Olympiad problems

1990 Irish Math Olympiad, 2

A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$ a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$ for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks

1986 IMO Longlists, 55

Tags: number theory unsolved , number theory

Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$

1983 IMO Longlists, 18

Tags: quadratics , algebra , number theory unsolved , number theory

Let $b \geq 2$ be a positive integer. (a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits. (b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.

2007 Estonia National Olympiad, 4

Tags: limit , number theory unsolved , number theory

Find all pairs $ (m, n)$ of positive integers such that $ m^n \minus{} n^m \equal{} 3$.

2007 Pre-Preparation Course Examination, 13

Tags: number theory unsolved , number theory

Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$ \[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]

2010 Postal Coaching, 4

Tags: number theory , relatively prime , number theory unsolved

For each $n\in \mathbb{N}$, let $S(n)$ be the sum of all numbers in the set $\{ 1, 2, 3, \cdots , n \}$ which are relatively prime to $n$. $(a)$ Show that $2 \cdot S(n)$ is not a perfect square for any $n$. $(b)$ Given positive integers $m, n$, with odd $n$, show that the equation $2 \cdot S(x) = y^n$ has at least one solution $(x, y)$ among positive integers such that $m|x$.

2007 Germany Team Selection Test, 2

Tags: LaTeX , number theory unsolved , number theory

Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]

2010 CHKMO, 4

Tags: quadratics , number theory unsolved , number theory

Find all non-negative integers $ m$ and $ n$ that satisfy the equation: \[ 107^{56}(m^2\minus{}1)\minus{}2m\plus{}5\equal{}3\binom{113^{114}}{n}\] (If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}\equal{}\frac{n}{r!(n\minus{}r)!}$ and $ \binom{n}{r}\equal{}0$ if $ r>n$.)

Bangladesh Mathematical Olympiad 2020 Final, #12

Tags: number theory , number theory unsolved

$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

2011 China Western Mathematical Olympiad, 4

Tags: number theory unsolved , number theory

Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$

2012 China Girls Math Olympiad, 3

Tags: number theory , greatest common divisor , modular arithmetic , relatively prime , number theory unsolved

Find all pairs $(a,b)$ of integers satisfying: there exists an integer $d \ge 2$ such that $a^n + b^n +1$ is divisible by $d$ for all positive integers $n$.

2004 China Team Selection Test, 1

Tags: number theory , prime factorization , number theory unsolved

Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

2004 Tournament Of Towns, 1

Tags: number theory unsolved , number theory

Is it possible to arrange numbers from 1 to 2004 in some order so that the sum of any 10 consecutive numbers is divisble by 10?

2007 South East Mathematical Olympiad, 3

Tags: ratio , number theory , prime numbers , number theory unsolved

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

2008 China Girls Math Olympiad, 8

Tags: search , number theory unsolved , number theory

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.

2005 China Team Selection Test, 2

Tags: number theory unsolved , number theory

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

2014 Spain Mathematical Olympiad, 1

Tags: geometry , number theory unsolved , number theory

Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.

2022 South Africa National Olympiad, 3

Tags: number theory , GCD , number theory unsolved

Let a, b, and c be nonzero integers. Show that there exists an integer k such that $$gcd\left(a+kb, c\right) = gcd\left(a, b, c\right)$$

1954 Putnam, B6

Tags: algorithm , number theory unsolved , number theory

Let $ x \in \mathbb{Q}^+$. Prove that there exits $\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}$ and pairwe distinct such that \[x= \sum_{i=1}^{k} \frac{1}{\alpha_i}\]

2000 Romania Team Selection Test, 1

Tags: number theory , greatest common divisor , number theory unsolved

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

2011 Estonia Team Selection Test, 5

Tags: number theory unsolved , number theory

Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.

2011 Czech-Polish-Slovak Match, 3

Tags: number theory , prime numbers , relatively prime , number theory unsolved

Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.

2006 Hungary-Israel Binational, 1

Tags: modular arithmetic , induction , number theory unsolved , number theory

If natural numbers $ x$, $ y$, $ p$, $ n$, $ k$ with $ n > 1$ odd and $ p$ an odd prime satisfy $ x^n \plus{} y^n \equal{} p^k$, prove that $ n$ is a power of $ p$.

2005 Morocco TST, 3

Tags: quadratics , calculus , integration , number theory unsolved , number theory

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

2007 All-Russian Olympiad, 8

Tags: number theory unsolved , number theory

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]