Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.

We call an isosceles trapezoid $PQRS$ [i]interesting[/i], if it is inscribed in the unit square $ABCD$ in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square. Find all interesting isosceles trapezoids and their areas.

In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is [i]below[/i] another rectangle if there is a line $g$ parallel to the $x$-axis such that the first rectangle is below $g$, the second one above $g$ and both rectangles do not touch $g$. Similarly, we say that one rectangle is [i]to the right of[/i] another rectangle if there is a line $h$ parallel to the $y$-axis such that the first rectangle is to the right of $h$, the second one to the left of $h$ and both rectangles do not touch $h$. Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$

A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop $X$ to bus stop $Y \neq X$, then we shall say [i]$Y$ can be reached from $X$[/i]. We shall use the phrase [i]$Y$ comes after $X$[/i] when we wish to express that every bus stop from which the bus stop $X$ can be reached is a bus stop from which the bus stop $Y$ can be reached, and every bus stop that can be reached from $Y$ can also be reached from $X$. A visitor to this town discovers that if $X$ and $Y$ are any two different bus stops, then the two sentences [i]“$Y$ can be reached from $X$”[/i] and [i]“$Y$ comes after $X$”[/i] have exactly the same meaning in this town. Let $A$ and $B$ be two bus stops. Show that of the following two statements, exactly one is true: [b] (i)[/b] $B$ can be reached from $A;$ [b] (ii) [/b] $A$ can be reached from $B.$

Let $G$ be the set of $2\times 2$ matrices that such $$ G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid\, a,b,c,d \in \mathbb{Z}, ad-bc = 1, c \text{ is a multiple of } 3 \right\} $$ and two matrices in $G$: $$ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\;\;\; B = \begin{pmatrix} -1 & 1 \\ -3 & 2 \end{pmatrix} $$ Show that any matrix in $G$ can be written as a product $M_1M_2\cdots M_r$ such that $M_i \in \{A, A^{-1}, B, B^{-1}\}, \forall i \leq r$

Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.

There are $2005$ young people sitting around a large circular table. Of these, at most $668$ are boys. We say that a girl $G$ has a strong position, if, counting from $G$ in either direction, the number of girls is always strictly larger than the number of boys ($G$ is herself included in the count). Prove that there is always a girl in a strong position.

The set $S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}$ contains arithmetic progressions of various lengths. For instance, $\frac{1}{20}$, $\frac{1}{8}$, $\frac{1}{5}$ is such a progression of length $3$ and common difference $\frac{3}{40}$. Moreover, this is a maximal progression in $S$ since it cannot be extended to the left or the right within $S$ ($\frac{11}{40}$ and $\frac{-1}{40}$ not being members of $S$). Prove that for all $n \in \mathbb{N}$, there exists a maximal arithmetic progression of length $n$ in $S$.

Let $a_1(x), a_2(x)$, and $a_3(x)$ be three polynomials with integer coefficients such that every polynomial with integer coefficients can be written in the form $p_1(x)a_1(x) + p_2(x)a_2(x) + p_3(x)a_3(x)$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients. Show that every polynomial is of the form $p_1(x)a_1(x)^2 + p_2(x)a_2(x)^2 + p_3(x)a_3(x)^2$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients.

Consider five-dimensional Cartesian space $R^5 = \{(x_1, x_2, x_3, x_4, x_5) | x_i \in R\}$, and consider the hyperplanes with the following equations: $\bullet$ $x_i = x_j$ for every $1 \le i < j \le 5$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = -1$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 0$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 1$. Into how many regions do these hyperplanes divide $R^5$ ?

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

Consider the set $S=\{ 1,2,\ldots ,100\}$ and the family $\mathcal{P}=\{ T\subset S\mid |T|=49\}$. Each $T\in\mathcal{P}$ is labelled by an arbitrary number from $S$. Prove that there exists a subset $M$ of $S$ with $|M|=50$ such that for each $x\in M$, the set $M\backslash\{ x\}$ is not labelled by $x$.

When a student multiplied the number $66$ by the repeating decimal, $$1. \underline{a} \underline{b} \underline{a} \underline{b} … = 1.\overline{ab},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1. \underline{a} \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$- digit integer $\underline{a} \underline{b}$? $\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

In Sikinia, there are $2024$ cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all $2023$ other cities. It is known, however, that there is a positive integer $n$ with the following property: For any $n$ cities in Sikinia, there is another city which is directly connected to all these cities. Determine the largest possible value of $n$.

Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers. (a) Is this filling possible for $n=4$? (b) Is this filling possible for $n=5$?

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turning the key in every $n^{\text{th}}$ cell, and he finished his mission after $2004$ trips. How many prisoners were released?

Let $L$ be the foot of the internal bisector of $\angle B$ in an acute-angled triangle $ABC.$ The points $D$ and $E$ are the midpoints of the smaller arcs $AB$ and $BC$ respectively in the circumcircle $\omega$ of $\triangle ABC.$ Points $P$ and $Q$ are marked on the extensions of the segments $BD$ and $BE$ beyond $D$ and $E$ respectively so that $\measuredangle APB=\measuredangle CQB=90^{\circ}.$ Prove that the midpoint of $BL$ lies on the line $PQ.$

Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?

In the $xyz$ space with the origin $O$, Let $K_1$ be the surface and inner part of the sphere centered on the point $(1,\ 0,\ 0)$ with radius 2 and let $K_2$ be the surface and inner part of the sphere centered on the point $(-1,\ 0,\ 0)$ with radius 2. For three points $P,\ Q,\ R$ in the space, consider points $X,\ Y$ defined by \[\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).\] (1) When $P,\ Q$ move every cranny in $K_1,\ K_2$ respectively, find the volume of the solid generated by the whole points of the point $X$. (2) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1$. (3) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1\cup K_2$.

[b]p1.[/b] Due to the crisis, the salaries of the company's employees decreased by $1/5$. By what percentage should it be increased in order for it to reach its previous value? [b]p2.[/b] Can the sum of six different positive numbers equal their product? [b]p3.[/b] Points$ A, B, C$ and $B$ are marked on the straight line. It is known that $AC = a$ and $BP = b$. What is the distance between the midpoints of segments $AB$ and $CB$? List all possibilities. [b]p4.[/b] Find the last three digits of $625^{19} + 376^{99}$. [b]p5.[/b] Citizens of five different countries sit at the round table (there may be several representatives from one country). It is known that for any two countries (out of the given five) there will be citizens of these countries sitting next to each other. What is the smallest number of people that can sit at the table? [b]p6.[/b] Can any rectangle be cut into $1999$ pieces, from which you can form a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2+(\log y)^2=\log(x^2)+\log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$.