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.

AND:
OR:
NO:

Found problems: 15460

2014 Iran Team Selection Test, 4
Tags: quadratic , number theory
$n$ 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.

2015 Purple Comet Problems, 21
Tags: number theory
Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.

2002 Iran MO (3rd Round), 20
Tags: number theory
$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

2002 Croatia National Olympiad, Problem 4
Tags: number theory , diophantine equation
Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.

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.

1995 AMC 12/AHSME, 27
Tags: modular arithmetic , number theory , greatest common divisor , arithmetic series
Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown. \begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular} Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 74$

2008 Argentina National Olympiad, 1
Tags: pigeonhole principle , modular arithmetic , number theory
$ 101$ positive integers are written on a line. Prove that we can write signs $ \plus{}$, signs $ \times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $ 16!$.

2014 Uzbekistan National Olympiad, 1
Tags: number theory
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

1987 Tournament Of Towns, (141) 1
Tags: number theory , sum of squares , perfect square , odd
Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2002 Spain Mathematical Olympiad, Problem 4
Tags: number theory , sum
Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.

2019 Mexico National Olympiad, 1
Tags: perfect power , number theory , number of divisors , function
An integer number $m\geq 1$ is [i]mexica[/i] if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$. Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$. [i]Proposed by Cuauhtémoc Gómez[/i]

2003 BAMO, 1
Tags: factorial , perfect number , number theory , divisor
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$. Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$. Find all perfect numbers greater than $1$ that are also factorials.

1999 Taiwan National Olympiad, 2
Tags: number theory
Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$. Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$.

2011 Greece JBMO TST, 3
Tags: integer , number theory , diophantine equation
Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

2015 Chile National Olympiad, 2
Tags: number theory , prime
Find all prime numbers that do not have a multiple ending in $2015$.

2023 German National Olympiad, 1
Tags: diophantine equation , number theory , algebra
Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

2021 Thailand Online MO, P3
Tags: sequence , number theory
Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and $$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$ for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.

2024 Belarusian National Olympiad, 8.3
Tags: number theory
Do there exist positive integer numbers $a$ and $b$, for which the number $(\sqrt{1+\frac{4}{a}}-1)(\sqrt{1+\frac{4}{b}}-1)$ is rational [i]V. Kamianetski[/i]

2014 Contests, 1
Tags: modular arithmetic , number theory
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$

2023 Thailand Mathematical Olympiad, 1
Tags: number theory
Let $A$ be set of 20 consecutive positive integers, Which sum and product of elements in $A$ not divisible by 23. Prove that product of elements in $A$ is not perfect square

1975 Dutch Mathematical Olympiad, 2
Tags: number theory , consecutive
Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$ Determine the gcd of the elements of $P$.

2000 Belarus Team Selection Test, 6.2
Tags: digit , perfect square , number theory
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

2016 Ecuador NMO (OMEC), 5
Tags: number theory , digit
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

2007 Danube Mathematical Competition, 4
Tags: prime number , number theory
Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

2005 Postal Coaching, 18
Tags: number theory , representation
Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.