An arithmetic progression has 1987 terms, first term $u_1=\pi/1987$ and common difference $d=\pi/3974$. Determine
$$S=\sum\cos(\pm u_1\pm u_2\pm\cdots\pm u_{1987}),$$
where the sum includes every possible independent choice of a plus or minus sign before each term.
Let $(x_n)_{n\ge0}$ and $(y_n)_{n\ge0}$ be defined by
$$x_0=365,\qquad x_{n+1}=x_n(x_n^{1986}+1)+1622,$$
$$y_0=16,\qquad y_{n+1}=y_n(y_n^3+1)-1952\quad(n\ge0).$$
Prove that $|x_n-y_k|\gt 0$ for every pair of natural-number indices $n,k$.
In the plane, $n$ distinct lines intersect pairwise but do not all pass through one point. Prove that there is a point incident with exactly two of these lines.
Let $a_i\gt 0$ for $i=1,\ldots,n$, where $n\ge2$, and put $S=\sum_{k=1}^n a_k$. Prove that
$$\sum_{k=1}^n\frac{a_k^{2^m}}{(S-a_k)^{2^t-1}}\ge\frac{S^{1+2^m-2^t}}{(n-1)^{2^t-1}n^{2^m-2^t}}$$
for all integers $m\ge t\ge0$. When does equality hold?
A function $f$ is defined and differentiable on $[0,+\infty)$. For every $x\ge0$, it satisfies:
(a) $|f(x)|\le5$;
(b) $f(x)f'(x)\ge\sin x$.
Does $\lim_{x\to+\infty}f(x)$ exist?
In three-dimensional space, five rays $Ox_k$ ($k=1,\ldots,5$) have a common endpoint $O$. Let $\alpha$ be the smallest angle between any pair $Ox_i,Ox_j$ with $i\ne j$. Prove that $\alpha\le90^\circ$.