Olympiad problems

2009 Indonesia TST, 4

Tags: number theory

Let $ n>1$ be an odd integer and define: \[ N\equal{}\{\minus{}n,\minus{}(n\minus{}1),\dots,\minus{}1,0,1,\dots,(n\minus{}1),n\}.\] A subset $ P$ of $ N$ is called [i]basis[/i] if we can express every element of $ N$ as the sum of $ n$ different elements of $ P$. Find the smallest positive integer $ k$ such that every $ k\minus{}$elements subset of $ N$ is basis.

2016 District Olympiad, 2

Tags: equation , algebra , perfect square , number theory , fractional part

If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $ 1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.

2006 MOP Homework, 1

Tags: set , rational , number theory , strong induction , rational number

Let $S$ be a set of rational numbers with the following properties: (a) $\frac12$ is an element in $S$, (b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$. Prove that $S$ contains all rational numbers in the interval $(0, 1)$.

2018 IMO Shortlist, N6

Tags: function , number theory

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

2023 Francophone Mathematical Olympiad, 4

Tags: number theory , divide , consecutive , product

Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?

2013 IFYM, Sozopol, 3

Tags: induction , relatively prime , number theory

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

2024 Chile TST IMO, 2

Tags: number theory

Find all natural numbers that have a multiple consisting only of the digit 9.

2005 Moldova Team Selection Test, 4

Tags: function , number theory

Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.

2022 Estonia Team Selection Test, 3

Tags: number theory

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

2019 Philippine TST, 3

Tags: number theory , divisor , bounding , infinite sequence

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

2008 Singapore Senior Math Olympiad, 2

Tags: number theory , perfect square

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2020/2021 Tournament of Towns, P5

Tags: number theory

Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others? [i]Mikhail Evdokimov[/i]

2023 Ecuador NMO (OMEC), 5

Tags: number theory

Find all positive integers $n$ such that $4^n + 4n + 1$ is a perfect square.

1996 Iran MO (3rd Round), 1

Tags: number theory

Find all non-negative integer solutions of the equation \[2^x + 3^y = z^2 .\]

2018 Moldova EGMO TST, 1

Tags: number theory

Find if there are solutions : $ a,b \in\mathbb{N} $ , $a^2+b^2=2018 $ , $ 7|a+b $ .

2015 Balkan MO Shortlist, N2

Tags: recurrence relation , number theory with sequences , sequence , divide , number theory , algebra

Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$, and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$. Prove that $n^2$ divides $a_n$ for infinite $n$. (Romania)

2016 BMT Spring, 1

Tags: number theory

A bag is filled with quarters and nickels. The average value when pulling out a coin is $10$ cents. What is the least number of nickels in the bag possible?

1976 Chisinau City MO, 122

Tags: cyclic , geometry , number theory , prime number , area

The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

2012 Iran MO (2nd Round), 3

Tags: number theory

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

2012 Moldova Team Selection Test, 10

Tags: number theory

Let $f:\mathbb{R}\rightarrow\mathbb{R}, f(x,y)=x^2-2y.$ Define the sequences $(a_n)_{n\geq1}$ and $(b_n)_{n\geq1}$ such that $a_{n+1}=f(a_n,b_n), b_{n+1}=f(b_n,a_n).$ If $4a_1-2b_1=7 :$ a) find the smallest $k\in\mathbb{N}$ for which the number $p=2^k\cdot(2^{512}a_9-b_9)$ is an integer. b) prove that $2^{2^{10}}+2^{2^9}+1$ divides $p.$

2001 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , perfect square , digit

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

2013 Grand Duchy of Lithuania, 4

Tags: combination , number theory , even

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

1989 IMO Longlists, 27

Tags: calculus , integration , modular arithmetic , number theory , relatively prime , combinatorics

Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

2010 Contests, 1

Tags: prime number , number theory

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

2009 All-Russian Olympiad Regional Round, 9.6

Tags: number theory

Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6? (Author: V. Senderov) (compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)