1998 Hong kong National Olympiad

Part of Hong Kong National Olympiad

Subcontests

(4)

1

Function problem

Define a function

f

on positive real numbers to satisfy

f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},

g(x)

is a function defined on real numbers and for all real numbers

y,z

0\leq t \leq 1

g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).

(1) Prove: for any integer

n

0 \leq t \leq 1

t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].

\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.

1

GOOD pair

s,t

are non-zero integers,

(x,y)

is an integer pair , A transformation is to change pair

(x,y)

(x+t,y-s)

. If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is

(s,t)

a good pair ? (2) Prove :for any

s

t

, there exists pair

(x,y)

which is " a good pair".

1

Paint the pyramid

The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )

1

HK bisect QS

In a concyclic quadrilateral

PQRS

\angle PSR=\frac{\pi}{2}

H,K

are perpendicular foot from

Q

PR,RS

HK

SQ