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\title{TJUSAMO Practice 10 - Random Funness!}
\author{PDiao06}
\date{\today}

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\begin{enumerate}

\item Prove that if $n$  is a positive integer such that $2n +1$ and 
$3n + 1$ are both perfect squares, then $n$ is divisible by $40$.

\item (Ross) What are all integer polynomials $p(x)$ that only take on
prime values when evaluated at every integer.

\item (USAMO 1974) $p(x)$ is a polynomial with integral coefficients. 
Show that there are no solutions to the equations $p(a) = b, p(b) = c, 
p(c) = a$, with $a, b, c$ distinct integers. 

\item (USAMO 1980) Find the maximum possible number of three term 
arithmetic progressions in a monotone sequence of $n$ distinct reals. 

\item (IMO 1977) Let $a$ and $b$ be positive integers. When $a^2 + b^2$ 
is divided by $a + b$, the quotient is $q$ and the remainder is $r$. 
Find all ordered pairs $(a, b)$ such that $q^2 + r = 1977$.

\end{enumerate}

\par I guess by random fun I meant NUMBER THEORY!  

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