatest common divisor is the largest natural number dividing both integers).
b. Prove that three of them $a,b,c$ can be found such that $\gcd(a,b)$ does not divide $c$.
Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0 \lt i \lt n$, $0 \lt j \lt n$, $i \neq j$ and $n$ divides $2i + j$, then $x_{i} \lt x_{j}$.
Let us call by an "edge" any segment of length $1$ which is common to two adjacent fields of a given chessboard $8 \times 8$. Consider all possible cuttings of the chessboard into $32$ pieces $2 \times 1$ and denote by $n(e)$ the total number of such cuttings that involve the given edge $e$. Determine the last digit of the sum of the numbers $n(e)$ over all the edges $e$.
Find the number of sequences of integers $(u_n)_{n\geq1}$ such that $u_n\neq-1$ for every $n\geq1$ and $$u_{n+2}=\frac{2014+u_n}{1+u_{n+1}}$$ for every $n\geq1$.
We are given an acute-angled triangle $ABC$. Denote by $k$ the circle with diameter $AB$. A circle touching the bisector of the angle $BAC$ at the point $A$ and passing through the point $C$ meets the circle $k$ in a point $P$, $P \neq A$. Similarly, a circle touching the bisector of the angle $ABC$ at the point $B$ and passing through the point $C$ meets the circle $k$ in a point $Q$, $Q \neq B$. Prove that the lines $AQ$ and $BP$ intersect each other on the bisector of the angle $ACB$. (Peter Novotný)
Consider a standard $8\times8$ chessboard. A zigzag is a path on the white squares starting at any square of the bottom row and moving upward row by row until it reaches any square of the top row; at each step it moves diagonally upward by one square. How many zigzags are there?
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition
$$
f\left(y^{2}+2 x f(y)+f(x)^{2}\right)=(y+f(x))(x+f(y))
$$
for all real numbers $x$ and $y$.
Prove that there are infinitely many positive integers $n$ such that the largest prime factor of $n^4+n^2+1$ equals the largest prime factor of $(n+1)^4+(n+1)^2+1$.
Let $D$ and $E$ be points in the interiors of sides $[A B]$ and $[A C]$ of a triangle $A B C$, such that $D B = B C = C E$. Let $F$ be the intersection of lines $C D$ and $B E$, $I$ the incenter of triangle $A B C$, $H$ the orthocenter of triangle $D E F$ and $M$ the midpoint of arc $B A C$ on the circumcircle of triangle $A B C$. Prove that $I, H$ and $M$ are collinear.
