4

Part of 2001 Hungary-Israel Binational

Problems(2)

$P (x) = x^3 − 3x + 1.$

Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

P (x) = x^{3}-3x+1.

Find the polynomial

Q

whose roots are the ﬁfth powers of the roots of

P

algebrapolynomialtrigonometryfunctionalgebra proposed

Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

G_{n}

denotes a simple undirected graph with

n

K_{n}

denotes the complete graph with

n

K_{n,m}

the complete bipartite graph whose components have

m

n

C_{n}

n

vertices. The number of edges in the graph

G_{n}

e(G_{n})

G_{n}

does not contain

K_{2,3}

e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n

n \geq 16

distinct points

P_{1}, . . . , P_{n}

in the plane, prove that at most

n\sqrt{n}

of the segments

P_{i}P_{j}

have unit length.

graph theorycombinatorics unsolvedcombinatorics