Place in the cells the boards measuring: a) $2 \times 2$, b) $4 \times 4$, c) $2n \times 2n$, numbers $0$, $1$ and $-1$ so that in each case all the sums of numbers in rows and columns are different.

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

Prove that a fifth-degree polynomial $$ P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6$$ is not the product of two lower-degree polynomials with integer coefficients.

There exists a unique triple of integers $(B,M, T)$ such that $B > T > M$ and $$3B^2(3T -M) + 8M^2(B - T) + 3T^2(5M - B) - (2B^3 + 3M^3 + 4T^3) + 15BMT = 2023.$$ Compute $B +M + T$.

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ $\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

Suppose that $n$ is a positive integer and let \[d_{1}<d_{2}<d_{3}<d_{4}\] be the four smallest positive integer divisors of $n$. Find all integers $n$ such that \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]

Show that the integral equation $$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$ has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$

Let $y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}$ where $\alpha \neq 0, y > 0, x_i > 0$ are real numbers, and let $\lambda \neq \alpha$ be a real number. Prove that $y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) > 0,$ and $y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}$ if $\alpha (\lambda - \alpha) < 0.$

Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]

A [i]9-cube[/i] is a nine-dimensional hypercube (and hence has $2^9$ vertices, for example). How many five-dimensional faces does it have? (An $n$ dimensional hypercube is defined to have vertices at each of the points $(a_1,a_2,\cdots ,a_n)$ with $a_i\in \{0,1\}$ for $1\le i\le n$) [i]Proposed by Evan Chen[/i]

Let $\Delta$ be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than $45^\circ,90^\circ$ and $135^\circ$. For each triangle $T\in\Delta$, $f(T)$ denotes the triangle with vertices at the second intersection points of the altitudes of $T$ with the circle. (a) Prove that there exists a natural number $n$ such that for every triangle $T\in\Delta$, among the triangles $T,f(T),\ldots,f^n(T)$ (where $f^0(T)=T$ and $f^k(T)=f(f^{k-1}(T))$) at least two are equal. (b) Find the smallest $n$ with the property from (a).

Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$. The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$. Show that the lines $AE$ and $CD$ are perpendicular.

Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ [i][b]perfect[/b][/i] if $|X| \in X$. Call a perfect subset $X$ [i][b]minimal[/b][/i] if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

Determine, if they exist, the real values of $x$ and $y$ that satisfy that $$\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0$$ such that $x + y <0.$

The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \begin{align*}x&=x^2+y^2, \\ y&=2xy\end{align*} is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

Find all functions $f:\mathbb N\to\mathbb N$ so that $$f^{2f(b)}(2a)=f(f(a+b))+a+b$$ holds for all positive integers $a,b$.

$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR \plus{} PS$ is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(3cm); pair D = origin; pair C = (2,0); pair B = (2,1); pair A = (0,1); pair P = waypoint(B--A,0.2); pair S = foot(P,D,B); pair R = foot(P,A,C); pair F = foot(A,D,B); pair Q = foot(P,A,F); pair T = intersectionpoint(P--Q,A--C); pair X = intersectionpoint(A--C,B--D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(P--S); draw(A--F); draw(P--R); draw(P--Q); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N); label("$S$",S,SE); label("$T$",T,N); label("$E$",X,SW+SE); label("$R$",R,SW); label("$F$",F,SE); label("$Q$",Q,SW);[/asy] $ \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT \plus{} AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$

Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation $$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$ Prove that $$f(x) =f(1)^{x^{2}}.$$

Let $S$ be the set of all positive integers $n$ such that $3n$ and $n+225$ share a divisor that is not $1$. Find the $100$th smallest element in $S$. [i]Tiebreaker #1[/i]

The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute \[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor. \]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 136$