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Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
1999 Hong kong National Olympiad
1999 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(3)
4
1
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Functional equation
Determine all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that f(x+yf(x))=f(x)+xf(y) \text{for all}\ x,y \in\mathbb{R}
2
1
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Angle AIO is less than 90 degrees iff 2a =<b+c
Let
I
I
I
be the incentre and
O
O
O
the circumcentre of a non-equilateral triangle
A
B
C
ABC
A
BC
. Prove that
∠
A
I
O
≤
9
0
∘
\angle AIO \le 90^{\circ}
∠
A
I
O
≤
9
0
∘
if and only if
2
B
C
≤
A
B
+
A
C
2BC\le AB+AC
2
BC
≤
A
B
+
A
C
.
1
1
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Find all rational r
Find all positive rational numbers
r
≠
1
r\not=1
r
=
1
such that
r
1
r
−
1
r^{\frac{1}{r-1}}
r
r
−
1
1
is rational.