Olympiad problems

2014 Chile National Olympiad, 4

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

2015 China Team Selection Test, 5

Tags: number theory , diophantine equation

FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$

PEN J Problems, 17

Tags: function , number theory , totient function , divisor function

Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.

2005 MOP Homework, 1

Tags: number theory

Find all triples $(x,y,z)$ such that $x^2+y^2+z^2=2^{2004}$.

2021 Final Mathematical Cup, 3

Tags: algebra , number theory

For a positive integer $n$ we define $f (n) = \max X_1^{X_2^{...^{X_k}}}$ where the maximum is taken over all possible decompositions of natural numbers $n = X_1X_2...X_k$. Determine $f(n)$.

1998 National Olympiad First Round, 18

Tags: modular arithmetic , number theory , prime number

Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$. $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$

2008 China Northern MO, 3

Tags: number theory , prime factor , divide

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

2021 Taiwan TST Round 1, N

Tags: floor function , sequence , number theory

For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers. [i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.

2004 Regional Competition For Advanced Students, 4

Tags: induction , number theory , prime number , algebra

The sequence $ < x_n >$ is defined through: $ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$ Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.

2013 Junior Balkan Team Selection Tests - Moldova, 2

Tags: number theory , set

Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.

2006 MOP Homework, 6

Tags: divisible , number theory , inequalities , subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

1990 French Mathematical Olympiad, Problem 1

Tags: algebra , number theory , sequence

Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$. (a) Calculate $u_{1990}$. (b) Find the number of indices $n\le1990$ for which $u_n=0$. (c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$.

2007 Bulgarian Autumn Math Competition, Problem 10.3

Tags: number theory , coprime , sum function

For a natural number $m>1$ we'll denote with $f(m)$ the sum of all natural numbers less than $m$, which are also coprime to $m$. Find all natural numbers $n$, such that there exist natural numbers $k$ and $\ell$ which satisfy $f(n^{k})=n^{\ell}$.

2008 Middle European Mathematical Olympiad, 4

Tags: number theory

Determine that all $ k \in \mathbb{Z}$ such that $ \forall n$ the numbers $ 4n\plus{}1$ and $ kn\plus{}1$ have no common divisor.

KoMaL A Problems 2019/2020, A. 761

Tags: number theory

Let $n\ge3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S|=n$, no element of S is a multiple of n, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly d nonempty subsets of $S$ the sum of whose elements is a multiple of $n$. Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria

2023 HMNT, 8

Tags: number theory , combinatorics

There are $n \ge 2$ coins, each with a different positive integer value. Call an integer $m$ [i]sticky [/i] if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value $100$.

2007 May Olympiad, 4

Tags: combinatorics , number theory , prime number

Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

2016 Korea - Final Round, 3

Tags: number theory , diophantine equation

Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.

2010 Contests, 2

Tags: induction , number theory

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

2015 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: binomial coefficient , number theory , greatest common divisor , lucas theorem

For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$. Find all possible values of $d(n)$

1971 Vietnam National Olympiad, 1

Tags: number theory , relatively prime , trigonometry

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$. i) Given $m, n$, find $p, q$. ii) Given $n, q$, find $m, p$. ii) Given $m, q$, find $n, p$.

2024 China Team Selection Test, 7

Tags: number theory

For coprime positive integers $a,b$,denote $(a^{-1}\bmod{b})$ by the only integer $0\leq m<b$ such that $am\equiv 1\pmod{b}$ (1)Prove that for pairwise coprime integers $a,b,c$, $1<a<b<c$,we have\[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a.\] (2)Prove that for any positive integer $M$,there exists pairwise coprime integers $a,b,c$, $M<a<b<c$ such that \[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.\]

2010 Benelux, 4

Tags: number theory , diophantine equation

Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and \[a^3 + b^3 = p^n\mbox{.}\] [i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]

2023 OMpD, 3

Tags: euler s totient function , phi function , number theory , parity

For each positive integer $x$, let $\varphi(x)$ be the number of integers $1 \leq k \leq x$ that do not have prime factors in common with $x$. Determine all positive integers $n$ such that there are distinct positive integers $a_1,a_2, \ldots, a_n$ so that the set: $$S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}$$ Have exactly $2n$ consecutive integers (in some order).

STEMS 2021 Math Cat C, Q2

Tags: algebraic number theory , number theory

Does there exist a nonzero algebraic number $\alpha$ with $|\alpha| \neq 1$ such that there exists infinitely many positive integers $n$ for which there's $\beta_n \in \mathbb{C}$ with $\beta_n \in \mathbb{Q}(\alpha)$ and $\beta_n^n = \alpha$?