UA-UMO 1999 Problem 117

Suppose that the function $f(x)=\tan(a_1x+1)+\ldots+\tan(a_{10}x+1)$ has the period $T>0$ , where $a_1,\ldots,a_{10}$ are positive numbers. Prove that $$T\ge\frac\pi{10}\min\left\{\frac1{a_1},\ldots,\frac1{a_{10}}\right\}.$$