Olympiad problems

2013 Kyiv Mathematical Festival, 2

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?

2017 Romanian Master of Mathematics, 1

Tags: representation , additive combinatorics , algebra

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

1980 IMO Shortlist, 9

Tags: modular arithmetic , pascal triangle , representation , number theory

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2011 Silk Road, 4

Tags: prime , representation , number theory

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

1997 Mexico National Olympiad, 6

Tags: representation , number theory

Prove that number $1$ has infinitely many representations of the form $$1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}$$ , where$ n$ and $a_i $ are positive integers with $5 < a_1 < a_2 < ... < a_n$.

2013 Spain Mathematical Olympiad, 4

Tags: representation , number theory

Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.

1999 IMO Shortlist, 2

Tags: representation , number theory , ratio

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

2007 Indonesia TST, 4

Tags: combinatorics , representation , subset , set theory

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

2013 Kyiv Mathematical Festival, 2

Tags: representation , sum , perfect square , number theory

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?

2005 Postal Coaching, 18

Tags: number theory , representation

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.

2005 All-Russian Olympiad, 1

Tags: representation , number theory

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.

Russian TST 2021, P1

Tags: number theory , representation

Do there exist infinitely many positive integers not expressible in the form \[(a+b)+\log_2(b+c)-2^{c+a},\]where $a,b,c$ are positive integers?

1980 IMO, 22

Tags: number theory , pascal triangle , representation , modular arithmetic

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2000 Romania Team Selection Test, 3

Tags: number theory , representation , ratio

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

1936 Moscow Mathematical Olympiad, 024

Tags: representation , number theory

Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs. (P. Dirac)

2017 Romanian Masters In Mathematics, 1

Tags: algebra , additive combinatorics , representation

[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

1957 Putnam, B4

Tags: combinatorics , representation , summation

Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$'s and $2$'s. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$.

1993 IMO Shortlist, 3

Tags: number theory , representation , divisibility

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

2007 Indonesia TST, 4

Tags: combinatorics , representation , set theory , subset

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

1980 IMO Longlists, 9

Tags: number theory , modular arithmetic , representation , pascal triangle

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.