Olympiad problems

1990 Romania Team Selection Test, 1

Let a,b,n be positive integers such that $(a,b) = 1$. Prove that if $(x,y)$ is a solution of the equation $ax+by = a^n + b^n$ then $$\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]$$

2009 AIME Problems, 6

Tags: floor function , number theory

How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)

2023-IMOC, N5

Tags: number theory

Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.

1995 India National Olympiad, 2

Tags: quadratic , number theory , relatively prime , diophantine equation , algebra

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

2021 Indonesia TST, N

Tags: number theory

For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.

1994 Baltic Way, 6

Tags: modular arithmetic , 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 $k$.

1970 Miklós Schweitzer, 4

Tags: modular arithmetic , quadratic , function , absolute value , number theory

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

2009 District Olympiad, 3

Tags: number theory , perfect square

Let $a$ and $b$ be non-negative integers. Prove that the number $a^2 + b^2$ is the difference of two perfect squares if and only if $ab$ is even.

2018 District Olympiad, 1

Tags: number theory , fractional part

Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.

2021 IMC, 3

Tags: number theory

We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find $\text{sup}\{d| d \text{is good}\}$.

2017 SDMO (High School), 4

Tags: number theory , relatively prime

For each positive integer $n$, let $\tau\left(n\right)$ be the number of positive divisors of $n$. It is well-known that if $a$ and $b$ are relatively prime positive integers then $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$. Does the converse hold? That is, if $a$ and $b$ are positive integers such that $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$, then is it necessarily true that $a$ and $b$ are relatively prime? Either give a proof, or find a counter-example.

2017 China Western Mathematical Olympiad, 1

Tags: number theory

Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.

2001 Moldova National Olympiad, Problem 7

Tags: set , number theory

Let $n$ be a positive integer. We denote by $S$ the sum of elements of the set $M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}$. (a) Show that $S$ is divisible by $6$. (b) Find all $n\in\mathbb N$ for which $S+(1-n)(1+n)=2001$.

2021 Philippine MO, 3

Tags: functional equation , number theory

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies • $f(p) = 1$ for all primes $p$, and • $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $. For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?

2024 Simon Marais Mathematical Competition, A2

Tags: number theory

A positive integer $n$ is [i] tripariable [/i] if it is possible to partition the set $\{1, 2, \dots, n\}$ into disjoint pairs such that the sum of two elements in each pair is a power of $3$. For example $6$ is tripariable because $\{1, 2, \dots, n\}=\{1,2\}\cup\{3,6\}\cup\{4,5\}$ and $$1+2=3^1,\quad 3+6 = 3^2\quad\text{and}\quad4+5=3^2$$ are all powers of 3. How many positive integers less than or equal to 2024 are tripariable?

2001 District Olympiad, 1

Tags: modular arithmetic , number theory

A positive integer is called [i]good[/i] if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Prove that: a)2001 is [i]good[/i], but 3001 isn't [i]good[/i]. b)the product of two [i]good[/i] numbers is a [i]good[/i] number. c)if the product of two numbers is [i]good[/i], then at least one of the numbers is [i]good[/i]. [i]Bogdan Enescu[/i]

2005 Estonia National Olympiad, 3

Tags: digit , number theory , divisible

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

2016 AMC 10, 25

Tags: number theory , least common multiple

How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$? $\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$

2014 Contests, 2

Tags: pigeonhole principle , number theory

Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.

Kvant 2019, M2549

Tags: number theory

For each non-negative integer $n$ find the sum of all $n$-digit numbers with the digits in a decreasing sequence. [I]Proposed by P. Kozhevnikov[/I]

2010 Albania Team Selection Test, 4

Tags: algebra , polynomial , function , inequalities , prime number , number theory

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2025 NEPALTST, 2

Tags: number theory

Find all integers $n$ such that if \[ 1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n \] are the divisors of $n$, then the sequence \[ d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1} \] forms a permutation of an arithmetic progression. [i](Kritesh Dhakal, Nepal)[/i]

2023 UMD Math Competition Part II, 3

Tags: number theory , combination , divisibility

Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

1999 Hong kong National Olympiad, 1

Tags: number theory

Find all positive rational numbers $r\not=1$ such that $r^{\frac{1}{r-1}}$ is rational.