Found problems: 1766
2010 Contests, 1
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?
1992 Hungary-Israel Binational, 4
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Prove that $F_{n-1}F_{n}F_{n+1}L_{n-1}L_{n}L_{n+1}(n \geq 2)$ is not a perfect square.
2007 Iran MO (3rd Round), 4
Find all integer solutions of \[ x^{4}\plus{}y^{2}\equal{}z^{4}\]
2022 German National Olympiad, 4
Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation
\[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]
1989 Turkey Team Selection Test, 1
Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that
[list=i]
[*] $f(m,m)=m$
[*] $f(m,k) = f(k,m)$
[*] $f(m, m+k) = f(m,k)$[/list] , for each $m,k \in \mathbb{Z}^+$.
2012 Paraguay Mathematical Olympiad, 4
Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$.
($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is).
2002 India IMO Training Camp, 21
Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.
2001 Junior Balkan Team Selection Tests - Romania, 4
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
2009 Indonesia TST, 3
Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.
2013 Iran MO (3rd Round), 5
$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow:
$L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$
[i]a)[/i] For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points)
[i]b)[/i] Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant.
Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix}
a & & &x \in A \\
b& & &x \in B \\
c& & & x \in C
\end{matrix}\right.$ . (7 points)
[i]c)[/i] Prove that $a+b+c = -3$. (4 points)
[i]d)[/i] Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points)
[i]e)[/i] Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points)
(${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)
2010 Contests, 1
Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.
2011 Argentina Team Selection Test, 4
Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.
1998 Balkan MO, 4
Prove that the following equation has no solution in integer numbers: \[ x^2 + 4 = y^5. \] [i]Bulgaria[/i]
1996 All-Russian Olympiad, 5
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
2010 Iran MO (3rd Round), 1
suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)
2005 Turkey MO (2nd round), 4
Find all triples of nonnegative integers $(m,n,k)$ satisfying $5^m+7^n=k^3$.
2011 India National Olympiad, 2
Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ and $n=a+b+c.$
$(i)$ Show that all but a finite number of natural numbers are faithful.
$(ii)$ Find the sum of all natural numbers which are not faithful.
2005 Silk Road, 1
Let $n \geq 2$ be natural number.
Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.
2010 Contests, 3
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that
\[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\]
where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
2007 Indonesia TST, 4
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.
2012 Romania Team Selection Test, 1
Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.
2005 Morocco TST, 1
Find all the positive primes $p$ for which there exist integers $m,n$ satisfying :
$p=m^2+n^2$ and $m^3+n^3-4$ is divisible by $p$.
2013 All-Russian Olympiad, 1
Does exist natural $n$, such that for any non-zero digits $a$ and $b$ \[\overline {ab}\ |\ \overline {anb}\ ?\]
(Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.)
[i]V. Senderov[/i]
2003 India IMO Training Camp, 10
Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.
2013 China Team Selection Test, 2
Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying:
$(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $;
$(2)$ For any positive integer $n$, $a_n<1.01^n K$;
$(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.