Olympiad problems

2011 Regional Olympiad of Mexico Center Zone, 4

Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

2004 Federal Competition For Advanced Students, P2, 2

Tags: prime number , number theory

Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $ \frac{1}{2004}$ is among this summands? Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$

2016 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: system of equations , algebra , number theory

Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.

1995 Baltic Way, 3

Tags: number theory , relatively prime

The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.

2014 Postal Coaching, 1

Tags: number theory

Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.

1987 All Soviet Union Mathematical Olympiad, 444

Tags: divisible , number theory

Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.

2023 Moldova Team Selection Test, 12

Tags: arithmetic sequence , number theory

The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and $$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$

2021 Nigerian Senior MO Round 2, 4

Tags: number theory , relatively prime , algebra , inequalities

let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$

2023 Korea Summer Program Practice Test, P1

Tags: number theory

A natural number $n$ is given. For all integer triplets $(a,b,c)$ such that $0 < |a| , |b|, |c| < 2023$ and satisfying below, show that the product of all possible integer $a$ is a perfect square. (The value of $a$ allows duplication) $$(a+nb)(a-nc) + abc = 0$$

2020 Centroamerican and Caribbean Math Olympiad, 1

Tags: digit , number theory

A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.

2017 Baltic Way, 19

Tags: number theory

For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that $$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$

1988 Romania Team Selection Test, 9

Tags: induction , number theory

Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number. [i]Laurentiu Panaitopol[/i].

2004 Chile National Olympiad, 4

Tags: sum of digits , digit , number theory

Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

2003 Mexico National Olympiad, 5

Tags: greatest common divisor , combinatorics , number theory

Some cards each have a pair of numbers written on them. There is just one card for each pair $(a,b)$ with $1 \leq a < b \leq 2003$. Two players play the following game. Each removes a card in turn and writes the product $ab$ of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to $1$ loses. Which player has a winning strategy?

1981 Putnam, B3

Tags: prime number , number theory

Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

1998 China National Olympiad, 3

Tags: number theory

Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

2013 Online Math Open Problems, 5

Tags: number theory , probability , relatively prime , analytic geometry

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

2018 Czech-Polish-Slovak Junior Match, 4

Tags: digit , divisible , number theory

Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

2023 CMIMC Algebra/NT, 10

Tags: algebra , number theory

For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$. [i]Proposed by David Tang[/i]

2004 Bulgaria Team Selection Test, 2

Tags: number theory , floor function

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

1982 Tournament Of Towns, (015) 1

Tags: divisible , divisor , number theory

Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. (M Levin)

2016 Switzerland Team Selection Test, Problem 11

Tags: divisibility , number theory

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2017 ELMO Shortlist, 1

Tags: greatest common divisor , number theory

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

2008 Argentina Iberoamerican TST, 3

Tags: number theory

Show that exists a sequence of $ 100$ terms such that: 1)Every term is a perfect square 2) every term is greater than the one before it ( it is strictly increasing) 3)Every two terms of the sequence are relative prime 4) The average between two consecutive terms is also a perfect square Daniel

1999 Chile National Olympiad, 5

Tags: multiple , number theory

Consider the numbers $x_1, x_2,...,x_n$ that satisfy: $\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$ $\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$ Prove that $n$ is a multiple of $4$.