Olympiad problems

2007 Balkan MO Shortlist, A7

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

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

2006 China Team Selection Test, 3

Tags: number theory unsolved , number theory

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

2007 Mexico National Olympiad, 1

Tags: number theory unsolved , number theory

The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$. Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions.

1992 Cono Sur Olympiad, 1

Tags: calculus , integration , number theory unsolved , number theory

Prove that there aren't any positive integrer numbers $x,y,z$ such that $x^2+y^2=3z^2$.

2002 Iran MO (3rd Round), 20

Tags: number theory unsolved , number theory

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

2011 Nordic, 4

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

Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$. (Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)

2013 Moldova Team Selection Test, 1

Tags: number theory unsolved , number theory

Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying: $1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$; $2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$; Prove that $m$ is odd.

1988 USAMO, 1

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

By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$. Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.

2000 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

2014 India IMO Training Camp, 2

Tags: number theory unsolved , number theory

Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

2007 Cono Sur Olympiad, 1

Tags: number theory unsolved , number theory

Find all pairs $(x,y)$ of nonnegative integers that satisfy \[x^3y+x+y=xy+2xy^2.\]

1987 Bundeswettbewerb Mathematik, 3

Tags: number theory unsolved , number theory

Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.

2006 Korea - Final Round, 2

Tags: search , arithmetic sequence , number theory unsolved , number theory

For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist inﬁnitely many primes that are not contained in $S_{a}$.

1987 IMO Longlists, 25

Tags: induction , number theory unsolved , number theory

Numbers $d(n,m)$, with $m, n$ integers, $0 \leq m \leq n$, are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and \[md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.\] Prove that all the $d(n,m)$ are integers.

1991 Canada National Olympiad, 2

Tags: number theory unsolved , number theory

Let $n$ be a fixed positive integer. Find the sum of all positive integers with the property that in base $2$ each has exactly $2n$ digits, consisting of $n$ 1's and $n$ 0's. (The first digit cannot be $0$.)

1976 IMO Longlists, 20

Tags: inequalities , induction , number theory unsolved , number theory

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and \[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\] Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$ \[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\] and give one such $q$ explicitly.

2015 International Zhautykov Olympiad, 1

Tags: floor function , number theory , least common multiple , number theory unsolved

Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.

2004 IberoAmerican, 3

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

Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$

2010 Postal Coaching, 4

Tags: number theory unsolved , number theory

How many ordered triples $(a, b, c)$ of positive integers are there such that none of $a, b, c$ exceeds $2010$ and each of $a, b, c$ divides $a + b + c$?

2017 Azerbaijan EGMO TST, 4

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

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2002 Czech and Slovak Olympiad III A, 3

Tags: number theory , modular arithmetic , number theory unsolved

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences \[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\] is divisible by $n$.

2014 Postal Coaching, 1

Tags: modular arithmetic , number theory unsolved , number theory

Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

1995 Turkey MO (2nd round), 1

Tags: number theory unsolved , number theory

Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$. Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences \[x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.\]

1991 USAMO, 3

Tags: induction , modular arithmetic , strong induction , number theory , relatively prime , prime numbers , number theory unsolved

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

1979 IMO Longlists, 52

Tags: number theory unsolved , number theory

Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$.