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: 15460

2002 Tournament Of Towns, 3
Tags: inequalities , induction , 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.

2004 Federal Competition For Advanced Students, P2, 4
Tags: number theory , perfect square , sequence , recurrence relation
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.

2001 Tournament Of Towns, 2
Tags: least common multiple , number theory , induction
Do there exist positive integers $a_1<a_2<\ldots<a_{100}$ such that for $2\le k\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is greater than the least common multiple of $a_k$ and $a_{k+1}$?

2012 USA TSTST, 5
Tags: relatively prime , modular arithmetic , number theory , pigeonhole principle , cigarettes
A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties: (a) $x_0=x$; (b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$; (c) $x_n$ is an integer for some $n$.

1967 Dutch Mathematical Olympiad, 2
Tags: arithmetic sequence , number theory , perfect cube
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.

2023 Myanmar IMO Training, 5
Tags: floor function , function , modular arithmetic , number theory , relatively prime
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2017 Azerbaijan EGMO TST, 1
Tags: number theory , set
$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

2015 Finnish National High School Mathematics Comp, 3
Tags: number theory , factorial , factor , exponential , divide
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

1997 IMO Shortlist, 17
Tags: equation , number theory , prime factorization
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2005 Paraguay Mathematical Olympiad, 4
Tags: number theory
In the expression $t=\frac{8a+ 1}{b}$ where $a, b, t$ are positive integers, where $b <7$. Determine the values of $a$ and$ b$ that allow to obtain $t$ under the established conditions.

2001 Romania Team Selection Test, 1
Tags: relatively prime , number theory , polynomial , algebra
Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.

1987 Bundeswettbewerb Mathematik, 3
Tags: number theory
Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.

1976 IMO Shortlist, 10
Tags: number theory , product , maximization
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

2007 Croatia Team Selection Test, 2
Tags: number theory , floor function , irrational number
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

2014 Iran Team Selection Test, 4
Tags: number theory , quadratic
$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

2022 Durer Math Competition Finals, 1
Tags: number theory , euler s phi function , phi function , sequence
Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

2018 Federal Competition For Advanced Students, P1, 4
Tags: number theory , positive integer
Let $M$ be a set containing positive integers with the following three properties: (1) $2018 \in M$. (2) If $m \in M$, then all positive divisors of m are also elements of $M$. (3) For all elements $k, m \in M$ with $1 < k < m$, the number $km + 1$ is also an element of $M$. Prove that $M = Z_{\ge 1}$. [i](Proposed by Walther Janous)[/i]

1949 Kurschak Competition, 3
Tags: number theory , consecutive
Which positive integers cannot be represented as a sum of (two or more) consecutive integers?

2021 Polish Junior MO First Round, 5
Tags: number theory , sum , product
Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.

2005 Switzerland - Final Round, 4
Tags: greatest common divisor , number theory , gcd
Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.

1981 Bundeswettbewerb Mathematik, 1
Tags: number theory , digit , power
Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

2009 Iran MO (2nd Round), 2
Tags: number theory , induction , inequalities
Let $ a_1<a_2<\cdots<a_n $ be positive integers such that for every distinct $1\leq{i,j}\leq{n}$ we have $ a_j-a_i $ divides $ a_i $. Prove that \[ ia_j\leq{ja_i} \qquad \text{ for } 1\leq{i}<j\leq{n} \]

2015 Iran MO (2nd Round), 3
Tags: algebra , number theory , prime number
Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.

2004 Czech and Slovak Olympiad III A, 4
Tags: number theory
Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.

2006 Princeton University Math Competition, 5
Tags: number theory
Find the largest integer $k$ such that $12^k | 66!$.