2

Part of 2019 All-Russian Olympiad

Problems(2)

Source: All-Russian Math Olympiad 2019

Find minimal natural

n

for which there exist integers

a_1, a_2,\ldots, a_n

such that quadratic trinom

x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)

has at least one integral root.

2019 All Russian MO Grade 11 P2

Source:

Is it true, that for all pairs of non-negative integers

a

b

, the system \begin{align*} \tan{13x} \tan{ay} =& 1 \\ \tan{21x} \tan{by}= & 1 \end{align*} has at least one solution?

algebratrigonometry