Prove that every prime of the form $4k+1$ is the hypotenuse of a rectangular triangle with integer sides.

Suppose that $p|(2t^2-1)$ and $p^2|(2st+1)$. Prove that $p^2|(s^2+t^2-1)$

Circles $\Omega_a$ and $\Omega_b$ are externally tangent at $D$, circles $\Omega_b$ and $\Omega_c$ are externally tangent at $E$, circles $\Omega_a$ and $\Omega_c$ are externally tangent at $F$. Let $P$ be an arbitrary point on $\Omega_a$ different from $D$ and $F$. Extend $PD$ to meet $\Omega_b$ again at $B$, extend $BE$ to meet $\Omega_c$ again at $C$ and extend $CF$ to meet $\Omega_a$ again at $A$. Show that $PA$ is a diameter of circle $\Omega_a$.

Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$

Let $a$, $b$, and $c$ be real numbers such that $abc=1$. prove that $\frac{1+a+ab}{1+b+ab}$ +$\frac{1+b+bc}{1+c+bc}$ + $\frac{1+c+ac}{1+a+ac}$ $>=3$

Let $N=4^KL$ where $L\equiv\ 7\pmod 8$. Prove that $N$ cannot be written as a sum of 3 squares