MathDB

1

closed bounded interval, non-constant monic polynomial function, max inequality

I \subset R

is a closed bounded interval, and

f : I \to R

is a non-constant monic polynomial function such that

max_{x\in I}|f(x)|< 2

, then there exists a non-constant monic polynomial function

g : I \to R

such that

max_{x\in I} |g(x)| < 1

. (b) Show that there exists a closed bounded interval

I \subset R

such that

max_{x\in I}|f(x)| \ge 2

for every non-constant monic polynomial function

f : I \to R

.

2

1

simple graphs, k(v) maximal cardinality of an independent set of neighbours of v

Let

n

be a positive integer and let

G_n

be the set of all simple graphs on

n

vertices. For each vertex

v

of a graph in

G_n

, let

k(v)

be the maximal cardinality of an independent set of neighbours of

v

. Determine

max_{G \in G_n} \Sigma_{v\in V (G)}k(v)

and the graphs in

G_n

that achieve this value.

3

1

another romanian concurrency

Let

ABC

be a triangle, let

A_1, B_1, C_1

be the antipodes of the vertices

A, B, C

, respectively, in the circle

ABC

, and let

X

be a point in the plane

ABC

, collinear with no two vertices of the triangle

ABC

. The line through

B

, perpendicular to the line

XB

, and the line through

C

, perpendicular to the line

XC

, meet at

A_2

, the points

B_2

and

C_2

are defined similarly. Show that the lines

A_1A_2, B_1B_2

and

C_1C_2

are concurrent.

1

P145. Romania Imar Mathematical Competition

Determine all positive integers expressible, for every integer n \geq 3 , in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where

a_1, a_2, \ldots, a_n

are pairwise distinct positive integers.

2015 IMAR Test

Subcontests

closed bounded interval, non-constant monic polynomial function, max inequality

simple graphs, k(v) maximal cardinality of an independent set of neighbours of v

another romanian concurrency

P145. Romania Imar Mathematical Competition