Olympiad problems

2019 IMO Shortlist, N5

Tags: number theory

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

2006 Croatia Team Selection Test, 1

Tags: number theory

Find all natural numbers that can be expressed in a unique way as a sum of ﬁve or less perfect squares.

2015 IFYM, Sozopol, 5

Tags: divisibility , prime , number theory

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

1983 Spain Mathematical Olympiad, 8

Tags: algebra , system of equations , number theory

In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?

2014 Dutch IMO TST, 2

Tags: number theory

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

2005 Junior Tuymaada Olympiad, 8

Tags: sequence , number theory with sequences , number theory

The sequence of natural numbers is based on the following rule: each term, starting with the second, is obtained from the previous addition works of all its various simple divisors (for example, after the number $12$ should be the number $18$, and after the number $125$ , the number $130$). Prove that any two sequences constructed in this way have a common member.

1988 Tournament Of Towns, (190) 3

Tags: permutation , number theory

Let $a_1 , a_2 ,... , a_n$ be an arrangement of the integers $1,2,..., n$. Let $$S=\frac{a_1}{1}+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{1}.$$ Find a natural number $n$ such that among the values of $S$ for all arrangements $a_1 , a_2 ,... , a_n$ , all the integers from $n$ to $n + 100$ appear .

2010 Spain Mathematical Olympiad, 3

Tags: number theory

Let $p$ be a prime number and $A$ an infinite subset of the natural numbers. Let $f_A(n)$ be the number of different solutions of $x_1+x_2+\ldots +x_p=n$, with $x_1,x_2,\ldots x_p\in A$. Does there exist a number $N$ for which $f_A(n)$ is constant for all $n<N$?

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 2

Tags: prime number , number theory

Let $p$ be a prime number and $F=\left \{0,1,2,...,p-1 \right \}$. Let $A$ be a proper subset of $F$ that satisfies the following property: if $a,b \in A$, then $ab+1$ (mod $p$) $ \in A$. How many elements can $A$ have? (Justify your answer.)

2000 Slovenia National Olympiad, Problem 1

Tags: number theory

The sequence $(a_n)$ is given by $a_1=2$, $a_2=500$, $a_3=2000$ and $$\frac{a_{n+2}+a_{n+1}}{a_{n+1}+a_{n-1}}=\frac{a_{n+1}}{a_{n-1}}\qquad\text{for }n\ge2$$Prove that all terms of this sequence are positive integers and that $a_{2000}$ is divisible by $2^{2000}$.

2024 Mongolian Mathematical Olympiad, 1

Tags: diophantine equation , number theory , factorial

Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

2014 ELMO Shortlist, 3

Tags: binomial theorem , polynomial , induction , algebra , number theory

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

1992 Bundeswettbewerb Mathematik, 2

Tags: sum , product , number theory

A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.

2010 Putnam, A4

Tags: putnam number theory , logarithm , greatest common divisor , modular arithmetic , number theory

Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

2010 China Team Selection Test, 3

Tags: pigeonhole principle , ceiling function , number theory

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

2013 Estonia Team Selection Test, 5

Tags: power , number theory

Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled: 1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$ 2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$. Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.

2014 Contests, 2

Tags: 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.

1990 Vietnam Team Selection Test, 1

Tags: least common multiple , greatest common divisor , ring theory , number theory , relatively prime

Let $ T$ be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$. 2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$. For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.

2007 JBMO Shortlist, 1

Tags: least common multiple , greatest common divisor , number theory

Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ , where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.

2012 Austria Beginners' Competition, 1

Tags: divide , divisible , number theory

Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$. Prove that the product $a\cdot b$ is always divisible by $14$.

2018 Thailand TST, 3

Tags: sequence , floor function , number theory

Let $n$ be a fixed odd positive integer. For each odd prime $p$, define $$a_p=\frac{1}{p-1}\sum_{k=1}^{\frac{p-1}{2}}\bigg\{\frac{k^{2n}}{p}\bigg\}.$$ Prove that there is a real number $c$ such that $a_p = c$ for infinitely many primes $p$. [i]Note: $\left\{x\right\} = x - \left\lfloor x\right\rfloor$ is the fractional part of $x$.[/i]

2009 Iran MO (3rd Round), 6

Tags: algorithm , function , induction , number theory , trigonometry

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$. Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$. Time allowed for this problem was 1 hour.

2022 Denmark MO - Mohr Contest, 2

Tags: palindrome , digit , number theory

A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.

2022 Iran-Taiwan Friendly Math Competition, 1

Tags: floor function , number theory

Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$ $$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$ Proposed by Navid Safaei

2005 CentroAmerican, 2

Tags: number theory

Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions. [i]Arnoldo Aguilar, El Salvador[/i]