SCHWEITZER 1982 Problem 6

For every positive $\alpha$ , natural number $n$ , and at most $\alpha n$ points $x_i$ , construct a trigonometric polynomial $P(x)$ of degree at most $n$ for which $$P(x_i) \leq 1, \int_0^{2 \pi} P(x)dx=0,\ \textrm{and}\ \max P(x) > cn\ ,$$ where the constant $c$ depends only on $\alpha$ .

G. Halasz