Olympiad problems

2023-24 IOQM India, 29

Tags: number theory

A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.

2010 China Western Mathematical Olympiad, 8

Tags: vieta jumping , number theory

Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.

2005 Flanders Math Olympiad, 4

Tags: number theory

If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.

2005 Hungary-Israel Binational, 1

Tags: number theory

Does there exist a sequence of $2005$ consecutive positive integers that contains exactly $25$ prime numbers?

1981 All Soviet Union Mathematical Olympiad, 322

Tags: consecutive , greatest common divisor , number theory

Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.

2016 Kosovo National Mathematical Olympiad, 1

Tags: number theory

Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .

2019 Purple Comet Problems, 23

Tags: number theory

Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$

2019 Junior Balkan Team Selection Tests - Moldova, 6

Tags: number theory

Let $p$ and $q$ be integers. If $k^2+pk+q>0$ for every integer $k$, show that $x^2+px+q>0$ for every real number $x$.

2009 Romanian Masters In Mathematics, 2

Tags: analytic geometry , number theory , combinatorial geometry , combinatorics , modular arithmetic

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

2001 Austrian-Polish Competition, 6

Tags: sequence , integer , floor function , algebra , number theory

Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

2013 Gheorghe Vranceanu, 2

Tags: grade 2 , number theory

Given two natural numbers $ n\ge 2,a, $ prove that there exists another natural number $ v\ge 2 $ such that: $$ \frac{v+\sqrt{v^2-4}}{2} =\left( \frac{n+\sqrt{n^2-4}}{2} \right)^a $$

2000 Turkey MO (2nd round), 2

Tags: polynomial , relatively prime , number theory , algebra

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

2021 Brazil Team Selection Test, 6

Tags: sequence , number theory

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

2010 All-Russian Olympiad Regional Round, 9.3

Tags: number theory , combinatorics

Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers? [hide=original wording beacuse it doesn't make much sense]Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?[/hide]

2008 All-Russian Olympiad, 5

Tags: inequalities , calculus , integration , number theory

The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?

2012 Albania Team Selection Test, 4

Tags: least common multiple , number theory , greatest common divisor , relatively prime

Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that \[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]

2020 Taiwan TST Round 1, 2

Tags: number theory

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

2010 Chile National Olympiad, 4

Tags: number theory , algebra

Let $m, n$ integers such that satisfy $$m + n\sqrt2 = \left(1 +\sqrt2\right)^{2010} .$$ Find the remainder that is obtained when dividing $n$ by $5$.

2002 Tournament Of Towns, 3

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

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

2023 Federal Competition For Advanced Students, P2, 6

Tags: number theory

Does there exist a real number $r$ such that the equation $$x^3-2023x^2-2023x+r=0$$ has three distinct rational roots?

1998 Brazil Team Selection Test, Problem 2

Tags: number theory , ratio

There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

2015 China Girls Math Olympiad, 5

Tags: combinatorics , number theory

Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter. (Congruent triangles are considered identical.)

2023 Assara - South Russian Girl's MO, 3

Tags: number theory , combinatorics

In equality $$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$ Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?

2012 Indonesia TST, 1

Tags: polynomial , least common multiple , number theory , algebra

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

2015 Princeton University Math Competition, A1/B2

Tags: number theory

What is the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$? (when written in base $10$).