Olympiad problems

1984 IMO Longlists, 59

Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.

1983 IMO Longlists, 56

Tags: symmetry , number theory , greatest common divisor , function , number theory unsolved

Consider the expansion \[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\] [b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ [b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$

2005 Mexico National Olympiad, 3

Tags: modular arithmetic , number theory , relatively prime , number theory unsolved

Already the complete problem: Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers: $\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$ One idea? :arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]

2014 Indonesia MO Shortlist, A2

Tags: number theory unsolved , number theory

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.

2007 China Team Selection Test, 2

Tags: algebra , polynomial , Euler , number theory unsolved , number theory

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

1992 IMTS, 3

Tags: number theory unsolved , number theory

In a mathematical version of baseball, the umpire chooses a positive integer $m$, $m \leq n$, and you guess positive integers to obtain information about $m$. If your guess is smaller than the umpire's $m$, he calls it a "ball"; if it is greater than or equal to $m$, he calls it a "strike." To "hit" it you must state the the correct value of $m$ after the 3rd strike or the 6th guess, whichever comes first. What is the largest $n$ so that there exists a strategy that will allow you to bat 1.000, i.e. always state $m$ correctly? Describe your strategy in detail.

2005 Taiwan TST Round 2, 1

Tags: number theory unsolved , number theory

Positive integers $a,b,c,d$ satisfy $a+c=10$ and \[\displaystyle S=\frac{a}{b} + \frac{c}{d} <1.\] Find the maximum value of $S$.

2010 South East Mathematical Olympiad, 2

Tags: algebra , polynomial , modular arithmetic , LaTeX , number theory , number theory unsolved

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

2002 China Team Selection Test, 3

Tags: modular arithmetic , induction , number theory unsolved , number theory

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

1985 Federal Competition For Advanced Students, P2, 5

Tags: number theory unsolved , number theory

A sequence $ (a_n)$ of positive integers satisfies: $ a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.

1978 IMO Longlists, 15

Tags: number theory unsolved , number theory

Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.

2001 China Western Mathematical Olympiad, 3

Tags: number theory , relatively prime , number theory unsolved

Let $ n, m$ be positive integers of different parity, and $ n > m$. Find all integers $ x$ such that $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$ is a perfect square.

2003 France Team Selection Test, 1

Tags: analytic geometry , modular arithmetic , number theory unsolved , number theory

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

2007 Estonia National Olympiad, 2

Tags: inequalities , triangle inequality , number theory unsolved , number theory

Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.

1998 Greece JBMO TST, 3

Tags: number theory unsolved , number theory

Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?

1992 China National Olympiad, 3

Tags: number theory unsolved , number theory

Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions: 1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ; 2) $2a_1=a_0+a_2-2$ ; 3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares. Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.

2011 Tournament of Towns, 5

Tags: number theory unsolved , number theory

Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.

2009 Germany Team Selection Test, 3

Tags: invariant , number theory unsolved , number theory

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

2006 China Team Selection Test, 2

Tags: modular arithmetic , number theory unsolved , number theory

Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.

1985 Vietnam National Olympiad, 1

Tags: number theory unsolved , number theory

Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.

2013 India Regional Mathematical Olympiad, 6

Tags: quadratics , modular arithmetic , number theory , Diophantine equation , number theory unsolved

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

2013 Romania Team Selection Test, 1

Tags: geometry , 3D geometry , modular arithmetic , symmetry , algebra , polynomial , number theory unsolved

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

2006 Turkey MO (2nd round), 3

Tags: algebra , polynomial , number theory unsolved , number theory

Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where \[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]

1998 China National Olympiad, 3

Tags: number theory unsolved , 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.

2005 France Team Selection Test, 4

Tags: floor function , ceiling function , induction , inequalities , logarithms , number theory unsolved , number theory

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.