Found problems: 2008
2013 Brazil Team Selection Test, 2
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
2013 ELMO Problems, 3
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
1991 India National Olympiad, 1
Find the number of positive integers $n$ for which
(i) $n \leq 1991$;
(ii) 6 is a factor of $(n^2 + 3n +2)$.
2001 IMO Shortlist, 3
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
PEN D Problems, 21
Determine the last three digits of \[2003^{2002^{2001}}.\]
2013 China Girls Math Olympiad, 4
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:
(i) $a,b\in\{1,2,\ldots,2013\}$;
(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.
2000 239 Open Mathematical Olympiad, 5
Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.
2008 National Olympiad First Round, 14
What is the last three digits of $49^{303}\cdot 3993^{202}\cdot 39^{606}$?
$
\textbf{(A)}\ 001
\qquad\textbf{(B)}\ 081
\qquad\textbf{(C)}\ 561
\qquad\textbf{(D)}\ 721
\qquad\textbf{(E)}\ 961
$
2003 France Team Selection Test, 1
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
2000 Tuymaada Olympiad, 3
Polynomial $ P(t)$ is such that for all real $ x$,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
1969 IMO Shortlist, 43
$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$
2003 China Team Selection Test, 2
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.
2013 Waseda University Entrance Examination, 2
For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$.
Answer the following questions.
(1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$
(2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$.
(3) Show that $a_n$ is not a multiple of $5$ for any $n$.
(4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.
1984 Balkan MO, 3
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.
1998 National Olympiad First Round, 6
Find the number of primes$ p$, such that $ x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right)$ has no three distinct integer roots in $ \left[0,\left. p\right)\right.$.
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
2007 Princeton University Math Competition, 1
Find the last three digits of
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]
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 $k$.
2005 National Olympiad First Round, 6
Which of the following divides $3^{3n+1} + 5^{3n+2}+7^{3n+3}$ for every positive integer $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 53
$
2010 Contests, 1
[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number?
[b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.
1978 Canada National Olympiad, 1
Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?
2009 China Girls Math Olympiad, 8
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
2000 JBMO ShortLists, 10
Prove that there are no integers $x,y,z$ such that
\[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]
1997 Putnam, 5
Let us define a sequence $\{a_n\}_{n\ge 1}$. Define as follows:
\[ a_1=2\text{ and }a_{n+1}=2^{a_n}\text{ for }n\ge 1 \]
Show this :
\[ a_{n}\equiv a_{n-1}\pmod n \]
2005 Georgia Team Selection Test, 6
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:
1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;
2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$;
Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
2008 AIME Problems, 15
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.