1898 Eotvos Mathematical Competition

Part of Eotvos Mathematical Competition (Hungary)

Subcontests

(3)

1

Let $A, B, C, D$ be four given points on a straight line $e$. Construct a square

A, B, C, D

be four given points on a straight line

e

. Construct a square such that two of its parallel sides (or their extensions) go through

A

B

respectively, and the other two sides (and their extensions) go through

C

D

1

Prove the following theorem: If two triangles have a common angle, then the sum

Prove the following theorem: If two triangles have a common angle, then the sum of the sines of the angles will be larger in that triangle where the difference of the remaining two angles is smaller.
On the basis of this theorem, determine the shape of that triangle for which the sum of the sines of its angles is a maximum.

1

Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.

Determine all positive integers

n

2^n + 1

is divisible by

3