Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$ the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.

In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: [list=disc] [*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests. [/list] Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.

[b]6.[/b] Let $\{ n_k \}_{k=1}^{\infty}$ be a stricly increasing sequence of positive integers such that $\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty$ Show that the sum of the series $\sum_{k=1}^{\infty} \frac {1}{n_k} $ is an irrational number. [b](N. 19)[/b]

We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

Rakesh is flipping a fair coin repeatedly. If $T$ denotes the event where the coin lands on tails and $H$ denotes the event where the coin lands on heads, what is the probability Rakesh flips the sequence $HHH$ before the sequence $THH$?

A positive integer $K$ is given. Define the sequence $(a_n)$ by $a_1 = 1$ and $a_n$ is the $n$-th positive integer greater than $a_{n-1}$ which is congruent to $n$ modulo $K$. [b](a)[/b] Find an explicit formula for $a_n$. [b](b)[/b] What is the result if $K = 2$?

Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? [i]A. Zaslavsky[/i]

A car drove from one city to another. She drove the first third of the journey at a speed of $50$ km/h, the second third at $60$ km/h, and the last third at $70$ km/h. What is the average speed of the car along the entire journey?

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

A $100 \times 100$ table is given. For each $k, 1 \le k \le 100$, the $k$-th row of the table contains the numbers $1,2,\dotsc,k$ in increasing order (from left to right) but not necessarily in consecutive cells; the remaining $100-k$ cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least $19$ times as large as the sum of the numbers in the other column.

Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$. Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality $$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$ holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$, then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying $$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0$$ such that $f$ can be represented as $$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B$$. [Gy. Maksa]

A student divides an integer $m$ by a positive integer $n$, where $n \le 100$, and claims that $\frac{m}{n}=0.167a_1a_2...$ . Show the student must be wrong.