A computer generates even integers half of the time and another computer generates even integers a third of the time. If $a_i$ and $b_i$ are the integers generated by the computers, respectively, at time $i$, what is the probability that $a_1b_1 +a_2b_2 +\cdots + a_kb_k$ is an even integer.

Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.

If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]

Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$, (i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$. If $123\star 456 = 789$, find $246\star 135$.

There are exactly $120$ Twitter subscribers from National Science High School. Statistics show that each of $10$ given celebrities has at least $85$ followers from National Science High School. Prove that there must be two students such that each of the $10$ celebrities is being followed in Twitter by at least one of these students.