Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white. a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares? b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares? c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?

Sherry and Val are playing a game. Sherry has a deck containing $2011$ red cards and $2012$ black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins $1$ dollar; otherwise, she loses $1$ dollar. In addition, Val must guess red exactly $2011$ times. If Val plays optimally, what is her expected profit from this game?

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments

Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose? $\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22$

Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?

In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row). (a) Show that $f(C) \in \mathcal F$. (b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$.

Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 80 $

On the extension of the hypotenuse $AB$ of the right-angled triangle $ABC$, the point $D$ after the point B is marked so that $DC = 2BC$. Let the point $H$ be the foot of the altitude dropped from the vertex $C$. If the distance from the point $H$ to the side $BC$ is equal to the length of the segment $HA$, prove that $\angle BDC = 18$.

How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $

Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.

Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

Let $n$ be a positive integer. Prove that $$\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.$$ (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that \[ \left(\frac{CE}{BF} \right)^2 \equal{} \frac{AJ \cdot JE}{AK \cdot KF}.\]

let m and n be natural numbers such that: $3m|(m+3)^n+1$ Prove that $\frac{(m+3)^n+1}{3m}$ is odd

Find the remainder when $2^{2019}$ is divided by $7$.

What is the sum of the prime factors of 20!?

Consider a $2^k$-tuple of numbers $(a_1,a_2,\dots,a_{2^k})$ all equal to $1$ or $-1$. In one step, we transform it to $(a_1a_2,a_2a_3,\dots,a_{2^k}a_1)$. Prove that eventually, we will obtain a $2^k$-tuple consisting only of $1$'s.

Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that \[ \liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0 \] holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?

Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible values of $$S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}$$