Find the number of integers $n$ that satisfy $\textit{both}$ of the following conditions: [list=i] [*]$208<n<2008$, [*]$n$ has the same remainder when divided by $24$ or by $30$.[/list]

For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied: (a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$; (b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$. Determine $N(n)$ for all $n\ge 2$.

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that: i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$; ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$. Find the maximal value of $M$.

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

For which of the following integers $n$, there is at least one integer $x$ such that $x^2 \equiv -1 \pmod{n}$? $ \textbf{(A)}\ 97 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 99 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled? [i]Proposed by A. Golovanov[/i]

A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property: For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form: \[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\] Prove that for each $ m$ we have a hyper-primitive root.

Find the remainder when $A=3^3\cdot 33^{33}\cdot 333^{333}\cdot 3333^{3333}$ is divided by $100$.

Find all intengers n such that $2^n - 1$ is a multiple of 3 and $(2^n - 1)/3$ is a divisor of $4m^2 + 1$ for some intenger m.

A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals. For every vertex the number of sides issuing from it is even, except for the vertices $A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and \[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\] if $k>0$ and \[n\equiv0\pmod3\mbox{ for }k=0.\] Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?