Olympiad problems

1990 IMO Shortlist, 13

Tags: counting , frobenius , relatively prime , number theory , recurrence relation

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

2014 IMO Shortlist, N1

Tags: induction , frobenius , additive combinatorics , additive number theory , additive representation , number theory

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: freshman s dream , frobenius , nilpotence

Let be two $ n\times n $ complex matrices $ A,B $ satisfying the equations $ (A+B)^2=A^2+B^2 $ and $ (A+B)^4=A^4+B^4. $ Show that $ (AB)^2=0. $

1969 IMO Longlists, 18

Tags: coprime , frobenius , additive number theory , number theory

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

2015 Belarus Team Selection Test, 1

Tags: frobenius , additive combinatorics , additive number theory , additive representation , induction , number theory

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1983 IMO, 3

Tags: frobenius , additive number theory , additive combinatorics , number theory , modular arithmetic

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

2015 Peru IMO TST, 11

Tags: frobenius , additive combinatorics , additive number theory , additive representation , number theory , induction

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1983 IMO Longlists, 27

Tags: frobenius , additive number theory , additive combinatorics , number theory , modular arithmetic

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

1990 IMO Longlists, 37

Tags: recurrence relation , counting , relatively prime , frobenius , number theory

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

2015 India IMO Training Camp, 1

Tags: induction , frobenius , additive combinatorics , additive number theory , additive representation , number theory

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1969 IMO Shortlist, 25

Tags: divisibility , frobenius , additive number theory , number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

2001 Romania Team Selection Test, 3

Tags: combinatorics , number theory , additive number theory , additive combinatorics , frobenius , modular arithmetic

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

2015 India IMO Training Camp, 1

Tags: induction , frobenius , additive combinatorics , additive number theory , additive representation , number theory

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

1969 IMO Shortlist, 18

Tags: coprime , frobenius , additive number theory , number theory

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

1969 IMO Longlists, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

1983 IMO Shortlist, 18

Tags: modular arithmetic , number theory , frobenius , additive number theory , additive combinatorics

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

2000 IMO Shortlist, 6

Tags: number theory , modular arithmetic , combinatorics , additive number theory , additive combinatorics , frobenius

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$