2024 SSMO Relay Round 5 Problems/Problem 3

Problem

Let $T = TNYWR.$ Let $k$ be the maximum prime factor that divides $T.$ How many values of $x$ satisfy both $\sin x^2+\cos x^2=\sin^2 x + \cos^2 x$ and $-k \le x \le k?$

Solution

Note that $k = 13.$ Now, $\sin^2x+\cos^2x = 1,$ so we are seeking to find the number of solutions to $\sin x^2+\cos x^2 = 1.$ The only solutions are when $x^2 = 2k\pi,\frac{\pi}{2}+2k\pi,$ for integer $k.$ Now, for $-13\le x\le 13,$ we have $x^2\le 169.$ Now, note that $52\pi+0.5\le 169< 54\pi.$ So, there are $54$ possible values for $x^2,$ each giving two unique solutions, except $x^2=0.$ In conclusion, the answer is $54\cdot2-1 = \boxed{107}.$

~SMO_Team

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2024 SSMO Relay Round 5 Problems/Problem 3

Problem

Solution