Olympiad problems

2011 IMAC Arhimede, 5

Solve in set of integers the following equation $x^5+y^5+z^5+t^5=93$.

2014 District Olympiad, 2

Tags: floor function , number theory proposed , number theory

Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]

2014 ELMO Shortlist, 11

Tags: algebra , polynomial , modular arithmetic , number theory proposed , number theory

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2001 District Olympiad, 2

Tags: number theory proposed , number theory

Consider the number $n=123456789101112\ldots 99100101$. a)Find the first three digits of the number $\sqrt{n}$. b)Compute the sum of the digits of $n$. c)Prove that $\sqrt{n}$ isn't rational. [i]Valer Pop[/i]

2009 Canadian Mathematical Olympiad Qualification Repechage, 8

Tags: number theory proposed , number theory

Determine an infinite family of quadruples $(a, b, c, d)$ of positive integers, each of which is a solution to $a^4+b^5+c^6=d^7$.

2012 China Team Selection Test, 1

Tags: inequalities , induction , number theory proposed , number theory

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

2010 Baltic Way, 20

Tags: induction , number theory proposed , number theory

Determine all positive integers $n$ for which there exists an infinite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.

1996 Taiwan National Olympiad, 1

Tags: trigonometry , number theory proposed , number theory

Suppose that $a,b,c$ are real numbers in $(0,\frac{\pi}{2})$ such that $a+b+c=\frac{\pi}{4}$ and $\tan{a}=\frac{1}{x},\tan{b}=\frac{1}{y},\tan{c}=\frac{1}{z}$ , where $x,y,z$ are positive integer numbers. Find $x,y,z$.

2011 German National Olympiad, 5

Tags: number theory proposed , number theory

Prove or disprove: $\exists n\in N$ , s.t. $324 + 455^n$ is prime.

2000 Iran MO (3rd Round), 2

Tags: function , number theory proposed , number theory

Find all f:N $\longrightarrow$ N that: [list][b]a)[/b] $f(m)=1 \Longleftrightarrow m=1 $ [b]b)[/b] $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ [b]c)[/b] $ f^{2000}(m)=f(m) $[/list]

2009 Portugal MO, 1

Tags: number theory proposed , number theory

João calculated the product of the non zero digits of each integer from $1$ to $10^{2009}$ and then he summed these $10^{2009}$ products. Which number did he obtain?

2008 All-Russian Olympiad, 1

Tags: number theory proposed , number theory

Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?

2010 Contests, 3

Tags: inequalities , number theory , prime numbers , number theory proposed

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

1998 Greece National Olympiad, 1

Tags: number theory proposed , number theory

Prove that for any integer $n>3$ there exist infinitely many non-constant arithmetic progressions of length $n-1$ whose terms are positive integers whose product is a perfect $n$-th power.

2012 Spain Mathematical Olympiad, 1

Tags: number theory proposed , number theory

Determine if the number $\lambda_n=\sqrt{3n^2+2n+2}$ is irrational for all non-negative integers $n$.

2008 Romania Team Selection Test, 2

Tags: number theory , prime numbers , number theory proposed

Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?

2012 Greece Team Selection Test, 1

Tags: number theory proposed , number theory

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

2000 Tuymaada Olympiad, 4

Tags: factorial , number theory proposed , number theory

Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.

2008 Hong Kong TST, 3

Tags: modular arithmetic , number theory proposed , number theory

Show that the equation $ y^{37} \equal{} x^3 \plus{}11\pmod p$ is solvable for every prime $ p$, where $ p\le 100$.

2006 JBMO ShortLists, 5

Tags: quadratics , number theory proposed , number theory

Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$. EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]

2007 Ukraine Team Selection Test, 6

Tags: geometry , 3D geometry , number theory proposed , number theory

Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes). E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$ only.

1995 Iran MO (2nd round), 1

Tags: geometry , 3D geometry , sphere , vector , number theory proposed , number theory

Prove that for every positive integer $n \geq 3$ there exist two sets $A =\{ x_1, x_2,\ldots, x_n\}$ and $B =\{ y_1, y_2,\ldots, y_n\}$ for which [b]i)[/b] $A \cap B = \varnothing.$ [b]ii)[/b] $x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.$ [b]ii)[/b] $x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.$

2011 Canada National Olympiad, 4

Tags: number theory proposed , number theory

Show that there exists a positive integer $N$ such that for all integers $a>N$, there exists a contiguous substring of the decimal expansion of $a$, which is divisible by $2011$. Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.

2013 China Team Selection Test, 1

Tags: symmetry , number theory proposed , number theory

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

2010 IMAR Test, 4

Tags: number theory proposed , number theory

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.