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
Prove that the squares with sides $1/1,1/2,1/3,\ldots$ may be put into the square with side $3/2$ in such a way that no two of them have any interior point in common.
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 $P$.
The sum of the squares of five real numbers $a_1,a_2,a_3,a_4,a_5$ equals $1$. Prove that the least of the numbers $(a_i-a_j)^2$, where $i,j=1,2,3,4,5$ and $i\ne j$, does not exceed $1/10$.
Let $A_r,B_r,C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\triangle_r$, the triangle $\triangle_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1},C_{r+1}$ on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\triangle_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\triangle_1,\triangle_2,\ldots,\triangle_{15}$ are congruent.
Let $a_i,b_i$ be coprime positive integers for $i=1,2,\ldots,k$, and $m$ the least common multiple of $b_1,\ldots,b_k$. Prove that the greatest common divisor of $a_1\frac m{b_1},\ldots,a_k\frac m{b_k}$ equals the greatest common divisor of $a_1,\ldots,a_k$.
Let $x,y,z$ be real numbers each of whose absolute value is different from $1/\sqrt3$ such that $x+y+z=xyz$. Prove that
$$\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}=\frac{3x-x^3}{1-3x^2}\cdot\frac{3y-y^3}{1-3y^2}\cdot\frac{3z-z^3}{1-3z^2}.$$
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 $\sqrt{\sin A\sin B}\le\sin\frac C2$.
Consider a partition of an $8\times8$ chessboard into $p$ rectangles whose interiors are disjoint such that each of them has 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 $a_1,a_2,\ldots,a_p$.
In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any nonallowed word.