$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function $b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$

Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.

A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Find all natural number which can be expressed in $ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$ where $ a,b,c\in \mathbb{N}$ satisfy $ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$

Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds: $$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$ For which largest $k$ do there exist $k$-similar numbers? [i]Proposed by Oleksiy Masalitin[/i]

In triangle ABC,X,Y are on the angle bisector of ∠BAC and ∠ABX=∠ACY.BX intersects CY at P and circles (BYP) and (CXP) intersect at Q different from P. Prove that A,P,Q are on a line.

A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$. $ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$ $ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

Compute $$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$?

We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.

In a school there $b$ teachers and $c$ students. Suppose that a) each teacher teaches exactly $k$ students, and b)for any two (distinct) students , exactly $h$ teachers teach both of them. Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.

Let $c$ be the number of ways to choose three vertices of an $6$-dimensional cube that form an equilateral triangle. Find the remainder when $c$ is divided by $2007$.

The continent of Trianglandia is an equilateral triangle of side length $9$, divided into $81$ triangular countries of side length $1$. Each country has the resources to choose at most $1$ of its $3$ sides and build a “wall” covering that entire side. However, since all the countries are at war, no two countries are willing to have their walls touch, even at a corner. What is the maximum number of walls that can be built in Trianglandia?

Let $I$ be the incenter of a triangle $ABC$; $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the orthocenters of the triangles $BIC$, $AIC$, $AIB$; $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of $BC$, $CA$, $AB$, and $S_{a}$, $S_{b}$, $S_{c}$ be the midpoints of $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$. Prove that $M_{a}S_{a}$, $M_{b}S_{b}$, $M_{c}S_{c}$ concur. Proposed by: S Kuznetsov

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.

Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].