Found problems: 1362
1995 APMO, 2
Let $a_1$, $a_2$, $\ldots$, $a_n$ be a sequence of integers with values between 2 and 1995 such that:
(i) Any two of the $a_i$'s are relatively prime,
(ii) Each $a_i$ is either a prime or a product of primes.
Determine the smallest possible values of $n$ to make sure that the sequence will contain a prime number.
2001 Balkan MO, 1
Let $a,b,n$ be positive integers such that $2^n - 1 =ab$. Let $k \in \mathbb N$ such that $ab+a-b-1 \equiv 0 \pmod {2^k}$ and $ab+a-b-1 \neq 0 \pmod {2^{k+1}}$. Prove that $k$ is even.
2010 Contests, 1
For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that
\[n = S(n) + U(n)^2.\]
1985 IMO Longlists, 68
Show that the sequence $\{a_n\}_{n\geq1}$ defined by $a_n = [n \sqrt 2]$ contains an infinite number of integer powers of $2$. ($[x]$ is the integer part of $x$.)
1984 IMO Longlists, 30
Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.
2006 MOP Homework, 3
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
$\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.
1990 Polish MO Finals, 3
Prove that for all integers $n > 2$,
\[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]
2014 Contests, 2
Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.
2003 Indonesia MO, 7
Let $k,m,n$ be positive integers such that $k > n > 1$ and $(k,n) = 1$. If $k-n | k^m - n^{m-1}$, prove that $k \le 2n - 1$.
2006 Italy TST, 2
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$.
a) Find all $n$ such that $A_{n}\neq \emptyset$
b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero.
c) Is there $n$ such that $|{A_{n}}| = 130$?
2007 All-Russian Olympiad, 8
Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment?
[i]A. Golovanov[/i]
1988 Federal Competition For Advanced Students, P2, 3
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.
1994 USAMO, 1
Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.
2004 India IMO Training Camp, 2
Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$
2010 Contests, 1
Let $n$ be a positive integer. Let $T_n$ be a set of positive integers such that:
\[{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}\]
Find all $n$ for which there don't exist two distinct positive integers $a, b \in T_n$ such that $a\equiv b \pmod{110}$
2013 Dutch IMO TST, 3
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
2014 Indonesia MO, 2
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.
2006 Rioplatense Mathematical Olympiad, Level 3, 1
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$.
(b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]
1995 IberoAmerican, 1
Find all the possible values of the sum of the digits of all the perfect squares.
[Commented by djimenez]
[b]Comment: [/b]I would rewrite it as follows:
Let $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)$ is the sum of all the digits of the number $n^2$. Find the image of $f$ (where, by image it is understood the set of all $x$ such that exists an $n$ with $f(n)=x$).
2007 Korea National Olympiad, 3
Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set.
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
2019 Latvia Baltic Way TST, 14
Let $m$ be a positive integer and $p$ be a prime, such that $m^2 - 2$ is divisible by $p$. Suppose that there exists positive integer $a$ such that $a^2+m-2$ is divisible by $p$. Prove that there exists positive integer $b$ such that $b^2- m -2$ is divisible by $p$.
2008 Federal Competition For Advanced Students, Part 2, 2
(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$?
(b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?
2012 Morocco TST, 1
Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that :
$$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$
2010 Contests, 3
Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.