Olympiad problems

2000 National High School Mathematics League, 2

Tags: number theory

Two sequences $(a_n)$ and $(b_n)$ satisfy that $a_0=1,a_1=4,a_2=49$, and $\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases}$ for $n=0,1,2,\cdots,$. Prove that $a_n$ is a perfect square for $n=0,1,2,\cdots,$.

2019 Hanoi Open Mathematics Competitions, 7

Tags: number theory , diophantine , diophantine equation

Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$

2014 ELMO Shortlist, 8

Tags: function , greatest common divisor , number theory

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

2021 Iran RMM TST, 3

Tags: number theory , sum of cubes

Let $n$ be an integer greater than $1$ such that $n$ could be represented as a sum of the cubes of two rational numbers, prove that $n$ is also the sum of the cubes of two non-negative rational numbers. Proposed by [i]Navid Safaei[/i]

2015 Argentina National Olympiad Level 2, 4

Tags: algebra , number theory , combinatorics

Let $N$ be the number of ordered lists of $9$ positive integers $(a,b,c,d,e,f,g,h,i)$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}+\frac{1}{g}+\frac{1}{h}+\frac{1}{i}=1.$$ Determine whether $N$ is even or odd.

1984 IMO Longlists, 15

Tags: number theory

Consider all the sums of the form \[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\] where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?

2023 Princeton University Math Competition, B2

Tags: number theory

I have a four-digit palindrome $\underline{a} \ \underline{b} \ \underline{b} \ \underline{a}$ that is divisible by $b$ and is also divisible by the two-digit number $\underline{b} \ \underline{b}.$ Find the number of palindromes satisfying both of these properties.

1956 Moscow Mathematical Olympiad, 326

Tags: number theory , algebra

a) In the decimal expression of a positive number, $a$, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of $a$ with accuracy to $0.01$ with deficiency). The number obtained is divided by $a$ and the quotient is similarly approximated with the same accuracy by a number $b$. What numbers $b$ can be thus obtained? Write all their possible values. b) same as (a) but with accuracy to $0.001$ c) same as (a) but with accuracy to $0.0001$

1989 Federal Competition For Advanced Students, 1

Tags: number theory

Natural numbers $ a \le b \le c \le d$ satisfy $ a\plus{}b\plus{}c\plus{}d\equal{}30$. Find the maximum value of the product $ P\equal{}abcd.$

2011 Baltic Way, 18

Tags: number theory

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

2019 Centers of Excellency of Suceava, 2

Tags: floor function , integer part , number theory , algebra

For a natural number $ n\ge 2, $ calculate the integer part of $ \sqrt[n]{1+n}-\sqrt {2/n} . $ [i]Dan Nedeianu[/i]

2021 CMIMC, 2.3 1.1

Tags: number theory

How many multiples of $12$ divide $12!$ and have exactly $12$ divisors? [i]Proposed by Adam Bertelli[/i]

2023 Tuymaada Olympiad, 1

Tags: number theory , combinatorics

The numbers $1, 2, 3, \ldots$ are arranged in a spiral in the vertices of an infinite square grid (see figure). Then in the centre of each square the sum of the numbers in its vertices is placed. Prove that for each positive integer n the centres of the squares contain infinitely many multiples of $n$.

2015 Puerto Rico Team Selection Test, 4

Tags: number theory , last digit

Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

2024 Malaysian IMO Training Camp, 3

Tags: number theory

Find all primes $p$ such that for any integer $k$, there exist two integers $x$ and $y$ such that $$x^3+2023xy+y^3 \equiv k \pmod p$$ [i]Proposed by Tristan Chaang Tze Shen[/i]

2020 Hong Kong TST, 4

Tags: number theory , prime number , floor function

Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

1989 Austrian-Polish Competition, 3

Tags: number theory , digit , prime

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

2010 South East Mathematical Olympiad, 4

Tags: number theory

Let $a$ and $b$ be positive integers such that $1\leq a<b\leq 100$. If there exists a positive integer $k$ such that $ab|a^k+b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.

2013 Harvard-MIT Mathematics Tournament, 9

Tags: number theory , relatively prime

Let $m$ be an odd positive integer greater than $1$. Let $S_m$ be the set of all non-negative integers less than $m$ which are of the form $x+y$, where $xy-1$ is divisible by $m$. Let $f(m)$ be the number of elements of $S_m$. [b](a)[/b] Prove that $f(mn)=f(m)f(n)$ if $m$, $n$ are relatively prime odd integers greater than $1$. [b](b)[/b] Find a closed form for $f(p^k)$, where $k>0$ is an integer and $p$ is an odd prime.

2023 Moldova EGMO TST, 11

Tags: number theory

Find all three digit positive integers that have distinct digits and after their greatest digit is switched to $1$ become multiples of $30$.

1986 IMO Shortlist, 4

Tags: number theory , diophantine equation , equation , algebra

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

2010 Belarus Team Selection Test, 3.2

Tags: number theory , divisibility

Prove that there exists a positive integer $n$ such that $n^6 + 31n^4 - 900\vdots 2009 \cdot 2010 \cdot 2011$. (I. Losev, I. Voronovich)

2014 China Team Selection Test, 1

Tags: floor function , inequalities , number theory

Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$). Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.

2005 Irish Math Olympiad, 3

Tags: modular arithmetic , number theory

Let $ x$ be an integer and $ y,z,w$ be odd positive integers. Prove that $ 17$ divides $ x^{y^{z^w}}\minus{}x^{y^z}$.

II Soros Olympiad 1995 - 96 (Russia), 9.5

Tags: number theory , perfect power

Give an example of four pairwise distinct natural numbers $a$, $b$, $c$ and $d$ such that $$a^2 + b^3 + c^4 = d^5.$$