Olympiad problems

2007 Pre-Preparation Course Examination, 7

Let $p$ be a prime such that $p \equiv 3 \pmod 4$. Prove that we can't partition the numbers $a,a+1,a+2,\cdots,a+p-2$,($a \in \mathbb Z$) in two sets such that product of members of the sets be equal.

2003 China Girls Math Olympiad, 8

Tags: number theory unsolved , number theory

Let $ n$ be a positive integer, and $ S_n,$ be the set of all positive integer divisors of $ n$ (including 1 and itself). Prove that at most half of the elements in $ S_n$ have their last digits equal to 3.

2017 Germany, Landesrunde - Grade 11/12, 4

Tags: combinatorics , number theory , number theory unsolved , Germany , divisor

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

2001 Tournament Of Towns, 5

Tags: number theory unsolved , number theory

Let $a$ and $d$ be positive integers. For any positive integer $n$, the number $a+nd$ contains a block of consecutive digits which constitute the number $n$. Prove that $d$ is a power of 10.

1971 IMO Longlists, 11

Tags: modular arithmetic , number theory , number theory unsolved

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

2011 Postal Coaching, 1

Tags: modular arithmetic , number theory unsolved , number theory

Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any ﬁfth power of a positive integer? Justify your answer.

2016 Germany Team Selection Test, 2

Tags: inequalities , Integer , number theory , number theory unsolved , n-variable inequality

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

1995 Turkey MO (2nd round), 6

Tags: function , number theory unsolved , number theory

Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]

2007 Pre-Preparation Course Examination, 18

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

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

2002 Kurschak Competition, 2

Tags: number theory unsolved , number theory

The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

2005 China Team Selection Test, 3

Tags: Euler , floor function , number theory unsolved , number theory

Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2008 Turkey Team Selection Test, 4

Tags: induction , number theory unsolved , number theory

The sequence $ (x_n)$ is defined as; $ x_1\equal{}a$, $ x_2\equal{}b$ and for all positive integer $ n$, $ x_{n\plus{}2}\equal{}2008x_{n\plus{}1}\minus{}x_n$. Prove that there are some positive integers $ a,b$ such that $ 1\plus{}2006x_{n\plus{}1}x_n$ is a perfect square for all positive integer $ n$.

1987 IMO Longlists, 58

Tags: number theory unsolved , number theory

Find, with argument, the integer solutions of the equation \[3z^2 = 2x^3 + 385x^2 + 256x - 58195.\]

2009 Kyrgyzstan National Olympiad, 6

Tags: number theory unsolved , number theory

Find all natural $a,b$ such that $\left. {a(a + b) + 1} \right|(a + b)(b + 1) - 1$.

2006 Bulgaria Team Selection Test, 2

Tags: number theory unsolved , number theory

[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$ [i]Ivan Landgev[/i]

2007 South East Mathematical Olympiad, 3

Tags: function , number theory unsolved , number theory

Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$, determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$, where $n\ge 2$ . ($[x]$ denotes the greatest integer not exceeding x)

1983 USAMO, 5

Tags: AMC , USA(J)MO , USAMO , number theory unsolved , number theory

Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.

2011 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial , induction , floor function , modular arithmetic , number theory , number theory unsolved

The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

1998 China National Olympiad, 1

Tags: number theory unsolved , number theory

Find all natural numbers $n>3$, such that $2^{2000}$ is divisible by $1+C^1_n+C^2_n+C^3_n$.

1993 China National Olympiad, 1

Tags: geometry , geometric transformation , number theory unsolved , number theory

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

2012 Finnish National High School Mathematics Competition, 5

Tags: function , induction , number theory unsolved , number theory

The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying \[ f(x)=\begin{cases} 3x+1,& \mbox{as }x\mbox{ is odd}\\ x/2, & \mbox{as }x\mbox{ is even.}\\ \end{cases} \] In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying \[f^{40}(x)> 2012x.\]

2009 Hungary-Israel Binational, 3

Tags: induction , number theory unsolved , number theory

(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers? (b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?

2010 Tournament Of Towns, 2

Tags: ratio , number theory unsolved , number theory

Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio $1 : a$. His goal is to divide the cheese into two piles of equal masses. Can he do it if $(a) a$ is irrational? $(b) a$ is rational, $a \neq 1?$

2001 USA Team Selection Test, 9

Tags: number theory unsolved , number theory

Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and \[\prod_{x\in B} x = \sum_{x\in B} x^2.\]

2011 Serbia National Math Olympiad, 2

Tags: limit , number theory unsolved , number theory

Are there positive integers $a, b, c$ greater than $2011$ such that: $(a+ \sqrt{b})^c=...2010,2011...$?