2004 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Three numbers cannot be all squares

Show that there is no positive integer

n

2n^2-1, 3n^2-1, 6n^2-1

are all perfect squares.

1

Unicursal figure-like konigsberg bridge problem

In a land any towns are connected by roads with exactly other three towns. Last year we made a trip starting from a town coming back to the town by visiting exactly one time all towns in land. This year we are plan to make a trip in the same way as last year's trip. Note that you can't take such order completely same as last year's one or trace the order only reversely. Prove that this is possible.

1

2 planes and bisector of angle

Given two planes

\pi _1,\ \pi _2

intersecting orthogonally in space. Let

A,B

be two distinct points on the line of intersection of

\pi _1

\pi _2,

C

be the point which is on

\pi _2

\pi_1.

P

the intersection point of the bisector of

\angle {BCA}

AB

S

by the circumference on

\pi _1

with a diameter

AB.

For an arbiterary plane

\pi _3

CP,

D,E

are the intersection points of

\pi_3

S,

then prove that

CP

is the bisector of

\angle {DCE}.

1

F(xf(x)+f(y))=(f(x))^2+y

Find all functions

f : \mathbb{R} \mapsto \mathbb{R}

f(xf(x)+f(y)) =(f(x))^2+y

\ x,y \in \mathbb{R}.

1

Proof of inequality

For positive real numbers

a,b,c

a+b+c=1,

Prove that we have

\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leqq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).

Note that you don't need to state for the condition for which the equality holds.