Olympiad problems

2011 Estonia Team Selection Test, 2

Tags: number theory

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

2008 Singapore MO Open, 1

Tags: number theory

Find all pairs of positive integers $ (n,k)$ so that $ (n\plus{}1)^k\minus{}1\equal{}n!$.

2004 Postal Coaching, 6

Tags: number theory

Find the number of ordered palindromic partitions of an integer $n$.

1971 Polish MO Finals, 4

Tags: number theory , diophantine

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

2005 Postal Coaching, 26

Tags: number theory

Let $a_1,a_2,\ldots a_n$ be real numbers such that their sum is equal to zero. Find the value of \[ \sum_{j=1}^{n} \frac{1}{a_j (a_j +a _{j+1}) (a_j + a_{j+1} + a_{j+2}) \ldots (a_j + \ldots a_{j+n-2})}. \] where the subscripts are taken modulo $n$ assuming none of the denominators is zero.

2014 AIME Problems, 15

Tags: function , induction , number theory , prime factorization

For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.

2022 ELMO Revenge, 2

Tags: number theory , diophantine equation

Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$ [i]Proposed by Henry Jiang[/i]

2020 ISI Entrance Examination, 7

Tags: number theory

Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then [b](a)[/b] Prove that $d\leqslant 2$. [b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .

2007 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory

Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

2003 Germany Team Selection Test, 3

Tags: floor function , inequalities , least common multiple , ceiling function , number theory

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

1999 Chile National Olympiad, 1

Tags: number theory

Pedrito's lucky number is $34117$. His friend Ramanujan points out that $34117 = 166^2 + 81^2 = 159^2 + 94^2$ and $166-159 = 7$, $94- 81 = 13$. Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is $7$ and in the difference between the seconds it gives $13$. Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.

2023-IMOC, N4

Tags: number theory

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $af(a)^3+2abf(a)+bf(b)$ is a perfect square for all positive integers $a,b$.

2020 Thailand Mathematical Olympiad, 10

Tags: polynomial , number theory

Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.

2022 South East Mathematical Olympiad, 7

Tags: number theory

Let $a,b$ be positive integers.Prove that there are no positive integers on the interval $\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg)$.

2005 India IMO Training Camp, 2

Tags: number theory

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

2019 Dutch BxMO TST, 1

Tags: number theory

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$, (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$.

2007 Romania National Olympiad, 3

Tags: number theory

For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

2015 European Mathematical Cup, 1

Tags: number theory , combinatorics , pigeonhole principle

$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$ [i]Tomi Dimovski[/i]

1966 Poland - Second Round, 4

Tags: number theory , perfect square

Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.

KoMaL A Problems 2022/2023, A.838

Tags: number theory , analytic number theory

Sets $X\subset \mathbb{Z}^{+}$ and $Y\subset \mathbb{Z}^{+}$ are called [i]comradely[/i], if every positive integer $n$ can be written as $n=xy$ for some $x\in X$ and $y\in Y$. Let $X(n)$ and $Y(n)$ denote the number of elements of $X$ and $Y$, respectively, among the first $n$ positive integers. Let $f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}$ be an arbitrary function that goes to infinity. Prove that one can find comradely sets $X$ and $Y$ such that $\dfrac{X(n)}{n}$ and $\dfrac{Y(n)}{n}$ goes to $0$, and for all $\varepsilon>0$ exists $n \in \mathbb{Z}^+$ such that \[\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon. \]

2005 Postal Coaching, 7

Tags: number theory

Fins all ordered triples $ \left(a,b,c\right)$ of positive integers such that $ abc \plus{} ab \plus{} c \equal{} a^3$.

2007 Regional Olympiad of Mexico Center Zone, 4

Tags: number theory , power of 2

Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

1989 IMO Longlists, 22

Tags: algebra , irrational number , number theory , equation

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

2004 Bulgaria National Olympiad, 6

Tags: induction , number theory

Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}\plus{}b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that a) if $ m\plus{}n > p$ then $ k \equal{} p$; b) if $ m\plus{}n\leq p$ then $ k\geq m\plus{}n\minus{}1$.

2008 Swedish Mathematical Competition, 2

Tags: number theory , combinatorics

Determine the smallest integer $n \ge 3$ with the property that you can choose two of the numbers $1,2,\dots, n$ in such a way that their product is equal to the sum of the other $n - 2$ languages. What are the two numbers?