THE

MATHEMATICAL MONTHLY.

Vol. II....JULY, 1860.... No. X.

PRIZE PROBLEMS FOR STUDENTS.

I. Solve the equations

by quadratics.

x2 y2 (x* — y1) = 2340,

x y (x y3 — 1) (x2 + y2) = 1794,

Communicated by E. A. HOPKINS, Cleveland, Ohio.

II. Given the lengths of the three perpendiculars dropped from any point in the plane of an equilateral triangle upon the sides; to find the segments of the sides.- Communicated by F. E. ToWER, Amherst College.

III. If e denote the edge of any regular dodecahedron, and ß= 36°, prove that

Also obtain similar formulas for the solidity of the icosahedron.— Communicated by Prof. D. W. HOYT.

IV. A given cylindrical vessel, filled with water, is placed with its base upon a horizontal plane. It is required to determine the angle of inclination to which the plane must be raised before the vessel will fall, the water being at liberty to overflow its top. The base is supposed to be fixed so as to prevent it from sliding, but not from tilting when the plane is inclined. - Communicated by Professor KIRKWOOD.

V. Bisect the attraction which a sphere of varying density exerts

upon an exterior point; that is, divide the sphere so that the two parts shall exert the same attractive force in the same direction.

Solutions of these problems must be received by September 1, 1860.

REPORT OF THE JUDGES UPON THE SOLUTIONS OF THE PRIZE PROBLEMS IN No. VII., Vol. II.

THE first Prize is awarded to JOHN A. WINEBRENER, Princeton College, N. J.

The second Prize is awarded to GEORGE C. ROUND, Wesleyan University, Middletown, Ct.

The third Prize is awarded to LEWIS FOOTE, O. C. Seminary, Cazenovia, N. Y.

PRIZE SOLUTION OF PROBLEM I.

By Miss HARRIET S. HAZELTINE, Worcester, Mass. Prove that an arithmetic mean is greater than a geometric.

Let xy and xy denote the extremes; then x is the arithmetic and √(2-2) the geometric mean, and it is evident that x = √x2 > √(x2 — y2).

SECOND SOLUTION.-Let ab; then a-b>0; a2-2ab+b2>0; a2 + 2 a b + b2 > 4 ab; a + b > 2 √ ab; :: ž (a + b) > √ ab. G. S. MORISON, Harvard College.

PRIZE SOLUTION OF PROBLEM II.

Let three bodies with velocities V, V, V", move uniformly in the same direction, in the circumference of a circle. Required the time of their conjunction, supposing them to quit a given point at the same time.

Let C denote the circumference of the circle; then, since VV' and V-V" are respectively the gains of V upon V' and V" in the

will elapse between the instant of starting and the conjunctions of V, V' and V, V" respectively. And the least common multiple of