Let be an odd integer greater than Let be an symmetric matrix such that each row and column consists of some permutation of the integers Show that each of the integers must appear in the main diagonal of .

Is the infinite series
convergent?

The complete graph with 6 points and 15 edges has each edge colored red or blue. Show that we can find 3 points such that the 3 edges joining them are the same color.

A homogeneous solid body is made by joining a base of a circular cylinder of height and radius and the base of a hemisphere of radius This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if is large as compared to , the equilibrium will be stable; but if is small compared to , the equilibrium will be unstable. What is the critical value of the ratio which enables the body to rest in neutral equilibrium in any position?

Develop necessary and sufficient conditions that the equation shall have a multiple root.

A man has a rectangular block of wood by by inches ( and are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.)

The flag of the United Nations consists of a polar map of the world, with the North Pole as its center, extending to approximately South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?

Let and be relatively prime positive integers, even. For each positive integer , let be chosen so that
is a minimum. Prove that

Let be a permutation of the integers Call a big integer if for all Find the mean number of big integers over all permutations on the first postive integers.

Let be two complex matrices of the same rank, and let be a positive integer. Prove that if and only if .