Found problems: 15460
2007 Romania Team Selection Test, 1
Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.
2017 China Team Selection Test, 5
Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.
2019 LIMIT Category B, Problem 3
Let $d_1,d_2,\ldots,d_k$ be all factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+\ldots+d_k=72$ then $\frac1{d_1}+\frac1{d_2}+\ldots+\frac1{d_k}$ is
$\textbf{(A)}~\frac{k^2}{72}$
$\textbf{(B)}~\frac{72}k$
$\textbf{(C)}~\frac{72}n$
$\textbf{(D)}~\text{None of the above}$
OMMC POTM, 2023 7
Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$.
[i]Proposed by Evan Chang (squareman), USA[/i]
2010 CHMMC Winter, 2
The largest prime factor of $199^4 + 4$ has four digits. Compute the second largest prime factor.
2023 Singapore Junior Math Olympiad, 4
Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.
2000 Greece National Olympiad, 2
Find all prime numbers $p$ such that $1 +p+p^2 +p^3 +p^4$ is a perfect square.
1978 Kurschak Competition, 1
$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.
2015 Tournament of Towns, 1
A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer.
[i]($3$ points)[/i]
2005 Iran MO (3rd Round), 2
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]
1979 IMO Longlists, 66
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
2024 OlimphÃada, 1
Find all pairs of positive integers $(m,n)$ such that
$$lcm(1,2,\dots,n)=m!$$
where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.
1990 India National Olympiad, 2
Determine all non-negative integral pairs $ (x, y)$ for which
\[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]
2024 Nepal TST, P1
Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares.
[i](Proposed by Orestis Lignos, Greece)[/i]
2019 China Team Selection Test, 2
Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?
2006 Singapore Senior Math Olympiad, 1
Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.
VMEO III 2006, 10.2
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
2019 Korea National Olympiad, 4
Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.
2018 Romania National Olympiad, 2
Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
1987 Iran MO (2nd round), 1
Solve the following system of equations in positive integers
\[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]
2020 Brazil Cono Sur TST, 4
Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.
2016 China Northern MO, 3
$m(m>1)$ is an intenger, define $(a_n)$:
$a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$.
If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$.
Note: $\varphi(n)$ means Euler Function.
2021 China Team Selection Test, 4
Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that
$$f(a) \equiv g(a+m_p) \pmod p$$
holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that
$$f(x)=g(x+r).$$
2012 Junior Balkan Team Selection Tests - Romania, 3
Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.
2006 BAMO, 2
Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers.
(a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.
(b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.