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]

Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.

Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$

What is the greatest integer not exceeding the sum $\sum^{1599}_{n=1} \dfrac{1}{\sqrt{n}}$?

At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each $(m,n,k)=(1,2012,2014)$, $(2011,2011,2012)$, $(2011,2012,2013)$, $(2011,2012,2014)$, $(2011,2013,2013)$. In how many of them can Ayşe guarantee to win the game? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A set $P$ has the following property: “For any positive integer $k$, if $p$ is a prime factor of $k^3+6$, then $p$ belongs to $P$ ”. Prove that $P$ is infinite.

Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$? [asy]unitsize(5mm); defaultpen(linewidth(0.8pt)+fontsize(10pt)); pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0); fill(shift(0,2)*xscale(4)*unitsquare,grey); draw(Ep--F--G--H--cycle); draw(A--B--C--D); label("$A$",A,W); label("$B$",B,E); label("$C$",C,E); label("$D$",D,W); label("$E$",Ep,NW); label("$F$",F,NE); label("$G$",G,SE); label("$H$",H,SW);[/asy]$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$

Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.

Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?

Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that \[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]

Consider the Sierpinski triangle iterations drawn below. $S_0$ is a single triangle, and $S_{n+1}$ consists of three copies of $S_n.$ Let a [i]maximal line segment[/i] be line segment in the drawing of $S_k$ which cannot be extended any further while remaining in $S_k.$ For example, $S_0$ has three maximal line segments and $S_1$ has $6$ maximal line segments. How many maximal line segments are there in $S_5$? [center] [img]https://cdn.artofproblemsolving.com/attachments/6/2/51d83da65910cd32ce0b235a9615ec467870e1.png[/img] [/center]

Timi was born in $1999$. Ever since her birth how many times has it happened that you could write that day’s date using only the digits $0$, $1$ and $2$? For example, $2022.02.21$. is such a date.

Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.

A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]

On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.

Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.

A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.

Find all natural numbers $N$ for which $N(N-101)$ is a perfect square.

David wanted to calculate the volume of a prism with an equilateral triangular base. He was given the height of the prism $H=15$ and the height of the base $h=6$. He accidentally swapped the values of $H$ and $h$ in his calculations. By what number should he multiply his result to get the correct volume?

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.

Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$. Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D\rfloor$? (For real $x$, $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)

Let $ M,N,P,Q $ be the middlepoints of the segments $ AB,BC,CD,DA, $ respectively, of a convex quadrilateral $ ABCD. $ Prove that if $ ANP $ and $ CMQ $ are equilateral, then $ ABDC $ is a rhombus . Moreover, determine the angles of this rhombus.

$$\frac{p}{q} = \sum_{n = 1}^\infty \frac{1}{2^{n + 6}} \frac{(10 - 4\cos^2(\frac{\pi n}{24})) (1 - (-1)^n) - 3\cos(\frac{\pi n}{24}) (1 + (-1)^n)}{25 - 16\cos^2(\frac{\pi n}{24})}$$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown on the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \left \{1,2,3,4,5 \right \}$. The covering satisfies the following conditions: $1)$ It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid. $2)$ Every unit triangle in the grid is covered, no two figures overlap and every figure is fully contained in $T$. Determine the smallest possible number of figures of type $1$ that can be used in such a covering. [i]Proposed by Ilija Jovcheski[/i]