Olympiad problems

2024 AIME, 14

Let $b\ge 2$ be an integer. Call a positive integer $n$ $b$-[i]eautiful[/i] if it has exactly two digits when expressed in base $b$ and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-[i]eautiful[/i] because $81 = \underline{6} \ \underline{3}_{13} $ and $6 + 3 = \sqrt{81}$. Find the least integer $b\ge 2$ for which there are more than ten $b$-[i]eautiful[/i] integers.

1987 Dutch Mathematical Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Solve into $N$: $$a^2 = 2^b +c^4$$

2014 NIMO Problems, 4

Tags: relatively prime , number theory , ratio

Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

2014 IMO Shortlist, N6

Tags: divisibility , modulo arithmetic , counting in two ways , integer polynomial , number theory

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

2024 Canadian Junior Mathematical Olympiad, 4

Tags: number theory , combinatorics

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

2025 Nordic, 2

Tags: number theory , nt , prime

Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$ [size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]

2013 Czech-Polish-Slovak Junior Match, 4

Tags: divide , divisor , digit , number theory

Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

2010 Nordic, 1

Tags: function , number theory , relatively prime , inequalities , modular arithmetic , algebra

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

2018 Caucasus Mathematical Olympiad, 5

Tags: number theory

Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect square, while $bn$ is a perfect cube". Determine if the statement of Baron’s theorem is correct.

2019 Macedonia Junior BMO TST, 5

Tags: number theory , prime number

Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.

2023 Miklós Schweitzer, 6

Tags: number theory

Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$.

2010 Indonesia TST, 4

Tags: number theory

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

2005 Austrian-Polish Competition, 9

Tags: diophantine equation , sum of cubes , infinitely many solutions , number theory

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.

2000 Mongolian Mathematical Olympiad, Problem 6

Tags: number theory

Given distinct prime numbers $p_1,\ldots,p_s$ and a positive integer $n$, find the number of positive integers not exceeding $n$ that are divisible by exactly one of the $p_i$.

2015 Indonesia MO, 5

Tags: divisibility , number theory

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

2008 Princeton University Math Competition, A5

Tags: number theory

If $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.

2019 SIMO, Q1

Tags: invariant , grasshopper , reachability , number theory

[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.

2015 Postal Coaching, 4

Tags: number theory , induction , sequence , recurrence

The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$ $n \ge 1$. Prove that all the terms in the sequence are integers.

MathLinks Contest 7th, 7.1

Tags: algebra , number theory , quadratic , vieta , relatively prime , modular arithmetic

Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

2023 Austrian MO National Competition, 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?

2015 Junior Regional Olympiad - FBH, 4

Tags: number theory

On the market one seller is selling watermelons, melons and young corn cobs. Total number of watermelons, melons and corn cobs is $239$. One buyer bought $\frac{2}{3}$ of all watermelons, $\frac{3}{5}$ of all melons and $\frac{5}{7}$ of all corn cobs. Other buyer bought $\frac{1}{13}$ of all watermelons, $\frac{1}{4}$ of all melons and $\frac{1}{5}$ of all corn cobs. How many pieces in total bought second buyer and how many seller had at the beggining of each watermelons, melons and corn cobs?

2006 AMC 12/AHSME, 14

Tags: number theory , greatest common divisor , ratio

Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? $ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$

2021 BMT, 9

Tags: binomial coefficient , number theory

Let $p=101.$ The sum \[\sum_{k=1}^{10}\frac1{\binom pk}\] can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$

2014 Estonia Team Selection Test, 6

Tags: number theory , diophantine , diophantine equation

Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers

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?