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

2004 239 Open Mathematical Olympiad, 6
Tags: number theory , least common multiple
Given distinct positive integers $a_1,\,a_2,\,\dots,a_n$. Let $b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)$. Prove that the least common multiple $[b_1,b_2,\dots,b_n]$ is divisible by $(n-1)!.$

2015 Indonesia MO Shortlist, N6
Tags: divisible , number theory , set , subset
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.

2012 Dutch IMO TST, 3
Tags: number theory , integer
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

2022 239 Open Mathematical Olympiad, 8
Tags: combinatorics , number theory
There are several rational numbers written on a board. If the numbers $x{}$ and $y{}$ are present on the board, we can add the number $(x+y)/(1-xy)$ to it. Prove that there is a rational number that cannot ever appear on the board.

2021 Iran Team Selection Test, 2
Tags: function , number theory
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

2014 BMT Spring, 10
Tags: number theory
A [i]unitary [/i] divisor d of a number $n$ is a divisor $n$ that has the property $\gcd (d, n/d) = 1$. If $n = 1620$, what is the sum of all of the unitary divisors of $d$?

2021 Thailand Mathematical Olympiad, 5
Tags: number theory
Determine all triples $(p,m,k)$ of positive integers such that $p$ is a prime number, $m$ and $k$ are odd integers, and $m^4+4^kp^4$ divides $m^2(m^4-4^kp^4)$.

1989 Canada National Olympiad, 3
Tags: logarithm , number theory , algebra , binomial theorem , divisibility tests , modular arithmetic
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

2017 South East Mathematical Olympiad, 7
Tags: number theory , perfect square
Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$

2010 Argentina Team Selection Test, 5
Tags: algebra , number theory , polynomial
Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$. Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.

2009 Singapore Junior Math Olympiad, 3
Tags: prime , digit , number theory
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

2015 Latvia Baltic Way TST, 13
Tags: floor function , prime , number theory
Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ? $[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.

2010 Contests, 4
Tags: number theory
Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

1980 Dutch Mathematical Olympiad, 4
Tags: number theory , combinatorics
In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property: Any amount of money of $n$ ducats ($n$ positive and integer) where $n \le N$ may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for $N$. Determine those six values and provide proof that all conditions have been met." Solve the problem of those officials

2004 Iran Team Selection Test, 6
Tags: polynomial , limit , algebra , number theory
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2000 Greece JBMO TST, 3
Tags: number theory , polynomial , integer
Find $a\in Z$ such that the equation $2x^2+2ax+a-1=0$ has integer solutions, which should be found.

2014 Romania National Olympiad, 3
Tags: number theory
Find the smallest integer $n$ for which the set $A = \{n, n +1, n +2,...,2n\}$ contains five elements $a<b<c<d<e$ so that $$\frac{a}{c}=\frac{b}{d}=\frac{c}{e}$$

1995 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , perfect square
Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

1998 All-Russian Olympiad Regional Round, 11.8
Tags: number theory , combinatorics , inequalities , combinatorial number theory , algebra
A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

2023 Princeton University Math Competition, A5 / B7
Tags: number theory
You play a game where you and an adversarial opponent take turns writing down positive integers on a chalkboard; the only condition is that, if $m$ and $n$ are written consecutively on the board, $\gcd(m,n)$ must be squarefree. If your objective is to make sure as many integers as possible that are strictly less than $404$ end up on the board (and your opponent is trying to minimize this quantity), how many more such integers can you guarantee will eventually be written on the board if you get to move first as opposed to when your opponent gets to move first?

2007 Irish Math Olympiad, 4
Tags: number theory
Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit.

2004 India IMO Training Camp, 2
Tags: number theory , induction
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

2001 All-Russian Olympiad Regional Round, 9.8
Tags: number theory , perfect square , digit
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

1999 Portugal MO, 1
Tags: number theory , digit
A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

2017 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: set , number theory , arithmetic progression
Let $S$ be a set of $6$ positive real numbers such that $\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$ Prove that if we sort these numbers in ascending order, then they form an arithmetic progression