Given a prime number $p\ge 5$. Find the number of distinct remainders modulus $p$ of the product of three consecutive positive integers.

A number is called [i]trilegal[/i] if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?

Let $F_k$ denote the series of Fibonacci numbers shifted back by one index, so that $F_0 = 1$, $F_1 = 1,$ and $F_{k+1} = F_k +F_{k-1}$. It is known that for any fixed $n \ge 1$ there exist real constants $b_n$, $c_n$ such that the following recurrence holds for all $k \ge 1$: $$F_{n\cdot (k+1)} = b_n \cdot F_{n \cdot k} + c_n \cdot F_{n\cdot (k-1)}.$$ Prove that $|c_n| = 1$ for all $n \ge 1$.

On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$. Find the least $k$ with this property.

Find the largest prime factor of $8001$.

Let $n$ be the smallest positive integer such that the number obtained by taking $n$’s rightmost digit (decimal expansion) and moving it to be the leftmost digit is $7$ times $n$. Determine the number of digits in $n$.

For any non-empty numerical numbers $A$ and $B$, denote $$A + B = \{a + b | a \in A, b \in B\} $$ a) Determine the largest natural number not $p$ with the property: [i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B = p$ [i]and [/i] $A+B = \{0, 1, 2,..., 2012\}$ b) Determine the smallest natural number $n$ with the property: [i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B $ [i]and [/i] $A+B =\{0, 1, 2,..., 2012\}$

For positive integers $n, m$ define $f(n,m)$ as follows. Write a list of $ 2001$ numbers $a_i$, where $a_1 = m$, and $a_{k+1}$ is the residue of $a_k^2$ $mod \, n$ (for $k = 1, 2,..., 2000$). Then put $f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}$. For which $n \ge 5$ can we find m such that $2 \le m \le n/2$ and $f(m,n) > 0$?

Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.

Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.

Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$. [i]Proposed by Justin Stevens and Evan Chen[/i]

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$. [i](R. Salimov)[/i]

[b]p1.[/b] Solve the equation $x^2 - x - cos y+1.25 =0$. [b]p2.[/b] Solve the inequality: $\left| \frac{x - 2}{x - 3}\right| \le x$ [b]p3.[/b] Bilbo and Dwalin are seated at a round table of radius $R$. Bilbo places a coin of radius $r$ at the center of the table, then Dwalin places a second coin as near to the table’s center as possible without overlapping the first coin. The process continues with additional coins being placed as near as possible to the center of the table and in contact with as many coins as possible without overlap. The person who places the last coin entirely on the table (no overhang) wins the game. Assume that $R/r$ is an integer. (a) Who wins, Bilbo or Dawalin? Please justify your answer. (b) How many coins are on the table when the game ends? [b]p4.[/b] In the center of a square field is an orc. Four elf guards are on the vertices of that square. The orc can run in the field, the elves only along the sides of the square. Elves run $\$1.5$ times faster than the orc. The orc can kill one elf but cannot fight two of them at the same time. Prove that elves can keep the orc from escaping from the field. [b]p5.[/b] Nine straight roads cross the Mirkwood which is shaped like a square, with an area of $120$ square miles. Each road intersects two opposite sides of the square and divides the Mirkwood into two quadrilaterals of areas $40$ and $80$ square miles. Prove that there exists a point in the Mirkwood which is an intersection of at least three roads. PS. You should use hide for answers.

Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

For a positive integer $n$, let $6^{(n)}$ be the natural number whose decimal representation consists of $n$ digits $6$. Let us define, for all natural numbers $m$, $k$ with $1 \leq k \leq m$ \[\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} .\] Prove that for all $m, k$, $ \left[\begin{array}{ccc}m\\ k\end{array}\right] $ is a natural number whose decimal representation consists of exactly $k(m + k - 1) - 1$ digits.

A group of students play the following game: they are counting one by one from $00$ to $99$ taking turns, but instead of every number they only say one of its digits. (The numbers in order are $00$, $01$, $02$, $...$., meaning that one-digit numbers are regarded as two-digit numbers with a first digit $0$.) One way of starting the counting could be for example $0$, $1$, $2$, $0$, $4$, $0$, $6$, $7$, $8$, $9,$ $1$, $1$, $2$, $1$, $1$, $5$, $6$, $1$, $8$, $1$, $0$, $2$ etc. When they reach $99$, the counting restarts from $00$. At some point Csongor enters the room and after listening to the counting for a while, he discovers that he is able to tell what number the counting is at. How many digits has Csongor heard at least?

The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ . Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.

Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.

Maxi chose $3$ digits, and by writing down all possible permutations of these digits, he obtained $6$ distinct $3$-digit numbers. If exactly one of those numbers is a perfect square and exactly three of them are prime, find Maxi’s $3$ digits. Give all of the possibilities for the $3$ digits.

None of the $n$ (not necessarily distinct) digits selected are equal to $0$ or $7$. It turns out that every $n$-digit number formed using these digits is divisible by $7$. Prove that $n$ is divisible by $6$.

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]