Olympiad problems

1990 IMO Longlists, 18

Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.

2017 Dutch IMO TST, 1

Tags: number theory

Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number. $(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$. (b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.

2020 Macedonia Additional BMO TST, 3

Tags: number theory , divisibility

Does there exist a set of $2020$ distinct positive whole numbers with the property that the product of any $101$ of them is divisible by the sum of those $101$ numbers?

2006 CHKMO, 4

Tags: search , number theory

Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

2003 Cuba MO, 7

Tags: sum of digits , number theory

Let S(n) be the sum of the digits of the positive integer $n$. Determine $$S(S(S(2003^{2003}))).$$

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 $A\cup B$.

2013 Saint Petersburg Mathematical Olympiad, 7

Tags: modular arithmetic , number theory

Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov

PEN P Problems, 19

Tags: relatively prime , additive number theory , number theory

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

1966 IMO Shortlist, 54

Tags: decimal representation , digit , last digit , number theory , modular arithmetic

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

2023 Iberoamerican, 1

Tags: number theory

Let $n$ be a positive integer. The following $35$ multiplication are performed: $$1 \cdot n, 2 \cdot n, \dots, 35 \cdot n.$$ Show that in at least one of these results the digit $7$ appears at least once.

2023 Paraguay Mathematical Olympiad, 4

Tags: number theory

We say that a positive integer is [i]Noble [/i] when: it is composite, it is not divisible by any prime number greater than $20$ and it is not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?

1984 IMO Shortlist, 3

Tags: number theory , divisor , equation

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

1993 All-Russian Olympiad, 1

Tags: number theory

For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

2022 Azerbaijan Junior National Olympiad, A1

Tags: number theory , algebra

Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$

1995 Polish MO Finals, 3

Tags: number theory , induction , polynomial , algebra , modular arithmetic

Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$ Find the remainder when $x_{p^3}$ is divided by $p$.

1968 IMO Shortlist, 21

Tags: divisibility , number theory

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

2016 Bangladesh Mathematical Olympiad, 2

Tags: modular arithmetic , divisibility , number theory

(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

1995 Greece National Olympiad, 1

Tags: number theory

Find all positive integers $n$ such that $-5^4 + 5^5 + 5^n$ is a perfect square. Do the same for $2^4 + 2^7 + 2^n.$

2007 Indonesia TST, 2

Tags: divisor , number theory , prime divisor

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

1998 Turkey Team Selection Test, 3

Tags: polynomial , algebra , combinatorial nullstellensatz , number theory

Let $f(x_{1}, x_{2}, . . . , x_{n})$ be a polynomial with integer coeﬃcients of degree less than $n$. Prove that if $N$ is the number of $n$-tuples $(x_{1}, . . . , x_{n})$ with $0 \leq x_{i} < 13$ and $f(x_{1}, . . . , x_{n}) = 0 (mod 13)$, then $N$ is divisible by 13.

2014 Stars Of Mathematics, 2

Tags: number theory

Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. ([i]Dan Schwarz[/i])

2014 Romania National Olympiad, 3

Tags: number theory

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

2006 Federal Math Competition of S&M, Problem 2

Tags: diophantine equation , number theory

Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that $$\frac1x+\frac1y=\frac1p-\frac1q.$$

2000 Moldova National Olympiad, Problem 1

Tags: divisibility , determinant , number theory

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

1989 IMO Shortlist, 25

Tags: number theory , equation , diophantine equation , quadratic

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]