parmenides51 wrote: p1. Let $a$ and $ b$ be different positive real numbers so that $a +\sqrt{ab}$ and $b +\sqrt{ab}$ are rational numbers. Prove that $a$ and $ b$ are rational numbers. p2. Find the number of ordered pairs of natural numbers $(a, b, c, d)$ that satisfy $$ab + bc + cd + da = 2016.$$ p3. For natural numbers $k$, we say a rectangle of size $1 \times k$ or $k\times 1$ as strips. A rectangle of size $2016 \times n$ is cut into strips of all different sizes . Find the largest natural number $n\le 2016$ so we can do that. Note: $1\times k$ and $k \times 1$ strips are considered the same size. p4. Let $PA$ and $PB$ be the tangent of a circle $\omega$ from a point $P$ outside the circle. Let $M$ be any point on $AP$ and $N$ is the midpoint of segment $AB$. $MN$ cuts $\omega$ at $C$ such that $N$ is between $M$ and $C$. Suppose $PC$ cuts $\omega$ at $D$ and $ND$ cuts $PB$ at $Q$. Prove $MQ$ is parallel to $AB$. p5. Given a triple of different natural numbers $(x_0, y_0, z_0)$ that satisfy $x_0 + y_0 + z_0 = 2016$. Every $i$-hour, with $i\ge 1$, a new triple is formed $$(x_i,y_i, z_i) = (y_{i-1} + z_{i-1} x_{i-1}, z_{i-1} + x_{i-1}-y_{i-1}, x_{i-1} + y_{i-1}- z_{i-1})$$Find the smallest natural number $n$ so that at the $n$-th hour at least one of the found $x_n$, $y_n$, or $z_n$ are negative numbers. 1. Put $a+\sqrt{ab}=x$ with x is a rational number, then we have $a^{2}+a\sqrt{ab}+a\sqrt{ab}+ab=x^{2}$ so $(a(a+\sqrt{ab}+b+{ab})=x^{2}$ and $a+\sqrt{ab}$, $b+\sqrt{ab}$, $x^{2}$ are rational numbers Hence a is a rational number