Let $\alpha$ vary freely and put $x=\cos\alpha$, $y=\cos n\alpha$.
(a) Prove that each $x$ in $-1\le x\le1$ determines exactly one value of $y$. Regard $y$ as a function $T_n(x)$. Calculate $T_1(x),T_2(x)$ and prove
$$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).$$
Deduce that $T_n$ is a polynomial of degree $n$.
(b) Prove that $T_n$ has $n$ distinct real roots in $[-1,1]$.
For a positive integer $N$, sum $(-1)^{(n-1)/2}$ over all odd positive divisors $n$ of $N$, and call the sum $f(N)$.
(a) Prove, for an integer $r$ and a prime $p$, that $f(2)=1$, $f(2^r)=1$, and that $f(p)=2$ if $p=4k+1$, and $f(p)=0$ if $p=4k-1$. Prove also
$$f(p^r)=\begin{cases}1+r,&p=4k+1,\\1,&p=4k-1\text{ and }r\text{ even},\\0,&p=4k-1\text{ and }r\text{ odd}.\end{cases}$$
(b) Prove multiplicativity: if $M,N$ are positive integers and $\gcd(M,N)=1$, then $f(MN)=f(M)f(N)$. Calculate $f(5^4\cdot11^{28}\cdot17^{19})$ and $f(1980)$ and state a general rule for $f(N)$.
Let $ABC$ be a triangle and let $d$ pass through $A$ without being parallel to $BC$.
(a) Construct parallelogram $CEFG$ with $E,F,G$ respectively on the lines $d,AB,BC$, and $CE$ parallel to the median $AI$. Then construct parallelogram $EAKH$ with $K,H$ respectively on $AB,BC$. Put $U=BC\cap d$ and let $V$ be the reflection of $U$ in $A$. Prove that $V,I,FG\cap KH$ are collinear, and also collinear with $B',C'$, the other two vertices of the parallelogram $BB'CC'$ whose opposite vertices are $B,C$ and whose successive sides are parallel to $d,AI$.
(b) How are the pairs of intersection points of $FG,KH$ with the three lines $BC,CA,AB$ arranged? State a general theorem. Given a point $O$ in the plane of $ABC$, can two such lines $FG,KH$ be constructed through $O$?
A regular tetrahedron $ABCD$ has edge $a$. On $AB$ choose $E,E'$ with $AE=a/6$, $AE'=5a/6$; on $AC$ choose $F,F'$ with $AF=a/4$, $AF'=3a/4$; on $AD$ choose $G,G'$ with $AG=a/3$, $AG'=2a/3$.
(a) Determine the intersection lines of the plane $BCD$ with the planes $EFG$ and $E'F'G'$. Study their relative position with respect to triangle $BCD$. State a general theorem.
(b) Calculate the volume of $EFGE'F'G'$ in terms of $a$, and calculate the angles that “the two planes $EFG$” make with $AB,AC,AD$. [The quoted wording is as printed; only $EFG$ is named.]