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3 problems match “intermediate value theorem”. Clear

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1. Number of solutions of a trig-root equation ●●○○○ 20 ptsFind the number of real solutions $x$ of the equation $\sin\left(\frac{\pi\sqrt x}{4}\right)+\cos\left(\frac{\pi\sqrt{2-x}}{4}\right)=\sqrt 2$.
2. Maximum integer k in a derivative inequality ●●●●○ 100 ptsLet $f(x)=e^{x}-x-2$. For $x>0$, the inequality $(x-k)f'(x)+x+1>0$ always holds, where $k$ is an integer. Find the maximum value of $k$.
3. Real roots of a sine-power equation ●●●●○ 100 ptsFind the number of real roots of the equation $\sin(2013\pi x)=x^{2013}$.