Question bank IMO 1974

Alice, Betty, and Carol took the same series of examinations. There was one grade of $A$, one grade of $B$, and one grade of $C$ for each examination, where $A,B,C$ are different positive integers. The final test scores were

$$\begin{array}{ccc}\text{Alice}&\text{Betty}&\text{Carol}\\20&10&9\end{array}$$

If Betty placed first in the arithmetic examination, who placed second in the spelling examination?

Let $\triangle ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ such that $CD$ is the geometric mean of $AD$ and $BD$ if and only if

$$\sqrt{\sin A\sin B}\le\sin\frac C2.$$

Prove that there does not exist a natural number $n$ for which the number

$$\sum_{k=0}^n\binom{2n+1}{2k+1}2^{3k}$$

is divisible by $5$.

Consider a partition of an $8\times8$ chessboard into $p$ rectangles whose interiors are disjoint such that each rectangle contains an equal number of white and black cells. Assume that $a_1\lt a_2\lt \cdots\lt a_p$, where $a_i$ denotes the number of white cells in the $i$th rectangle. Find the maximal $p$ for which such a partition is possible and for that $p$ determine all possible corresponding sequences $a_1,a_2,\ldots,a_p$.

If $a,b,c,d$ are arbitrary positive real numbers, find all possible values of
$$S=\frac a{a+b+d}+\frac b{a+b+c}+\frac c{b+c+d}+\frac d{a+c+d}.$$

Let $P(x)$ be a polynomial with integer coefficients. If $n(P)$ is the number of (distinct) integers $k$ such that $P^2(k)=1$, prove that $n(P)-\deg(P)\le2$, where $\deg(P)$ denotes the degree of the polynomial $P$.