1985 Kurschak Competition

Part of Kürschák Math Competition

Subcontests

(3)

1

Silly construction problem - power sum of n

n\in\mathbb{N}

, define the power sum of

n

as follows. For every prime divisor

p

n

, consider the largest positive integer

k

p^k\le n

, and sum up all the

p^k

's. (For instance, the power sum of

100

2^6+5^2=89

.) Prove that the power sum of

n

n

for infinitely many positive integers

n

1

Labelling triangles of an (n+1)-gon

We have triangulated a convex

(n+1)

P_0P_1\dots P_n

(i.e., divided it into

n-1

n-2

non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers

1,2,\dots,n-1

such that for any

i\in\{1,2,\dots,n-1\}

P_i

is a vertex of the triangle with label

i

1

Reflect the vertices of a triangle on the opposite sides

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.