olymmath

DE-MO 1965

German Mathematical Olympiad 1965.

Other problems

Problem 1

For a given positive real parameter pp , solve the equation p+x+p−x=x\sqrt{p+x}+\sqrt{p-x }= x .

Problem 2

Determine which of the prime numbers 2,3,5,7,11,13,109,151,4912,3,5,7,11,13,109,151,491 divide z=19631965−1963z =1963^{1965} -1963 .

Problem 3

Two parallelograms ABCDABCD and A′B′C′D′A'B'C'D' are given in space. Points A′′,B′′,C′′,D′′A'',B'',C'',D'' divide the segments AA′,BB′,CC′,DD′AA',BB',CC',DD' in the same ratio. What can be said about the quadrilateral A′′B′′C′′D′′A''B''C''D'' ?

Problem 4

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.

Problem 5

Determine all triples of nonzero decimal digits (x,y,z)(x,y,z) for which the equality xxx...x⏟2n−yy...y⏟n=zzz...z⏟n\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n} holds for at least two different natural numbers nn .

Problem 6

Let α,β,γ\alpha,\beta, \gamma be the angles of a triangle. Prove that cos⁡α,+cos⁡β+cos⁡γ≤32\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} and find the cases of equality.