Found problems: 15460
2019 Kazakhstan National Olympiad, 3
Let $p$ be a prime number of the form $4k+1$ and $\frac{m}{n}$ is an irreducible fraction such that
$$\sum_{a=2}^{p-2} \frac{1}{a^{(p-1)/2}+a^{(p+1)/2}}=\frac{m}{n}.$$
Prove that $p|m+n$.
(Fixed, thanks Pavel)
2007 China Team Selection Test, 3
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.
2018 SG Originals, Q5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
2024 Indonesia TST, N
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
2008 Hanoi Open Mathematics Competitions, 1
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
2011 Ukraine Team Selection Test, 3
Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.
2000 IMO Shortlist, 6
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
1983 All Soviet Union Mathematical Olympiad, 350
Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$.Is it possible to admit that the initial numbers were
a) $\{2, 2, 2\}$?
b) $\{3, 3, 3\}$?
2011 Portugal MO, 1
A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?
2022 Junior Balkan Team Selection Tests - Moldova, 4
Rational number $\frac{m}{n}$ admits representation
$$\frac{m}{n} = 1+ \frac12+\frac13 + ...+ \frac{1}{p-1}$$
where p $(p > 2)$ is a prime number. Show that the number $m$ is divisible by $p$.
2004 China Western Mathematical Olympiad, 4
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$).
Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.
2006 Australia National Olympiad, 1
Find all positive integers $m$ and $n$ such that $1 + 5 \cdot 2^m = n^2$.
2024 Mozambique National Olympiad, P1
Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils and $216$ erasers were distributed. This distribution was made so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. In this case, how many notebooks, pencils and erasers did each family receive?
2013 NZMOC Camp Selection Problems, 11
Show that we cannot find $171$ binary sequences (sequences of $0$’s and $1$’s), each of length $12$ such that any two of them differ in at least four positions.
2019 IMO Shortlist, N3
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
1999 Nordic, 3
The infinite integer plane $Z\times Z = Z^2$ consists of all number pairs $(x, y)$, where $x$ and $y$ are integers. Let $a$ and $b$ be non-negative integers. We call any move from a point $(x, y)$ to any of the points $(x\pm a, y \pm b)$ or $(x \pm b, y \pm a) $ a $(a, b)$-knight move. Determine all numbers $a$ and $b$, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only $(a, b)$-knight moves.
2004 Abels Math Contest (Norwegian MO), 1b
Let $a_1,a_2,a_3,...$ be a strictly increasing sequence of positive integers. A number $a_n$ in the sequence is said to be [i]lucky [/i] if it is the sum of several (not necessarily distinct) smaller terms of the sequence, and [i]unlucky [/i]otherwise. (For example, in the sequence $4,6,14,15,25,...$ numbers $4,6,15$ are [i]unlucky[/i], while $14 = 4+4+6$ and $25 = 4+6+15$ are [i]lucky[/i].) Prove that there are only finitely many [i]unlucky [/i]numbers in the sequence.
2010 Malaysia National Olympiad, 6
A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.
2004 IberoAmerican, 3
Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$
1999 Baltic Way, 19
Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.
2014 Romania Team Selection Test, 4
Let $f$ be the function of the set of positive integers into itself, defi ned by $f(1) = 1$,
$f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the
number of positive odd integers m such that $f(m) = n$ is equal to the number of positive
integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$.
[color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]
1972 IMO Longlists, 1
Find all integer solutions of the equation
\[1 + x + x^2 + x^3 + x^4 = y^4.\]
1998 Korea Junior Math Olympiad, 1
Show that there exist no integer solutions $(x, y, z)$ to the equation
$$x^3+2y^3+4z^3=9$$
2014 Dutch BxMO/EGMO TST, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2008 Mongolia Team Selection Test, 2
Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m \plus{} c^m d^n$ is composite number