Found problems: 15460
2016 Danube Mathematical Olympiad, 2
Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i]
2016 Romania National Olympiad, 4
For $n \in N^*$ we will say that the non-negative integers $x_1, x_2, ... , x_n$ have property $(P)$ if
$$x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 + ...+ nx_n.$$
a) Show that for every $n \in N^*$ there exists $n$ positive integers with property $(P)$.
b) Find all integers $n \ge 2$ so that there exists $n$ positive integers $x_1, x_2, ... , x_n$ with $x_1< x_2<x_3< ... <x_n$, having property $(P)$.
2005 Georgia Team Selection Test, 6
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:
1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;
2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$;
Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
2010 Thailand Mathematical Olympiad, 10
Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.
2009 China Team Selection Test, 2
Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.
1997 Belarusian National Olympiad, 2
A sequence $(a_n)_{-\infty}^{-\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} +...+ a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0,a_1,...,a_{97}$.
2024 Lusophon Mathematical Olympiad, 2
For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\} $, let $P_S$ be the product of all differences between two of the elements, namely
$$P_S=(a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$
Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
2009 Germany Team Selection Test, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2004 Bulgaria Team Selection Test, 3
Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.
2018 Thailand TST, 3
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
2012 Saint Petersburg Mathematical Olympiad, 2
Natural $a,b,c$ are $>100$ and $(a,b,c)=1$. $c|a+b,a|b+c$ Find minimal $b$
2008 Iran MO (3rd Round), 3
a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{\minus{}1,0,1\}$.
b) Does there exist a multiple of $ x^2\minus{}3x\plus{}1$ such that all of its coefficient are in the set $ \{\minus{}1,0,1\}$
1963 Putnam, A2
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$
for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.
2005 Indonesia MO, 2
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)\equal{}n^2\minus{}2005$.
2019 Thailand TST, 1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2014 AIME Problems, 14
In $\triangle ABC$, $AB=10$, $\angle A=30^\circ$, and $\angle C=45^\circ$. Let $H,D$, and $M$ be points on line $\overline{BC}$ such that $\overline{AH}\perp\overline{BC}$, $\angle BAD=\angle CAD$, and $BM=CM$. Point $N$ is the midpoint of segment $\overline{HM}$, and point $P$ is on ray $AD$ such that $\overline{PN}\perp\overline{BC}$. Then $AP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2022 239 Open Mathematical Olympiad, 4
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.
2017 South East Mathematical Olympiad, 2
Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$
$(1)$ Find the number of $n$-var Boole function;
$(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$,
Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that
$\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$
2007 France Team Selection Test, 1
For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$.
Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$:
\[a_{n+1}=a_{n}+a'_{n}. \]
Can $a_{7}$ be prime?
2012 IMO Shortlist, N2
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
2021 Brazil Team Selection Test, 6
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
2004 Harvard-MIT Mathematics Tournament, 1
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
2004 Alexandru Myller, 1
Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $
[i]Mihai Piticari[/i]
2025 EGMO, 1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$ for all $1 \leqslant i \leqslant m-1$
\\[i]Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.[/i]
\\ [i]Proposed by Paulius Aleknavičius, Lithuania[/i]
2006 MOP Homework, 5
Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that
$$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$