Olympiad problems

2007 Korea National Olympiad, 3

Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set.

2012 Online Math Open Problems, 6

Tags: induction

An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have? [i]Ray Li.[/i]

1994 APMO, 1

Tags: function , induction , algebra

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that (i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \] (ii) For all $x \in [0,1)$, $f(0) \geq f(x)$, (iii) $-f(-1) = f(1) = 1$. Find all such functions $f$.

1999 Turkey MO (2nd round), 6

Tags: induction , combinatorics

We wish to ﬁnd the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to ﬁnd the desired sum.

2003 USAMO, 1

Tags: induction , number theory

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

1997 Canada National Olympiad, 3

Tags: induction , limit , floor function , inequalities

Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.

2009 CHKMO, 1

Tags: function , polynomial , induction , algebra

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

2005 Romania Team Selection Test, 1

Tags: function , induction , algebra

Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$. [i]Dan Schwartz[/i]

2012 Indonesia TST, 2

Tags: induction , combinatorics

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

2005 Brazil Undergrad MO, 5

Tags: real analysis , calculus , integration , function , induction , logarithm

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

2005 CentroAmerican, 6

Tags: modular arithmetic , induction , number theory

Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.

2008 China Team Selection Test, 2

Tags: induction , number theory , algebra , sequence , recurrence relation

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

2005 Romania Team Selection Test, 3

Tags: modular arithmetic , floor function , quadratic , induction , functional equation , number theory , algebra

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that \[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \] [i]Călin Popescu[/i]

PEN K Problems, 3

Tags: functional equation , function , induction

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]

2014 Online Math Open Problems, 18

Tags: floor function , probability , induction

We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$. [i]Proposed by Evan Chen[/i]

2024 Thailand October Camp, 5

Tags: combinatorics , induction

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

1994 Irish Math Olympiad, 1

Tags: induction , algebra

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

2014 Contests, 3

Tags: induction , inequalities , graph theory , extremal combinatorics , combinatorics

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

1990 IMO Longlists, 80

Tags: function , induction , algebra

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

2010 Junior Balkan MO, 2

Tags: induction , inequalities , prime factorization , number theory

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2002 Mexico National Olympiad, 3

Tags: modular arithmetic , induction , strong induction , number theory , bijection

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

2014 Bulgaria National Olympiad, 2

Tags: function , induction , limit , geometric series , algebra

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

2008 Junior Balkan Team Selection Tests - Moldova, 8

Tags: induction , algebra

Archipelago consists of $ n$ islands : $ I_1,I_2,...,I_n$ and $ a_1,a_2,...,a_n$ - number of the roads on each island. $ a_1 \equal{} 55$, $ a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1)$, ($ k \equal{} 2,3,...,n$) a) Does there exist an island with 2008 roads? b) Calculate $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n.$

2006 Romania Team Selection Test, 1

Tags: induction , quadratic , algebra

Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \[ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . \] a) Prove that all the terms of the sequence are positive integers. b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$. [i]Valentin Vornicu[/i]

2014 Contests, 1

Tags: induction , symmetry , algorithm , euler , combinatorics

Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef". Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.