5

Part of 2019 Iran Team Selection Test

Problems(3)

Beautiful combinatorial geometry (Iran TST)

Source: Iranian TST 2019, second exam day 2, problem 5

P

be a simple polygon completely in

C

, a circle with radius

1

P

does not pass through the center of

C

. The perimeter of

P

36

. Prove that there is a radius of

C

that intersects

P

6

times, or there is a circle which is concentric with

C

and have at least

6

common points with

P

.
Proposed by Seyed Reza Hosseini

combinatorial geometrycombinatoricsperimeter

Source: Iranian TST 2019, third exam day 2, problem 5

A sub-graph of a complete graph with

n

vertices is chosen such that the number of its edges is a multiple of

3

and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero.
Proposed by Morteza Saghafian

Source: Iranian TST 2019, first exam day 2, problem 5

Find all functions

f:\mathbb{R}\rightarrow \mathbb{R}

such that for all

x,y\in \mathbb{R}

f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2

Proposed by Ali Behrouz - Mojtaba Zare Bidaki

functionalgebraTSTIranIranian TSTfunctional equation