Olympiad problems

2014 China National Olympiad, 1

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.

2009 Korea National Olympiad, 3

Tags: number theory , Diophantine equation , number theory proposed

Let $n$ be a positive integer. Suppose that the diophantine equation \[z^n = 8 x^{2009} + 23 y^{2009} \] uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.

2006 QEDMO 3rd, 2

Tags: number theory , number theory proposed

Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$. Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$. An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either: If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.

2011 Preliminary Round - Switzerland, 2

Tags: function , number theory proposed , number theory

Find all positive integers $n$ such that $n^3$ is the product of all divisors of $n$.

2005 Indonesia MO, 5

Tags: function , floor function , number theory proposed , number theory

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

2005 Danube Mathematical Olympiad, 1

Tags: trigonometry , number theory proposed , number theory

Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

2024 All-Russian Olympiad, 2

Tags: number theory , number theory proposed

A positive integer has exactly $50$ divisors. Is it possible that no difference of two different divisors is divisible by $100$? [i]Proposed by A. Chironov[/i]

1998 Federal Competition For Advanced Students, Part 2, 2

Tags: algebra , polynomial , greatest common divisor , number theory proposed , number theory

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

1994 Baltic Way, 6

Tags: modular arithmetic , number theory proposed , number theory

Prove that any irreducible fraction $p/q$, where $p$ and $q$ are positive integers and $q$ is odd, is equal to a fraction $\frac{n}{2^k-1}$ for some positive integers $n$ and $k$.

2011 Middle European Mathematical Olympiad, 7

Tags: modular arithmetic , number theory proposed , number theory , case view

Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.

2004 Bulgaria Team Selection Test, 1

Tags: algebra , polynomial , induction , number theory proposed , number theory

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

2014 Contests, 1

Tags: number theory proposed , number theory

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

2014 Costa Rica - Final Round, 2

Tags: number theory , number theory unsolved , number theory proposed

Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.

2014 Brazil National Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

2016 APMC, 8

Tags: number theory , number theory proposed

Let be $n\geq 3$ fixed positive integer.Let be real numbers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ such that satisfied this conditions: [b]$i)$[/b] $ $ $a_n\geq a_{n-1}$ and $b_n\geq b_{n-1}$ [b]$ii)$[/b] $ $ $0<a_1\leq b_1\leq a_2\leq b_2\leq ... \leq a_{n-1}\leq b_{n-1}$ [b]$iii)$[/b] $ $ $a_1+a_2+...+a_n=b_1+b_2+...+b_n$ [b]$iv)$[/b] $ $ $a_{1}\cdot a_2\cdot ...\cdot a_n=b_1\cdot b_2\cdot ...\cdot b_n$ Show that $a_i=b_i$ for all $i=1,2,...,n$

2025 Bundeswettbewerb Mathematik, 2

Tags: number theory , number theory proposed , factorial , Digits

For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

2003 India IMO Training Camp, 2

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

Find all triples $(a,b,c)$ of positive integers such that (i) $a \leq b \leq c$; (ii) $\text{gcd}(a,b,c)=1$; and (iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.

2007 Indonesia MO, 8

Tags: number theory proposed , number theory

Let $ m$ and $ n$ be two positive integers. If there are infinitely many integers $ k$ such that $ k^2\plus{}2kn\plus{}m^2$ is a perfect square, prove that $ m\equal{}n$.

2014 Contests, 2

Tags: number theory proposed , number theory

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

2015 JBMO Shortlist, NT5

Tags: number theory , prime numbers , modular arithmetic , number theory proposed , number theory unsolved

Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$

1996 All-Russian Olympiad, 1

Tags: geometry , 3D geometry , number theory proposed , number theory

Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be? [i]A. Golovanov[/i]

2021 Bundeswettbewerb Mathematik, 2

Tags: number theory , number theory proposed , reciprocal sum

The fraction $\frac{3}{10}$ can be written as a sum of two reciprocals in exactly two ways: \[\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}\] a) In how many ways can $\frac{3}{2021}$ be written as a sum of two reciprocals? b) Is there a positive integer $n$ not divisible by $3$ with the property that $\frac{3}{n}$ can be written as a sum of two reciprocals in exactly $2021$ ways?

2011 CentroAmerican, 3

Tags: inequalities , number theory proposed , number theory

A [i]slip[/i] on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$ Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.

2012 India IMO Training Camp, 2

Tags: pigeonhole principle , number theory proposed , number theory

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

2004 Polish MO Finals, 6

Tags: number theory proposed , number theory

An integer $ m > 1$ is given. The inﬁnite sequence $ (x_n)_{n\ge 0}$ is deﬁned by $ x_i\equal{}2^i$ for $ i<m$ and $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$ for $ i\ge m$. Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$.