2018 Nordic

Part of Nordic

Subcontests

(4)

1

Representations of three-variable polynomial

f = f(x,y,z)

be a polynomial in three variables

x

y

z

f(w,w,w) = 0

w \in \mathbb{R}

. Show that there exist three polynomials

A

B

C

in these same three variables such that

A + B + C = 0

f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x).

Is there any polynomial

f

for which these

A

B

C

are uniquely determined?

1

Orthocenter of BCI lies on DE

ABC

be a triangle with

AB < AC

D

E

be on the lines

CA

BA

, respectively, such that

CD = AB

BE = AC

A

D

E

lie on the same side of

BC

I

be the incenter of triangle

ABC

H

be the orthocenter of triangle

BCI

D

E

H

1

Sequence contains finitely many primes

A sequence of primes

p_1, p_2, \dots

is given by two initial primes

p_1

p_2

p_{n+2}

being the greatest prime divisor of

p_n + p_{n+1} + 2018

n \ge 1

. Prove that the sequence only contains finitely many primes for all possible values of

p_1

p_2

1

Ray intersects at least k lines

k

be a positive integer and

P

a point in the plane. We wish to draw lines, none passing through

P

, in such a way that any ray starting from

P

intersects at least

k

of these lines. Determine the smallest number of lines needed.