Russian TST 2017

Part of Russian Team Selection Tests

Subcontests

(3)

2

From a Graph to the Plane

K=(V, E)

be a finite, simple, complete graph. Let

\phi: E \to \mathbb{R}^2

be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set

V

, and the points assigned to the edges of any triangle are collinear. Show that the range of

\phi

is contained in a line.

Similar to Bulgaria 2000

a_1,\ldots , a_{p-2}{}

be nonzero residues modulo an odd prime

p{}

d\mid p - 1

there are at least

\lfloor(p - 2)/d\rfloor

i{}

p{}

does not divide

a_i^d-1

. Prove that the product of some of

a_1,\ldots , a_{p-2}

gives the remainder two modulo

p{}

4

5