Olympiad problems

2007 Abels Math Contest (Norwegian MO) Final, 3

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

2002 China Team Selection Test, 2

Tags: number theory

Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

2011 Akdeniz University MO, 1

Tags: prime number , number theory

Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

2015 CHMMC (Fall), 4

Tags: number theory

The following number is the product of the divisors of $n$. $$46, 656, 000, 000$$ What is $n$?

2025 ISI Entrance UGB, 8

Tags: number theory

Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$. Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.

Russian TST 2014, P1

Tags: divisibility , prime number , number theory

Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$

2012 Macedonia National Olympiad, 1

Tags: number theory , parameterization

Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.

2018 Stars of Mathematics, 1

Tags: number theory

Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal? [i]Proposed by Belarus for the 1999th IMO[/i]

2007 Bundeswettbewerb Mathematik, 1

Tags: quadratic , number theory , induction

For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$

1966 IMO Longlists, 42

Tags: sequence , divisibility , number theory

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

2008 Nordic, 4

Tags: number theory , 3d geometry , linear algebra , matrix , geometry

The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.

2021 Purple Comet Problems, 15

Tags: number theory

Let $m$ and $n$ be positive integers such that $$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$ Find the maximum possible value of $m^3 + n^3$.

2019 SG Originals, Q4

Tags: number theory , prime number , algebra , polynomial

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

2019 Saudi Arabia Pre-TST + Training Tests, 1.2

Tags: divide , arithmetic sequence , number theory

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

2016 Saint Petersburg Mathematical Olympiad, 7

Tags: integer sequence , sequence , number theory with sequences , number theory

A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: perfect square , digit , number theory

The last four digits of a perfect square are equal. Prove that all of them are zeros.

2015 South East Mathematical Olympiad, 4

Tags: number theory

Given $8$ pairwise distinct positive integers $a_1,a_2,…,a_8$ such that the greatest common divisor of any three of them is equal to $1$. Show that there exists positive integer $n\geq 8$ and $n$ pairwise distinct positive integers $m_1,m_2,…,m_n$ with the greatest common divisor of all $n$ numbers equal to $1$ such that for any positive integers $1\leq p<q<r\leq n$, there exists positive integers $1\leq i<j\leq 8$ that $a_ia_j\mid m_p+m_q+m_r$.

2014 Vietnam National Olympiad, 3

Tags: 3d geometry , absolute value , geometry , number theory

Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.

1994 IMO Shortlist, 1

Tags: subset , perfect square , extremal combinatorics , number theory

$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$

2007 All-Russian Olympiad Regional Round, 8.6

Tags: number theory

A number $ B$ is obtained from a positive integer number $ A$ by permuting its decimal digits. The number $ A\minus{}B\equal{}11...1$ ($ n$ of $ 1's$). Find the smallest possible positive value of $ n$.

2015 Estonia Team Selection Test, 3

Tags: rational , number theory

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

2012 China Team Selection Test, 1

Tags: logarithm , function , number theory , floor function

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2013 Cuba MO, 3

Tags: number theory , sum of digits

Find all the natural numbers that are $300$ times the sum of its digits.

2020 Brazil Team Selection Test, 1

Tags: number theory

Determine if there is a positive integer $n$ such that for any $n$ consecutive positive integers, there is [b]one[/b] of them(denote $c$) such that $c$ can be written as sum of consecutive integers(not necessarily all positive) of at most $2020$ distinct ways.

2003 Bulgaria Team Selection Test, 6

Tags: number theory

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$