Olympiad problems

2005 Germany Team Selection Test, 1

Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.

1991 IMTS, 4

Tags: analytic geometry , geometry , geometric transformation , number theory solved , number theory

Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?

2005 Germany Team Selection Test, 1

Tags: number theory solved , number theory

Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.

2005 India IMO Training Camp, 2

Tags: number theory solved , number theory

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

2016 Israel Team Selection Test, 2

Tags: number theory , number theory solved

Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?

2005 Taiwan TST Round 1, 1

Tags: Putnam , Diophantine equation , number theory solved , number theory

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

1993 India National Olympiad, 8

Tags: function , number theory solved , number theory

Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.

India EGMO 2021 TST, 6

Tags: number theory , Perfect Square , india , number theory solved

Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$. Show that $a^2+b^2-ab$ is not a square.

2001 India National Olympiad, 2

Tags: modular arithmetic , algebra , polynomial , arithmetic sequence , number theory solved , number theory

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2004 Singapore Team Selection Test, 3

Tags: number theory solved , number theory

Let $p \geq 5$ be a prime number. Prove that there exist at least 2 distinct primes $q_1, q_2$ satisfying $1 < q_i < p - 1$ and $q_i^{p-1} \not\equiv 1 \mbox{ (mod }p^2)$, for $i = 1, 2$.

2001 South africa National Olympiad, 3

Tags: number theory solved , number theory

For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.

2000 IMO Shortlist, 1

Tags: modular arithmetic , Euler , number theory , IMO Shortlist , number theory solved

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

1999 South africa National Olympiad, 4

Tags: number theory , number theory solved

The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by \[ L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2. \] Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: induction , number theory solved , number theory

Let $A$ be a set of positive integers such that a) if $a\in A$, the all the positive divisors of $a$ are also in $A$; b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$. Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.

1993 Polish MO Finals, 3

Tags: number theory solved , number theory

Denote $g(k)$ as the greatest odd divisor of $k$. Put $f(k) = \dfrac{k}{2} + \dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1, x_2, x_3, ...$ by $x_1 = 1$, $x_{n+1} = f(x_n)$. Find $n$ such that $x_n = 800$.

2003 Bundeswettbewerb Mathematik, 4

Tags: number theory solved , number theory

Determine all positive integers which cannot be represented as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a,b$ being positive integers.

1997 Romania Team Selection Test, 2

Tags: modular arithmetic , number theory solved , number theory , Divisibility , Multiplicative NT , Multiplicative order

Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2004 Olympic Revenge, 5

Tags: calculus , integration , induction , number theory solved , number theory

$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n. prove that $a_n$ is one integer for all positive integers n.

2002 Flanders Math Olympiad, 3

Tags: calculus , integration , search , number theory , number theory solved

show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory solved , number theory , BritishMathematicalOlympiad

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2005 India IMO Training Camp, 2

Tags: number theory solved , number theory

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

2004 Vietnam Team Selection Test, 3

Tags: number theory solved , number theory

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$, let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$. Let $a$ be greatest element from $S$. Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $ n > k $.

1996 India National Olympiad, 5

Tags: induction , number theory solved , number theory

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$. prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.