Found problems: 408
Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.
A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure:
In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.
Prove that if $ n $ is a non-negative integer, then number $$
2^{n+2} + 3^{2n+1}$$
is divisible by $7$.
Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.
The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that:
a) At least one of the numbers $s(k)$ is divisible by $n$.
b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.
For $n\ne 0$, let f(n) be the largest $k$ such that $3^k$ divides $n$. If $M$ is a set of $n > 1$ integers, show that the number of possible values for $f(m-n)$, where $m, n$ belong to $M$ cannot exceed $n-1$.
For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.
Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.
Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.
Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$
is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$.
How many integers between $1$ and $100$ are octal?
(A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$
Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with
$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).
Prove that
a) $(a-1)\vdots p_i$ for some $i=1,..,n$
b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?
I. Bliznets
a) Show that the product of $5$ consecutive even integers is divisible by $15$.
b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.
Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.