Olympiad problems

2021 Saudi Arabia JBMO TST, 3

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

2010 International Zhautykov Olympiad, 1

Tags: number theory

Find all primes $p,q$ such that $p^3-q^7=p-q$.

2011 Serbia National Math Olympiad, 2

Tags: number theory , euler

Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.

2012 Bundeswettbewerb Mathematik, 2

Tags: number theory , perfect square

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

2007 Balkan MO, 3

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

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.

2014 Dutch BxMO/EGMO TST, 4

Tags: number theory

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

2008 Stars Of Mathematics, 2

Tags: inequalities , number theory

Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers. i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$ ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples. [i]Dan Schwarz[/i]

1969 IMO Shortlist, 17

Tags: arithmetic sequence , equation , number theory , algebra

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

2020 LIMIT Category 1, 20

Tags: number theory , limit

How many integers $n$, satisfy $|n|<2020$ and the equation $11^3|n^3+3n^2-107n+1$ (A)$0$ (B)$101$ (C)$367$ (D)$368$

2005 Morocco TST, 1

Tags: greatest common divisor , quadratic , number theory

Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

2019 Greece Junior Math Olympiad, 3

Tags: number theory

Determine all positive integers equal to 13 times the sum of their digits.

2024 Thailand Mathematical Olympiad, 2

Tags: number theory

Find all pairs of positive integers $(m,n)$ such that $\frac{m^5+n}{m^2+n^2}$ and $\frac{m+n^5}{m^2+n^2}$ are integers.

2015 Peru IMO TST, 6

Tags: number theory , sequence , parity , floor function

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

MathLinks Contest 1st, 1

Tags: number theory

Let $a, m$ be two positive integers, $a \ne 10^k$, for all non-negative integers $k$ and $d_1, d_2, ... , d_m$ random decimal$^1$ digits with $d_1 > 0$. Prove that there exists some positive integer $n$ for which the representation in the decimal base of the number $a^n$ begins with the digits $d_1, d_2, ... , d_m$ in this order. $^1$ lesser or equal with $9$

2004 AMC 12/AHSME, 22

Tags: number theory

The square \[ \begin{tabular}{|c|c|c|} \hline 50&\textit{b}&\textit{c}\\ \hline \textit{d}&\textit{e}&\textit{f}\\ \hline \textit{g}&\textit{h}&2\\ \hline \end{tabular} \]is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $ g$? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 136$

1983 IMO Longlists, 18

Tags: algebra , number theory , quadratic

Let $b \geq 2$ be a positive integer. (a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits. (b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.

Russian TST 2019, P1

Tags: number theory

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

1955 Moscow Mathematical Olympiad, 299

Tags: prime , arithmetic progression , sequence , divisible , number theory

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

Russian TST 2018, P3

Tags: number theory , euclidean algorithm

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties: [list] [*] $f(1,1)=0$. [*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1; [*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$. [/list] Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$

2006 Junior Balkan MO, 1

Tags: number theory , floor function

If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

2003 China National Olympiad, 2

Tags: combinatorics , number theory , prime number

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

2004 France Team Selection Test, 3

Tags: number theory , prime number , induction

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2010 ELMO Shortlist, 2

Tags: number theory , difference of squares , special factorizations , modular arithmetic , algebra

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

PEN D Problems, 16

Tags: number theory , congruence , relatively prime , modular arithmetic

Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]

2023 CMWMC, R1

Tags: number theory

[u]Set 1[/u] [b]1.1[/b] How many positive integer divisors are there of $2^2 \cdot 3^3 \cdot 5^4$? [b]1.2[/b] Let $T$ be the answer from the previous problem. For how many integers $n$ between $1$ and $T$ (inclusive) is $\frac{(n)(n - 1)(n - 2)}{12}$ an integer? [b]1.3[/b] Let $T$ be the answer from the previous problem. Find $\frac{lcm(T, 36)}{gcd(T, 36)}$. PS. You should use hide for answers.