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14 problems match “symmetry”. Clear

Angle NCB in a triangle ●●●○○ 50 ptsIn $\triangle ABC$ , $\angle ABC=50^{\circ}$ and $\angle ACB=20^{\circ}$ . Point $N$ is inside $\triangle ABC$ with $\angle NAB=40^{\circ}$ and $\angle NBC=30^{\circ}$ . Find the measure of $\angle NCB$ in degrees.
Product after sum-replacement ●●●○○ 50 ptsOn a board there are 2023 distinct real numbers. Each number is simultaneously replaced by the sum of all the other numbers. After this replacement, the 2023 numbers on the board are the same as at the beginning (as a set of numbers). What value can the product of all the numbers on the board take?
Unique solution with a parameter ●●●○○ 50 ptsFind all values of the parameter $a$ for which the equation $a x^{2}+\sin ^{2} x=a^{2}-a$ has a unique solution. Give the value of $a$ .
Distinct values of a symmetric function ●●●●○ 100 ptsGiven a function $f$ for which $f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$ for all real $x$ , what is the largest number of different values that can appear in the list $f(0), f(1), f(2), \ldots, f(999)$ ?