VMO 1972

Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ( $n \in Z$ ). i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$ . Thus we can write $y$ as a function of $x, y = T_n(x)$ . Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ . From this it follows that $T_n(x)$ is a polynomial of degree $n$ . ii) Prove that the polynomial $T_n(x$ ) has $n$ distinct roots in $[-1, 1]$ .

For any positive integer n, let f(n) = ∑ (-1)$^{(d-1)/2}$ where the sum is taken over all odd d dividing n. Show that: f($2^n$) = 1 f(p) = 2 for p a prime congruent to 1 mod 4 f(p) = 0 for p a prime congruent to 3 mod 4 f($p^n$) = n+1 for p a prime congruent to 1 mod 4 f($p^n$) = 1 for p a prime congruent to 3 mod 4, and n even f($p^n$) = 0 for p a prime congruent to 3 mod 4, and n odd Show that f(mn) = f(m) f(n) for m and n relatively prime. Find f($5^411^{28}17^{19}$) and f(1980). Show how to calculate f(n).

$ABC$ is a triangle. $U$ is a point on the line $BC$ . $I$ is the midpoint of $BC$ . The line through $C$ parallel to $AI$ meets the line $AU$ at $E$ . The line through $E$ parallel to $BC$ meets the line $AB$ at $F$ . The line through $E$ parallel to $AB$ meets the line $BC$ at $H$ . The line through $H$ parallel to $AU$ meets the line $AB$ at $K$ . The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$ . Show that $V, T$ and $I$ are collinear.

Let $ABCD$ be a regular tetrahedron with side $a$ . Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$ . Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$ .