Olympiad problems

2005 BAMO, 1

An integer is called [i]formidable[/i] if it can be written as a sum of distinct powers of $4$, and [i]successful [/i] if it can be written as a sum of distinct powers of $6$. Can $2005$ be written as a sum of a [i]formidable [/i] number and a [i]successful [/i] number? Prove your answer.

2014 Cuba MO, 7

Tags: diophantine , diophantine equation , number theory

Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

2018 Junior Balkan MO, 1

Tags: number theory , algebra

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2014 Singapore Junior Math Olympiad, 1

Tags: number theory , multiple , digit

Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

2009 Bulgaria National Olympiad, 1

Tags: diophantine equation , exponential equation , coprime , number theory

The natural numbers $a$ and $b$ satisfy the inequalities $a > b > 1$ . It is also known that the equation $\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$. Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).

1989 USAMO, 1

Tags: number theory

For each positive integer $n$, let \begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*} Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.

2015 India PRMO, 7

Tags: number theory , digit

$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$

2016 Brazil Team Selection Test, 2

Tags: number theory

A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},$$ where $a_1,a_2, \cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2014 AMC 8, 4

Tags: modular arithmetic , number theory , prime number

The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

1991 IMO Shortlist, 16

Tags: number theory , arithmetic sequence , system of equations , eulers function , laurentiu panaitopol

Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

2020 JBMO Shortlist, 2

Tags: number theory

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

1999 Greece JBMO TST, 4

Tags: perfect cube , perfect power , number theory

Examine whether exists $n \in N^*$, such that: (a) $3n$ is perfect cube, $4n$ is perfect fourth power and $5n$ perfect fifth power (b) $3n$ is perfect cube, $4n$ is perfect fourth power, $5n$ perfect fifth power and $6n$ perfect sixth power

2015 Argentina National Olympiad, 2

Tags: power of 2 , number theory

Find all pairs of natural numbers $a,b$ , with $a\ne b$ , such that $a+b$ and $ab+1$ are powers of $2$.

2018 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , divisibility , sum of digits

For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions $$a \mid b + s(a)$$ $$b \mid a + s(b)$$ [i]Proposed by Victor Domínguez[/i]

1979 IMO Longlists, 78

Tags: inequalities , number theory

Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

1998 Belarus Team Selection Test, 2

Tags: number theory , diophantine , perfect square , diophantine equation

a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

2014 Turkey Team Selection Test, 1

Tags: modular arithmetic , number theory

Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.

2010 Gheorghe Vranceanu, 1

Tags: number theory

[b]a)[/b] Prove that any multiple of $ 6 $ is the sum of four cubes. [b]b)[/b] Show that any integer is the sum of five cubes.

1969 IMO Shortlist, 49

Tags: number theory , combinatorics , divisibility , invariant

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

2005 AIME Problems, 3

Tags: number theory , algebra , ratio , geometric series , relatively prime , system of equations

An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

2025 Thailand Mathematical Olympiad, 10

Tags: algebra , polynomial , number theory

Let $n$ be a positive integer. Show that there exist a polynomial $P(x)$ with integer coefficient that satisfy the following [list] [*]Degree of $P(x)$ is at most $2^n - n -1$ [*]$|P(k)| = (k-1)!(2^n-k)!$ for each $k \in \{1,2,3,\dots,2^n\}$ [/list]

2015 Dutch IMO TST, 2

Tags: diophantine equation , system of equations , diophantine , number theory

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

1996 All-Russian Olympiad, 1

Tags: number theory , 3d geometry , geometry

Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be? [i]A. Golovanov[/i]

2009 Estonia Team Selection Test, 2

Tags: number theory , mean , coprime

Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer. a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice. b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?

1993 AIME Problems, 15

Tags: pythagorean theorem , number theory , relatively prime , geometry

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$