Given natural numbers $N,n$, prove that for every nonnegative real number $\alpha\le N$ and every real $x$,
$$\left|\sum_{k=0}^n\frac{\sin((\alpha+k)x)}{N+k}\right|\le\min\left\{(n+1)|x|,\frac1{N|\sin(x/2)|}\right\}.$$
Consider the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$. Put $f(n)=1985n^2+1956n+1960$.
(a) Prove that infinitely many terms $F$ of the Fibonacci sequence satisfy $1989\mid f(F)$.
(b) Does there exist a term $G$ of this sequence such that $1989\mid f(G)+2$?
In the plane, let $ABCD$ be a square of side $2$, with $A,B,C,D$ in cyclic order. The segment $AB$ is moved continuously until it coincides with $CD$, with $A$ ending at $C$ and $B$ at $D$. Let $S$ be the area swept out by $AB$ during the motion. Prove that a motion can be found for which $S\lt 5\pi/6$ (a point swept over twice is counted only once).
Let the polynomial sequence $\{P_n(x)\}_{n=0}^{\infty}$ be defined by $P_0(x)=0$ and
$$P_{n+1}(x)=P_n(x)+\frac{x-P_n^2(x)}2\qquad(n\ge0).$$
Prove that for every $x\in[0,1]$ and every nonnegative integer $n$,
$$0\le\sqrt x-P_n(x)\le\frac2{n+1}.$$
Let $ABCD.A'B'C'D'$ be a rectangular box. Prove that if a line $\Delta$ intersects three of the four lines $AB'$, $BC'$, $CD'$ and $DA'$, then it must intersect or be parallel to the fourth.