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1. Real roots of an exponential-cosine equation ●●○○○ 20 ptsFind the number of real roots of the equation $e^{\cos x} - e^{-\cos x} - 4 = 0$.
2. Roots by completing the square ●●○○○ 20 ptsFind the roots of $2x^2 - 7x + 3 = 0$ by completing the square.
3. Roots of a symmetric odd function ●●●○○ 50 ptsLet $y=f(x)$ be an odd function defined on $\mathbf{R}$ whose graph is symmetric about the line $x=1$. When $0<x\leq 1$, $f(x)=\log_3 x$. Find the sum of all real roots of the equation $f(x)=-\frac{1}{3}+f(0)$ in the interval $(0,10)$.
4. Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$. Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$. Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$.
5. Real roots of a sine-power equation ●●●●○ 100 ptsFind the number of real roots of the equation $\sin(2013\pi x)=x^{2013}$.
6. Longest chain of quadratic equations ●●●●○ 100 ptsQuadratic equations are solved one after another. Each equation has leading coefficient 1, of the form $x^2+p_jx+q_j=0$. Whenever an equation has two distinct real roots, the next equation is formed as follows: its constant term is the larger root of the current equation, and its linear coefficient is the smaller root of the current equation. The process stops at the first equation that does not have two distinct real roots. Find the maximum possible number of quadratic equations in such a sequence.