Found problems: 1362
2013 India Regional Mathematical Olympiad, 2
Find all $4$-tuples $(a,b,c,d)$ of natural numbers with $a \le b \le c$ and $a!+b!+c!=3^d$
2008 South East Mathematical Olympiad, 4
Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions :
(i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$;
(ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$.
(1) Find the value of $f(10)$;
(2) Determine the remainder of $f(2008)$ upon division by $13$.
1976 IMO Longlists, 19
For a positive integer $n$, let $6^{(n)}$ be the natural number whose decimal representation consists of $n$ digits $6$. Let us define, for all natural numbers $m$, $k$ with $1 \leq k \leq m$
\[\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} .\]
Prove that for all $m, k$, $ \left[\begin{array}{ccc}m\\ k\end{array}\right] $ is a natural number whose decimal representation consists of exactly $k(m + k - 1) - 1$ digits.
2004 Korea - Final Round, 2
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
2006 Australia National Olympiad, 1
Find all positive integers $m$ and $n$ such that $1 + 5 \cdot 2^m = n^2$.
2004 Tournament Of Towns, 5
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.
2019 Romania Team Selection Test, 1
Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.
2014 Indonesia MO, 1
A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.
1984 IMO Longlists, 55
Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers.
$(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number.
$(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.
2003 China Team Selection Test, 3
Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.
2007 Estonia Math Open Senior Contests, 1
Let $ a_n \equal{} 1 \plus{} 2 \plus{} ... \plus{} n$ for every $ n \ge 1$; the numbers $ a_n$ are called triangular. Prove that if $ 2a_m \equal{} a_n$ then $ a_{2m \minus{} n}$ is a perfect square.
2006 Estonia Math Open Junior Contests, 9
A computer outputs the values of the expression $ (n\plus{}1) . 2^n$ for $ n \equal{} 1, n \equal{} 2, n \equal{} 3$, etc. What is the largest number of consecutive values that are perfect squares?
2014 South East Mathematical Olympiad, 1
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 $p$
2013 Benelux, 4
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\]
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]
2002 Silk Road, 4
Observe that the fraction $ \frac{1}{7}\equal{}0,142857$ is a pure periodical decimal with period $ 6\equal{}7\minus{}1$,and in one period one has $ 142\plus{}857\equal{}999$.For $ n\equal{}1,2,\dots$ find a sufficient and necessary
condition that the fraction $ \frac{1}{2n\plus{}1}$ has the same properties as above and find two such fractions other than $ \frac{1}{7}$.
2006 International Zhautykov Olympiad, 1
Solve in positive integers the equation
\[ n \equal{} \varphi(n) \plus{} 402 ,
\]
where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.
2014 Indonesia MO Shortlist, N4
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.
2003 Baltic Way, 2
Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.
2011 South East Mathematical Olympiad, 2
If positive integers, $a,b,c$ are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find $a,b,$ and $c$
2006 Federal Competition For Advanced Students, Part 1, 1
Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$.
For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?
2008 Canada National Olympiad, 4
Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which
\[ (f(n))^p \equiv n\quad {\rm mod}\; f(p)
\]
for all $ n \in {\bf N}$ and all prime numbers $ p$.
2011 Finnish National High School Mathematics Competition, 4
Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is $2011.$
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
1989 India National Olympiad, 5
For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which
(a) $ A(n)$ is an even number,
(b) $ A(n)$ is a multiple of $ 4$.
2010 Kyrgyzstan National Olympiad, 2
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.