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

2019 AIME Problems, 14
Tags: number theory
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.

2021 Taiwan TST Round 2, 2
Tags: sequence , number theory
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

2019 Grand Duchy of Lithuania, 4
Tags: perfect square , number theory
Determine all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a square of an integer.

2006 Tournament of Towns, 6
Tags: combinatorics , number theory
Let us say that a deck of $52$ cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is a) divisible by $12!$ (3) b) divisible by $13!$ (5)

2022 BMT, 26
Tags: number theory
Compute the number of positive integers $n$ less than $10^8$ such that at least two of the last five digits of $$ \lfloor 1000\sqrt{25n^2 + \frac{50}{9}n + 2022}\rfloor$$ are $6$. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^7\right\rfloor \right)$.

1994 ITAMO, 2
Tags: diophantine equation , number theory
solve this diophantine equation y^2 = x^3 - 16

2015 China Northern MO, 3
Tags: number theory , prime factorization
If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$

2018 lberoAmerican, 4
Tags: number theory , greatest common divisor
A set $X$ of positive integers is said to be [i]iberic[/i] if $X$ is a subset of $\{2, 3, \dots, 2018\}$, and whenever $m, n$ are both in $X$, $\gcd(m, n)$ is also in $X$. An iberic set is said to be [i]olympic[/i] if it is not properly contained in any other iberic set. Find all olympic iberic sets that contain the number $33$.

2003 AMC 10, 16
Tags: greatest common divisor , modular arithmetic , number theory , algorithm , euclidean algorithm
What is the units digit of $ 13^{2003}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2018 Puerto Rico Team Selection Test, 3
Tags: number theory , divide , divisible
Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

2021 CMIMC, 2.7
Tags: algebra , number theory
For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$ [i]Proposed by Kyle Lee[/i]

2018 Dutch IMO TST, 3
Tags: floor function , number theory , sum , perfect square
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

1998 Moldova Team Selection Test, 7
Tags: geometry , number theory , perimeter
Find all triangles with integer sidelenghts such that their perimeter and area are equal.

2011 Canadian Mathematical Olympiad Qualification Repechage, 8
Tags: induction , number theory , greatest common divisor
Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine , diophantine equation
Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

2017 Moldova Team Selection Test, 11
Tags: number theory
Find all ordered pairs of nonnegative integers $(x,y)$ such that \[x^4-x^2y^2+y^4+2x^3y-2xy^3=1.\]

2009 China Northern MO, 8
Tags: number theory , sum of digits , divide
Find the smallest positive integer $N$ satisfies : 1 . $209$│$N$ 2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )

2014 China Team Selection Test, 6
Tags: induction , number theory
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

2008 Indonesia TST, 3
Tags: greatest common divisor , gcd , number theory
Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).

2007 Denmark MO - Mohr Contest, 2
Tags: last digit , number theory
What is the last digit in the number $2007^{2007}$?

1996 Dutch Mathematical Olympiad, 5
Tags: greatest common divisor , number theory , perfect square
For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ . Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.

2014 All-Russian Olympiad, 3
Tags: number theory
Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least $15$. [i]A. Golovanov[/i]

1996 Irish Math Olympiad, 3
Tags: number theory
Suppose that $ p$ is a prime number and $ a$ and $ n$ positive integers such that: $ 2^p\plus{}3^p\equal{}a^n$. Prove that $ n\equal{}1$.

PEN A Problems, 73
Tags: number theory , divisibility theory
Determine all pairs $(n,p)$ of positive integers such that [list][*] $p$ is a prime, $n>1$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

1978 Romania Team Selection Test, 1
Tags: modular arithmetic , number theory
Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $