$ a,b,c>0 $ and $ abc=1 $. Prove that $\frac{1}{2}\ (\sqrt{a}\ +\sqrt{b}\ + \sqrt{c}\ ) +\frac{1}{1+a}\ + \frac{1}{1+b}\ + \frac{1}{1+c}\ge\ 3 $. ( The official problem is with $ abc = 1 $ but it can be proved without using it. )

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$. [i]Proposed by David Altizio[/i]

The points of intersection of $ xy \equal{} 12$ and $ x^2 \plus{} y^2 \equal{} 25$ are joined in succession. The resulting figure is: $ \textbf{(A)}\ \text{a straight line} \qquad\textbf{(B)}\ \text{an equilateral triangle} \qquad\textbf{(C)}\ \text{a parallelogram}$ $ \textbf{(D)}\ \text{a rectangle} \qquad\textbf{(E)}\ \text{a square}$

Determine the smallest positive integer n for that there are positive integers $x_1, x_2,. . . , x_n$ so that each natural number from 1001 to 2021 inclusive can be written as sum of one or more different terms $x_i$ (i = 1, 2,..., n).

In a triangle ABC, D is midpoint of BC . If $\angle ADB = 45 ^{\circ}$ and $\angle ACD = 30^{\circ}$, determine $\angle BAD.$

Prove that for any integer $a$ there exist infinitely many positive integers $n$ such that $a^{2^n}+2^n$ is not a prime. [b]proposed by S. Berlov[/b]

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

The quadrilateral $ABCD$ is inscribed in the circle $o$. Bisectors of angles $DAB$ and $ABC$ intersect at point $P$, and bisectors of angles $BCD$ and $CDA$ intersect in point $Q$. Point $M$ is the center of this arc $BC$ of the circle $o$ which does not contain points $D$ and $A$. Point $N$ is the center of the arc $DA$ of the circle $o$, which does not contain points $B$ and $C$. Prove that the points $P$ and $Q$ lie on the line perpendicular to $MN$.

[b]p1.[/b] In the correct identity $(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)$ two numbers were replaced with dots. What were these numbers? [b]p2.[/b] A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? [img]https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png[/img] [b]p3.[/b] Find the angle between the hour and minute hands at $7$ hours $38$ minutes. [b]p4.[/b] The lottery game is played as follows. A random number from $1$ to $1000$ is selected. If it is divisible by $2$, they pay a ruble, if it is divisible by $10$ - two rubles, by $12$ - four rubles, by $20$ - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options. [b]p5.[/b]The sum of the digits of a positive integer $x$ is equal to $n$. Prove that between $x$ and $10x$ you can find an integer whose sum of digits is $ n + 5$. [b]p6.[/b] $9$ people took part in the campaign, which lasted $12$ days. There were $3$ people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all $12$ days they were able to appoint three people on duty, taking into account this requirement. Could this be so? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$

Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\] [i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$. Let $P$ a point on $\Gamma$. Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$. Prove that when $P$ varies on $\Gamma$, the line $AB$ is tangent to a fixed circunference.