2018 Romanian Master of Mathematics

Part of Romanian Masters of Mathematics Collection

Subcontests

(6)

1

Combo with good configurations

n

be positive integer and fix

2n

distinct points on a circle. Determine the number of ways to connect the points with

n

arrows (oriented line segments) such that all of the following conditions hold: [*]each of the

2n

points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows

\overrightarrow{AB}

\overrightarrow{CD}

A

B

C

D

appear in clockwise order around the circle (not necessarily consecutively).

1

Number theory

a,b,c,d

be positive integers such that

ad \neq bc

gcd(a,b,c,d)=1

S

be the set of values attained by

\gcd(an+b,cn+d)

n

runs through the positive integers. Show that

S

is the set of all positive divisors of some positive integer.

1

Geometry with fix circle

\Gamma

\ell

\Gamma

, and another circle

\Omega

\ell

\Gamma

\Omega

lie on opposite sides of

\ell

. The tangents to

\Gamma

from a variable point

X

\Omega

\ell

Y

Z

. Prove that, as

X

\Omega

, the circumcircle of

XYZ

is tangent to two fixed circles.

1

Game theory question (RMM 2018 day 1 question 3)

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

1

Geometry with parallel lines.

ABCD

be a cyclic quadrilateral an let

P

be a point on the side

AB.

AC

meets the segments

DP

Q.

The line through

P

CD

mmets the extension of the side

CB

B

K.

The line through

Q

BD

meets the extension of the side

CB

B

L.

Prove that the circumcircles of the triangles

BKP

CLQ

1

Polynomials and powers

Determine whether there exist non-constant polynomials

P(x)

Q(x)

with real coefficients satisfying

P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.