Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$.

Let $r(x)$ be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials $p(x) $ and $q(x)$ with real coefficients satisfying the equation $(p(x))^3 + q(x^2) = r(x)$.

A particle is in the origin of the Cartesian plane. In each step the particle can go $1$ unit in any of the directions, left, right, up or down. Find the number of ways to go from $(0, 0)$ to $(0, 2)$ in $6$ steps. (Note: Two paths where identical set of points is traversed are considered different if the order of traversal of each point is different in both paths.)

Anett is drawing $X$-es on a $5 \times 5$ grid. For each newly drawn $X$ she gets points in the following way: She checks how many $X$-es there are in the same row (including the new one) that can be reached from the newly drawn $X$ with horizontal steps, moving only on fields that were previously marked with $X$-es. For the vertical $X$-es, she gets points the same way. a) What is the maximum number of points that she can get with drawing $25$ $X$-es? b) What is the minimum number of points that she can get with drawing $25$ $X$-es? For example, if Anett put the $X$ on the field that is marked with the circle, she would get $3$ points for the horizontal fields and $1$ point for the vertical ones. Thus, she would get $4$ points in total. [img]https://cdn.artofproblemsolving.com/attachments/5/c/662c2e4c3dea8d78e2f6397489b277daee0ad0.png[/img]

Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.

On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.

In one country there are coins of value $1,2,3,4$ or $5$. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has $100$ coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?

Each of the distinct letters in the following addition problem represents a different digit. Find the number represented by the word MEET. $ \begin{array}{cccccc}P&U&R&P&L&E\\&C&O&M&E&T\\&&M&E&E&T\\ \hline Z&Z&Z&Z&Z&Z\end{array} $

Find all real numbers $y$, for which there exists at least one real number $x$ such that $y=\frac{\sqrt{x^2+4}}{\sqrt{x^2+1}+\sqrt{x^2+9}}.$

At what index the harmonic series has a fractional part of $ 1/12? $

Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$. Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.

Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied: 1) there is at least $1$ marked point on each side, 2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$

The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all: $\textbf{(A)}\ \text{ integers } x \qquad \textbf{(B)}\ \text{ rational } x \qquad \textbf{(C)}\ \text{ real } x\qquad \textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad \textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$

For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

The vertical mast located on the tower can be seen at the greatest angle from a point on the ground whose distance from the mast axis is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast.

A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows: [list] [*]It has $3$ digits. [*]It is a palindrome. [*]The middle digit is a multiple of $4$. [*]It is divisible by $11$. [/list] What is the sum of all possible numbers of cows that the herder has? $\textbf{(A) }343\qquad\textbf{(B) }494\qquad\textbf{(C) }615\qquad\textbf{(D) }635\qquad\textbf{(E) }726$

The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.

$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$. [i]Author: Alex Zhu[/i]

A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]