Found problems: 15460
1983 IMO Shortlist, 10
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
\[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$
2014 VTRMC, Problem 3
Find the least positive integer $n$ such that $2^{2014}$ divides $19^n-1$.
2012 Indonesia TST, 4
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
2010 Serbia National Math Olympiad, 3
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\]
[i]Proposed by Milos Milosavljevic[/i]
2020 Denmark MO - Mohr Contest, 3
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.
2015 Hanoi Open Mathematics Competitions, 2
The last digit of number $2017^{2017} - 2013^{2015}$ is
(A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.
2006 All-Russian Olympiad, 2
Show that there exist four integers $a$, $b$, $c$, $d$ whose absolute values are all $>1000000$ and which satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}$.
2009 Purple Comet Problems, 18
On triangle $ABC$ let $D$ be the point on $AB$ so that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ so that $AE$ bisects angle $BAC.$ Let $G$ be the intersection of $AE$ and $CD,$ and let point $F$ be the intersection of side $AC$ and the ray $BG.$ If $AB$ has length $28,$ $AC$ has length $14,$ and $CD$ has length $10,$ then the length of $CF$ can be written as $\tfrac{k-m\sqrt{p}}{n}$ where $k, m, n,$ and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any
prime. Find $k - m + n + p.$
2022 Nordic, 3
Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$. Britta wins if
$(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$
otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.
2013 India IMO Training Camp, 1
For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.
1997 Vietnam National Olympiad, 2
Prove that for evey positive integer n, there exits a positive integer k such that $ 2^n | 19^k \minus{} 97$
2005 iTest, 31
Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.
2003 May Olympiad, 4
Celia chooses a number $n$ and writes the list of natural numbers from $1$ to $n$: $1, 2, 3, 4, ..., n-1, n.$ At each step, it changes the list: it copies the first number to the end and deletes the first two. After $n-1$ steps a single number will be written.
For example, for $n=6$ the five steps are: $$ 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5$$
and the number $5$ is written.
Celia chose a number $n$ between $1000$ and $3000$ and after $n-1$ steps the number $1$ was written.
Determine all the values of $n$ that Celia could have chosen.
Justify why those values work, and the others do not.
2009 India IMO Training Camp, 8
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.
2010 Contests, 1
a) Factorize $xy - x - y + 1$.
b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.
2006 AIME Problems, 14
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
2019 Regional Olympiad of Mexico West, 5
Prove that for every integer $n > 1$ there exist integers $x$ and $y$ such that $$\frac{1}{n}=\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+...+\frac{1}{y(y+1)}.$$
1978 IMO Shortlist, 17
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
JOM 2013, 2.
Find all positive integers $a\in \{1,2,3,4\}$ such that if $b=2a$, then there exist infinitely many positive integers $n$ such that $$\underbrace{aa\dots aa}_\textrm{$2n$}-\underbrace{bb\dots bb}_\textrm{$n$}$$ is a perfect square.
2020/2021 Tournament of Towns, P1
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2019 CMIMC, 4
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \[\gcd(n,2^n - 1) = 3.\]
2021 Greece JBMO TST, 3
Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.
2011 All-Russian Olympiad, 4
Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers?
2021 Peru IMO TST, P3
For any positive integer $n$, we define
$$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$
Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.
2012 Rioplatense Mathematical Olympiad, Level 3, 5
Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.