JOM 2014

Part of JOM

Subcontests

(5)

1

Common point on line XO

\triangle ABC

with circumcircle

\Gamma

and circumcentre

O

X

\Gamma

XC_1

XB_1

to be feet of perpendiculars from

X

AB

AC

\omega_C

as the circle with centre the midpoint of

AB

and passing through

C_1

\omega_B

similarly.
Prove that

\omega_B

\omega_C

has a common point on

XO

1

Painting edges of a complete graph red or blue

There is a complete graph

G

4027

vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs

(r, b)

such that Ivan can paint the edges so that every vertex is connected to exactly

r

b

1

Special Polynomial

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after

2014

tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of

x^{2013}

2014

, and that the constant term is

1

. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)

1

Zschess & Ivanight

In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes through the edge and comes out from the other side of the board, and attacks that square instead. The ZS chessboard is an 8 \times 8 board, where cells are coloured with

n

distinct colours, where

n

is a natural number, such that a Ivanight placed on any square attacks

8

squares that consist of all

n

colours, and the colours appear equally many times in those

8

squares. For which values of

n

does such a ZS chess board exist?

1

Product of factors

f(n)

be the product of all factors of

n

. Find all natural numbers

n

f(n)

is not a perfect power of

n