Found problems: 15460
1978 IMO Shortlist, 5
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
2019 Serbia Team Selection Test, P5
Solve the equation in nonnegative integers:\\
$2^x=5^y+3$
2024 Romania Team Selection Tests, P1
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
1997 Vietnam National Olympiad, 3
Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying:
(i) $ f(1) \equal{} 1$
(ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.
2019 BMT Spring, 8
Let $(k_i)$ be a sequence of unique nonzero integers such that $x^2- 5x + k_i$ has rational solutions. Find the minimum possible value of $$\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}$$
2012 China Northern MO, 8
Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that :
(1) $(a+1, \frac{a^p+1}{a+1}) = p$
(2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$.
(3) $p!|(a +1) $.
2017 Hanoi Open Mathematics Competitions, 3
Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there?
(A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.
2020 Tournament Of Towns, 2
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.)
Boris Frenkin
2014 Iran MO (3rd Round), 6
Prove that there are 100 natural number $a_1 < a_2 < ... < a_{99} < a_{100}$ ( $ a_i < 10^6$) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ?
$A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}$
$2A = \{2a_i \vert 1\leq i\leq 100\}$
$A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}$
$A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}$
$2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}$
(20 ponits )
2016 Japan Mathematical Olympiad Preliminary, 2
For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.
2020 Korea Junior Math Olympiad, 1
The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ · · · +2000
Find all of the following positive integers that cannot be the value of n.
2013 USA TSTST, 5
Let $p$ be a prime. Prove that any complete graph with $1000p$ vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by $p$.
2005 Portugal MO, 4
A natural number $n$ is said to be [i]abundant [/i] if the sum of its divisors is greater than $2n$. For example, $18$ is abundant because the sum of its divisors, $1 + 2 + 3 + 6 + 9 + 18$, is greater than $36$. Write every even number greater than $46$ as a sum of two numbers abundant.
2021 Durer Math Competition Finals, 14
How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?
2003 Tournament Of Towns, 4
In the sequence $00, 01, 02, 03,\ldots , 99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19, 39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?
1986 IMO Longlists, 64
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
2013 Peru MO (ONEM), 2
The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.
Clarification: gcd stands for greatest common divisor.
2002 India IMO Training Camp, 8
Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$.
[list]
[b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$
[b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]
2018 Romanian Master of Mathematics Shortlist, N2
Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.)
[i]Proposed by Serbia[/i]
1974 IMO, 3
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
2022 Durer Math Competition Finals, 10
The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?
2001 Bosnia and Herzegovina Team Selection Test, 2
For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$.
Prove that $xyz\geq 3600$
2007 QEDMO 4th, 4
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$
2021 BMT, 9
Let $p=101.$ The sum
\[\sum_{k=1}^{10}\frac1{\binom pk}\]
can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$
2011 Romanian Masters In Mathematics, 1
Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$).
Prove the following two claims:
i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$;
ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$.
[i](Romania) Dan Schwarz[/i]