This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1782

2002 Tournament Of Towns, 6
Tags: induction , inequalities , number theory
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

PEN K Problems, 33
Tags: function , induction , strong induction , algebra , functional equation
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]

2011 Today's Calculation Of Integral, 752
Tags: calculus , integration , induction , calculus computations
Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$

1997 Polish MO Finals, 1
Tags: induction , number theory
The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = 0$, $a_n = a_{[n/2]} + (-1)^{n(n+1)/2}$. Show that for any positive integer $k$ we can find $n$ in the range $2^k \leq n < 2^{k+1}$ such that $a_n = 0$.

2014 District Olympiad, 4
Tags: function , induction , algebra
Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that \[ f(x+3f(y))=f(x)+f(y)+2y \quad \forall x,y\in \mathbb{Q}\]

2012 Putnam, 4
Tags: limit , induction , calculus , integration , logarithm , inequalities
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

2011 ELMO Shortlist, 3
Tags: induction , complex number , imaginary numbers , algebra
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

2013 USAMTS Problems, 2
Tags: induction
Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.

2014 Brazil National Olympiad, 4
Tags: polynomial , induction , algebra
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as \[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\] Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.

2010 All-Russian Olympiad, 2
Tags: pigeonhole principle , induction , combinatorics
Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.

2010 Indonesia TST, 3
Tags: function , induction , number theory
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

2006 Vietnam Team Selection Test, 3
Tags: induction , number theory
The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.

1991 USAMO, 3
Tags: induction , modular arithmetic , strong induction , relatively prime , prime number , number theory
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

2005 Korea - Final Round, 6
Tags: induction , graph theory , number theory
A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$

1992 IMO Longlists, 8
Tags: function , induction , algebra , functional equation
Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation \[f(f(x)) + af(x) = b(a + b)x.\] Prove that there exists a unique solution of this equation.

1971 Bundeswettbewerb Mathematik, 3
Tags: induction
Between any two cities of a country there is only one one-way road. Show that there is a city from that every other city can be reached directly or by going over only one intermediate city. [hide] I'm sure it was posted before but couldn't find it. [/hide]

1997 China National Olympiad, 3
Tags: induction , inequalities
Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$. Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.

2009 Balkan MO, 4
Tags: function , induction , algebra
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

2007 France Team Selection Test, 2
Tags: function , induction , functional equation , algebra
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

2006 APMO, 2
Tags: calculus , integration , ratio , induction , algorithm , algebra
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.

1988 Romania Team Selection Test, 4
Tags: induction , inequalities
Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \] [i]Viorel Vajaitu[/i]

1993 China National Olympiad, 6
Tags: function , induction , inequalities
Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.

1995 Baltic Way, 9
Tags: induction , algebra , logarithm
Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

1975 Canada National Olympiad, 2
Tags: induction
A sequence of numbers $ a_1, a_2, a_3, ...$ satisfies (i) $ a_1 \equal{} \frac{1}{2}$ (ii) $ a_1\plus{}a_2 \plus{} \cdots \plus{} a_n \equal{} n^2 a_n \ (n \geq 1)$ Determine the value of $ a_n \ (n \geq 1)$.

2003 Bulgaria National Olympiad, 3
Tags: induction , algebra
Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and \[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\] find all integers $k$ such that every term of the sequence is a perfect square.