There are only two letters in the Mumu tribe alphabet: M and $U$. The word in the Mumu language is any sequence of letters $M$ and $U$, in which next to each letter $M$ there is a letter $U$ (for example, $UUU$ and $UMMUM$ are words and $MMU$ is not). Let $f(m,u)$ denote the number of words in the Mumu language which have $m$ times the letter $M$ and $u$ times the letter $U$. Prove that $f (m, u) - f (2u - m + 1, u) = f (m, u - 1) - f (2u - m + 1, u - 1)$ for any $u \ge 2,3 \le m \le 2u$.

In convex quadrilateral $ABCD$, $\angle A \cong \angle C$, $AB = CD = 180$, and $AD \neq BC$. The perimeter of $ABCD$ is 640. Find $\lfloor 1000 \cos A \rfloor$. (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

Reflect each of the shapes $A,B$ over some lines $l_A,l_B$ respectively and rotate the shape $C$ such that a $4 \times 4$ square is obtained. Identify the lines $l_A,l_B$ and the center of the rotation, and also draw the transformed versions of $A,B$ and $C$ under these operations. [img]https://s8.uupload.ir/files/photo14908574605_i39w.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Prove that for all positive integers $n$ there exist $n$ distinct, positive rational numbers with sum of their squares equal to $n$. Proposed by Daniyar Aubekerov

In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations $1,2,2^2,2^3,..$ Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for $2$ Bens in two ways : $2$ one Ben notes or $1$ two Ben note. For $5$ Ben one can given $1$ one Ben and $1$ four Ben note or $1$ Ben note and $2$ two Ben notes. Using $5$ one Ben notes or $3$ one Ben notes and $1$ two Ben notes for a $5$ Ben transaction is prohibited. Find the number of ways in which one can give a change $100$ Bens following the rules of the Government.

A finite sequence of positive integers $ m_i$ for $ i\equal{}1,2,...,2006$ are defined so that $ m_1\equal{}1$ and $ m_i\equal{}10m_{i\minus{}1} \plus{}1$ for $ i>1$. How many of these integers are divisible by $ 37$?

Convex pentagon $PQRST$ has $PQ = T P = 5$, $QR = RS = ST = 6$, and $\angle QRS = \angle RST = 90^o$. Given that points $U$ and $V$ exist such that $RU =UV = VS = 2$, find the area of pentagon $PQUVT$ . [i]Proposed by Kira Tang[/i]

The number of nonnegative solutions to the equation $2x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}=3$ is________.

Circles $c_1$ and $c_2$ are tangent internally to circle $c$ at points $A$ and $B$ , respectively, as seen in the figure. The inner tangent common to $c_1$ and $c_2$ touches these circles in $P$ and $Q$ , respectively. Show that the $AP$ and $BQ$ lines intersect the circle $c$ at diametrically opposite points. [img]https://cdn.artofproblemsolving.com/attachments/0/a/9490a4d7ba2038e490a858b14ba21d07377c5d.gif[/img]

We are given triangle $ABC$. Two lines, symmetric with $AC$, relative to lines $AB$ and $BC$ are drawn, and meet at $K$ . Prove that the line $BK$ passes through the centre of the circumscribed circle of triangle $ABC$. (V.Y. Protasov)

Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.

A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.

Let $\triangle ABC$ be a triangle with $\angle BAC=60^\circ$, $M$ be a point in its interior and $A',\, B',\, C'$ be the orthogonal projections of $M$ on the sides $BC,\, CA,\, AB$. Determine the locus of $M$ when the sum $A'B+B'C+C'A$ is constant. [i]Horea Călin Pop[/i]

What is the greatest common factor of $12345678987654321$ and $12345654321$? [i]Proposed by Evan Chen[/i]

Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.

Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$. Show that \[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\] [i]Ioan Tomescu[/i]

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

Some $ m$ squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest $ m$ for which it is possible to mark all squares after several such operations.

Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\] $(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$. $(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.

Let $S$ be the number of bijective functions $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$ such that $f((m+n)\pmod{17})$ is divisible by $17$ if and only if $f(m)+f(n)$ is divisible by $17$. Compute the largest positive integer $n$ such that $2^n$ divides $S$.

In a game of [i]Chomp[/i], two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats'') all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.) [asy] defaultpen(linewidth(0.7)); fill((2,2)--(2,3)--(3,3)--(3,2)--cycle, mediumgray); int[] array={5, 5, 5, 4, 2, 2, 2, 0}; pair[] ex = {(2,3), (2,4), (3,2), (3,3)}; draw((3,5)--(7,5)^^(4,4)--(7,4)^^(4,3)--(7,3), linetype("3 3")); draw((4,4)--(4,5)^^(5,2)--(5,5)^^(6,2)--(6,5)^^(7,2)--(7,5), linetype("3 3")); int i, j; for(i=0; i<7; i=i+1) { for(j=0; j<array[i]; j=j+1) { draw((i,j+1)--(i,j)--(i+1,j)); } draw((i,array[i])--(i+1,array[i])); if(array[i]>array[i+1]) { draw((i+1,array[i])--(i+1,array[i+1])); }} for(i=0; i<4; i=i+1) { draw(ex[i]--(ex[i].x+1, ex[i].y+1), linewidth(1.2)); draw((ex[i].x+1, ex[i].y)--(ex[i].x, ex[i].y+1), linewidth(1.2)); }[/asy] The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.

In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur: a) $$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$ b) $$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$ c) $$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$ where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron. [i]Proposed by Marius Olteanu[/i]

Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.