Found problems: 15460
1998 USAMTS Problems, 2
For a nonzero integer $i$, the exponent of $2$ in the prime factorization of $i$ is called $ord_2 (i)$. For example, $ord_2(9)=0$ since $9$ is odd, and $ord_2(28)=2$ since $28=2^2\times7$. The numbers $3^n-1$ for $n=1,2,3,\ldots$ are all even so $ord_2(3^n-1)>0$ for $n>0$.
a) For which positive integers $n$ is $ord_2(3^n-1) = 1$?
b) For which positive integers $n$ is $ord_2(3^n-1) = 2$?
c) For which positive integers $n$ is $ord_2(3^n-1) = 3$?
Prove your answers.
1983 All Soviet Union Mathematical Olympiad, 370
The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.
2014 May Olympiad, 3
Ana and Luca play the following game. Ana writes a list of $n$ different integer numbers. Luca wins if he can choose four different numbers, $a, b, c$ and $d$, so that the number $a+b-(c+d)$ is multiple of $20$. Determine the minimum value of $n$ for which, whatever Ana's list, Luca can win.
2008 Kazakhstan National Olympiad, 1
Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation:
$ (2008\minus{}a_1)^2\plus{}(a_1\minus{}a_2)^2\plus{}\dots\plus{}(a_{2007}\minus{}a_{2008})^2\plus{}a_{2008}^2\equal{}2008$
2007 China National Olympiad, 3
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.
2023 China Northern MO, 6
A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.
2009 Croatia Team Selection Test, 4
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
2022 Bulgarian Autumn Math Competition, Problem 11.3
Find the largest positive integer $n$ of the form $n=p^{2\alpha}q^{2\beta}r^{2\gamma}$ for primes $p<q, r$ and positive integers $\alpha, \beta, \gamma$, such that $|r-pq|=1$ and $p^{2\alpha}-1, q^{2\beta}-1, r^{2\gamma}-1$ all divide $n$.
2023 Chile Junior Math Olympiad, 2
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
2003 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.
(H. Nestra)
2019 Brazil Team Selection Test, 2
Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$
For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.
2006 All-Russian Olympiad Regional Round, 11.7
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.
VMEO III 2006 Shortlist, N8
For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$.
Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.
2023 SG Originals, Q6
Let $p$ be a prime such that $\frac{p-1}{2}$ is also prime. A pair of integers $(x, y)$ with $1\le x, y \le p-1$ is called a [i]commuter[/i] if at least one of $x^y -y^x$ or $x^y +y^x$ is divisible by $p$. Show that the number of commuters is at most $4.2p\sqrt{p}$.
2023 Purple Comet Problems, 8
Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3 + 18 + 3$, $18 + 3 + 3$, and $3 + 6 + 3 + 9 + 3$, but not $18 + 6$ or $24$.
2022 Assara - South Russian Girl's MO, 1
Given three natural numbers $a$, $b$ and $c$. It turned out that they are coprime together. And their least common multiple and their product are perfect squares. Prove that $a$, $b$ and $c$ are perfect squares.
1986 IMO Longlists, 62
Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number.
2019 Austrian Junior Regional Competition, 1
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$
(Walther Janous)
1990 China National Olympiad, 4
Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution:
\[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]
2024 IFYM, Sozopol, 4
At the wedding of two Bulgarian nationals in mathematics, every guest who gave a positive integer \(n\), not yet given by another guest, which divides \(3^n-3\) but does not divide \(2^n-2\), received a prize. If there were an infinite number of guests, would the newlyweds theoretically need an infinite number of gifts?
Russian TST 2016, P1
Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]
2018 Brazil Team Selection Test, 4
Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.
2007 Silk Road, 1
On the board are written $2 , 3 , 5 ,... , 2003$ , that is, all the prime numbers of the interval $[2,2007]$ . The operation of [i]simplification [/i] is the replacement of two numbers $a , b$ by a maximal prime number not exceeding $\sqrt{a^2-a b+b^2}$ . First, the student erases the number $q, 2<q<2003$, then applies the [i]simplification [/i] operation to the remaining numbers until one number remains. Find the maximum possible and minimum possible values of the number obtained in the end. How do these values depend on the number $q$?
2007 Estonia National Olympiad, 2
Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.
2005 Junior Balkan Team Selection Tests - Moldova, 2
Prove that:
a) there are infinitely many natural numbers of the form 3p + 1, p is positive integer , which can be
represented as the difference of 2 cubes of positive integers;
b) there are infinitely many natural numbers of the form 5q + 1, q is positive integer , which can be
represented as the difference of two cubes of positive integers.