The last digit of the number A = $7^{2011}$ is ?

It is given that there exist unique integers $m_1,\ldots, m_{100}$ such that \[0\leq m_1 < m_2 < \cdots < m_{100}\quad\text{and}\quad 2018 = \binom{m_1}1 + \binom{m_2}2 + \cdots + \binom{m_{100}}{100}.\] Find $m_1 + m_2 + \cdots + m_{100}$.

Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$

For any positive integer $n$, define the subset $S_n$ of natural numbers as follow $$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$. [i]Proposed by Yahya Motevassel[/i]

Determine all natural numbers $n$ of the form $n=[a,b]+[b,c]+[c,a]$ where $a,b,c$ are positive integers and $[u,v]$ is the least common multiple of the integers $u$ and $v$.

Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$-tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$, then $(b_1, b_2, \ldots, b_n) \in S$. Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n | a + b + c + d$ and $n | a^2 + b^2 + c^2 + d^2. $ Show that $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27

The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime. (Proposed by Gerhard Woeginger, Austria)

Let $S(k)$ denote the digit-sum of a positive integer $k$(in base $10$). Determine the smallest positive integer $n$ such that \[S(n^2 ) = S(n) - 7\]

Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there? (Le Anh Vinh)

Denote $\textrm{SL}_2 (\mathbb{Z})$ and $\textrm{SL}_3 (\mathbb{Z})$ the sets of matrices with $2$ rows and $2$ columns, respectively with $3$ rows and $3$ columns, with integer entries and their determinant equal to $1$. $\textbf{(a)}$ Let $N$ be a positive integer and let $g$ be a matrix with $3$ rows and $3$ columns, with rational entries. Suppose that for each positive divisor $M$ of $N$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $N$ and a matrix $\gamma_M \in \textrm{SL}_3 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . \] Moreover, if $q_1 = 1$, prove that $\det (g) = N$ and $g$ has the following shape: \[ g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), \] where $a_{ij}$ are all integers, $i, j \in \{ 1, 2, 3 \} .$ $\textbf{(b)}$ Provide an example of a matrix $g$ with $2$ rows and $2$ columns which satisfies the following properties: $\bullet$ For each positive divisor $M$ of $6$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $6$ and a matrix $\gamma_M \in \textrm{SL}_2 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) \] and $q_1 = 1$. $\bullet$ $g$ does not have its determinant equal to $6$ and is not of the shape \[ g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), \] where $a_{ij}$ are all positive integers, $i, j \in \{ 2, 3 \}$. [i](Radu Toma)[/i]

Let $g(n)$ be defined as follows: \[ g(1) = 0, g(2) = 1 \] and \[ g(n+2) = g(n) + g(n+1) + 1, n \geq 1. \] Prove that if $n > 5$ is a prime, then $n$ divides $g(n) \cdot (g(n) + 1).$

Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer. $$x + y = 3$$ $$3xy -z^2 = 9$$ [b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much? [b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles. [b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$ [b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer. [b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile PS. You should use hide for answers.