Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $ f(x) $ be a polynomial, $ n $ - a natural number. Prove that if $ f(x^{n}) $ is divisible by $ x-1 $, then it is also divisible by $ x^{n-1} + x^{n-2} + \ldots + x + $1.

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

Let $a$, $b$, $c$ be the side lengths of a triangle. Prove that $$ \sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}. $$

Each cell of a $1000\times 1000$ table contains $0$ or $1$. Prove that one can either cut out $990$ rows so that at least one $1$ remains in each column, or cut out $990$ columns so that at least one $0$ remains in each row.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

A polynomial $P$ is called [i]level[/i] if it has integer coefficients and satisfies $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$. What is the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P\left(n\right)$ for all integers $n$? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }6\qquad\text{(E) }10$

Three positive reals $x,y,z$ are given so that $xy=\frac{z-x+1}{y}=\frac{z+1}2.$ Prove that one of the numbers is the arithmetic mean of the other two.

Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$

Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$. A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$, which is the symmetric point of $P_0$ with respect to $A$. From $P_1$, the grasshopper jumps to $P_2$, which is the symmetric point of $P_1$ with respect to $B$. Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$, and so on. Compare the distance $P_0P_1$ and $P_0P_n$. $n \in N$.

\[\int_0^1 \frac{1}{x+\sqrt x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$? $\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$

The sequence $a_{1},a_{2},\cdots,a_{2010}$ has the following properties: (1) each sum of the 20 successive values of the sequence is nonnegative, (2) $|a_{i}a_{i+1}| \leq 1$ for $i=1,2,\cdots,2009$. Determine the maximal value of the expression $\sum_{i=1}^{2010}a_{i}$.

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

For an integer $a \ge 2$, denote by $\delta_(a) $ the second largest divisor of $a$. Let $(a_n)_{n\ge 1}$ be a sequence of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$ for all $n \ge 1$. Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$.

In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$. Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$, respectively. Assume that $F$ lies between $A$ and $C$, that $A$ lies between $D$ and $E$, and that the line $EF$ passes through the midpoint of the segment $BD$. Prove that the quadrilateral $ABCD$ is cyclic.

A shepherd left, as an inheritance, to his children a flock of $k$ sheep, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all possible values for $k$.

Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square? (S Tokarev)

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and \[f(f(m) + f(n)) = f(f(m)) + f(n)\] for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.

A line in the $xy$-plane has positive slope, passes through the point $(x, y) = (0, 29)$, and lies tangent to the ellipse defined by $\frac{x^2}{100} +\frac{y^2}{400} = 1$. What is the slope of the line?

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

[asy] size(5cm); defaultpen(fontsize(6pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle); draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle); draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle); draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2); label("1", (-3.5,0), S); label("2", (-2,0), S); label("1", (-0.5,0), S); label("1", (3.5,0), S); label("2", (2,0), S); label("1", (0.5,0), S); label("1", (0,3.5), E); label("2", (0,2), E); label("1", (0,0.5), E); label("1", (0,-3.5), E); label("2", (0,-2), E); label("1", (0,-0.5), E); [/asy] Compute the area of the resulting shape, drawn in red above. [i]Proposed by Nathan Xiong[/i]