Found problems: 2008
2011 BMO TST, 4
Find all prime numbers p such that $2^p+p^2 $ is also a prime number.
1985 Dutch Mathematical Olympiad, 2
Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?
2001 Macedonia National Olympiad, 1
Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.
2013 AIME Problems, 11
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
2013 Princeton University Math Competition, 7
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$.
2008 ITest, 78
Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:
\[\text{How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009)} \ \text{have exactly 256 odd entries?}\]
What is the solution to Wendy's second problem?
2012 AMC 10, 4
When Ringo places his marbles into bags with $6$ marbles per bag, he has $4$ marbles left over. When Paul does the same with his marbles, he has $3$ marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with $6$ marbles per bag. How many marbles will be left over?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
2010 Polish MO Finals, 2
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.
2006 China Girls Math Olympiad, 8
Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: \[-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}\] making the product: \[\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}\] equals to $3^{m}$ where $m$ is a positive integer.
1998 AMC 12/AHSME, 30
For each positive integer $n$, let
\[a_n = \frac {(n + 9)!}{(n - 1)!}.\]
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 7\qquad
\textbf{(E)}\ 9$
1970 IMO Shortlist, 7
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
PEN D Problems, 9
Show that there exists a composite number $n$ such that $a^n \equiv a \; \pmod{n}$ for all $a \in \mathbb{Z}$.
2011 Middle European Mathematical Olympiad, 7
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.
2003 National Olympiad First Round, 10
Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 17
\qquad\textbf{(D)}\ 22
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2010 National Olympiad First Round, 34
Which one divides $2^{2^{2010}}+2^{2^{2009}}+1$?
$ \textbf{(A)}\ 19
\qquad\textbf{(B)}\ 17
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ \text{None}
$
2012 India IMO Training Camp, 2
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:
$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$
$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$
2004 Pan African, 1
Do there exist positive integers $m$ and $n$ such that:
\[ 3n^2+3n+7=m^3 \]
2011 Iran Team Selection Test, 12
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
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 $3.$
1973 USAMO, 2
Let $ \{X_n\}$ and $ \{Y_n\}$ denote two sequences of integers defined as follows:
\begin{align*} X_0 \equal{} 1,\ X_1 \equal{} 1,\ X_{n \plus{} 1} \equal{} X_n \plus{} 2X_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots), \\
Y_0 \equal{} 1,\ Y_1 \equal{} 7,\ Y_{n \plus{} 1} \equal{} 2Y_n \plus{} 3Y_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots).\end{align*}
Prove that, except for the "1", there is no term which occurs in both sequences.
2000 Manhattan Mathematical Olympiad, 2
How many zeroes are there at the end the number $9^{999} + 1$?
2011 All-Russian Olympiad, 3
There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme.
[i]A. Magazinov[/i]
2008 Saint Petersburg Mathematical Olympiad, 5
Given are distinct natural numbers $a$, $b$, and $c$. Prove that
\[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]
2014 Online Math Open Problems, 22
Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$.
[i]Proposed by Michael Kural and Evan Chen[/i]
2010 IMO Shortlist, 4
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$
[list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good.
[*][b](b)[/b] Show that all 2010-good pairs are very good.[/list]
[i]Proposed by Okan Tekman, Turkey[/i]