Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Let $n \geq 2$ be natural number. Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.

In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval? $\textbf{(A)}\ \frac{2\sqrt{2} \plus{}1}{4} \qquad\textbf{(B)}\ \frac{\sqrt{2} \plus{}1}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad\textbf{(D)}\ \sqrt{2} \minus{}1 \qquad\textbf{(E)}\ \frac{\sqrt{2} \minus{}1}{2}$

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

[b]p1.[/b] Two proper ordinary fractions are given. The first has a numerator that is $5$ less than the denominator, and the second has a numerator that is $1998$ less than the denominator. Can their sum have a numerator greater than its denominator? [b]p2.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in $365$ days on the next New Year's Eve? [b]p3.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p4.[/b] In the quadrilateral $ABCD$, the extensions of opposite sides $AB$ and $CD$ intersect at an angle of $20^o$; the extensions of opposite sides $BC$ and $AD$ also intersect at an angle of $20^o$. Prove that two angles in this quadrilateral are equal and the other two differ by $40^o$. [b]p5.[/b] Given two positive integers $a$ and $b$. Prove that $a^ab^b\ge a^ab^a.$ [b]p6.[/b] The square is divided by straight lines into $25$ rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $ 17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] In expression $$(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q)$$ opened the brackets. How many members will there be? How many of them will be preceded by a minus sign? [b]p9.[/b] In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of $100$ people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it? [b]p10.[/b] Vasya and Petya play such a game on a $10 \times 10 board$. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right? [img]https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

You are given $n \geq 3$ distinct real numbers. Prove that one can choose either $3$ numbers with positive sum, or $2$ numbers with negative sum. [i]Proposed by Mykhailo Shtandenko[/i]

On the table there are $2016$ coins. Two players play the following game making alternating moves. In one move it is allowed to take $1, 2$ or $3$ coins. The player who takes the last coin wins. Which player has a winning strategy?

What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $

A regular hexagon is inscribed in a circle of radius $ 10$ inches. Its area is: $ \textbf{(A)}\ 150\sqrt{3} \text{ sq. in.} \qquad \textbf{(B)}\ \text{150 sq. in.} \qquad \textbf{(C)}\ 25\sqrt{3} \text{ sq. in.} \qquad \textbf{(D)}\ \text{600 sq. in.} \qquad \textbf{(E)}\ 300\sqrt{3} \text{ sq. in.}$

A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product. [i]Proposed by Péter Nagy Tibor[/i]

Let $m$ be an integer and $\mathbb{Z}_m$ the set of integer modulo $m$. An equivalence relation is defined in $\mathbb{Z}_m$ given by, $x \sim y$ if there exists a natural $t$ such that $y \equiv 2^tx \, (\bmod m)$ . Find al values of $m$ such that the number of equivalent classes is even.