Several trigonometry problems

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6. At what values ​​of $a$ the system of equation
   $$\begin{cases} x^2 - 4y^2 = 1 \\ ax + y = b \end{cases}$$
   has real solutions for any values ​​of $b$?

7. 24 chess players took part in the international tournament in Palma de Mallorca. To fulfill the grandmaster norm, you had to score at least $14\frac{1}{2}$ points. How many chess players could fulfill this norm? (In chess tournaments, the winner receives one point; if the game ends in a draw, each player receives $\frac{1}{2}$ points).

8. Solve the equation
   $$\sqrt{45x^2 - 30x + 1} = 7 + 6x - 9x^2.$$
   (Only non-negative values ​​of the radical are considered).

9. All plane angles of a trihedral angle are equal to $\alpha$. Point $M$ is located inside the angle at the same distance from all its faces. Find this distance if the distance from the point $M$ to the vertex of the angle is $d$.

10. Find the smallest value of an expression
    $$\frac{1 + x^4}{1 + x^2}.$$

ANSWERS TO THE CROSSWORD (see page 31)
Vertical: 1. Vampire.
2. Nuance. 3. Opera. 4. Carbon. 5. Kober.
9. Pine. 10. Weight. 12. Scale.
15. Krulier. 16. Flood. 18. Auntie. 19. Ninny.

Several trigonometry problems

1. Given a triangle $ABC$ with side $a, b, c$. Prove that

1. if $C = \frac{\sqrt{ab}}{a - \sqrt{ab} + b}$, then
   $$\frac{tg A}{tg B} = -\frac{a}{b} \sqrt{\frac{a}{b}}$$;
2. if $tg C, tg A, tg B$ form a geometric progression, then its value is equal to $b^2/c^2$, and $A \leqslant \frac{\pi}{3}$;
3. if $tg \frac{C}{2}, tg \frac{A}{2}, tg \frac{B}{2}$ form a geometric progression, then its value is $b/c$, and $A \leqslant \frac{\pi}{3}$.

2. Let $tg \alpha, tg \beta, tg \gamma$ be the roots of the equation $ax^3 + 3bx^2 + 3cx + d = 0$. Write an equation having roots $tg \alpha$ tg $(\beta + \gamma), tg \beta$ tg $(\alpha + \gamma), tg \gamma$ tg $(\alpha + \beta)$.
3. Let $\alpha + \beta + \gamma$ be a multiple of $\pi$ and $x^2\sin r\alpha + y^2\sin r\beta + z^2\sin r\gamma$ to zero at $r = 1$ and $r = 2$. Prove that this expression is equal to zero for all natural values ​​of $r$.
4. On the sides $BC, CA, AB$ of the spherical triangle $ABC$ equal lengths $BA_1, CB_1, AC_1$ are laid out. Prove that the locus of the intersection points of the medians of the triangle $A_1B_1C_1$ is a great circle.
   5*. Let the points $A, B, C, D$ lie on the sphere, $P$ a point distinct from them on this sphere, and $\pi$ the plane tangent to the sphere at the point $P$. Let us denote by $\alpha$ the angle formed by the intersection of the planes $PAC$ and $PBC$ with the plane $\pi$, and by $\beta$ the angle formed by the intersection with the plane $PAD, PBD$. Prove that $|\alpha - \beta|$ does not depend on the choice of point $P$.

*) This is the problem of the famous chess player E. Lasker.