Mathematical Olympiad in China
82
hence
tan2
el > tan2e2 > 72 , -
so
Thus
cos
el + cos e2 +cos e3 < @+ 3 1< 2,
that is, 0 holds too. Hence we can get cos
el + cos e2 + cos e3 + ...+ cos en <
On the other hand, if we take 0,
= 0, =
m.0
-
I.
= 0, = a
>0, a-
0,
then
Obviously, 0, cos
-
iT
-
2
, thus
el + cos e2 + cos e3 + ...+ cos en
Consequently, we get A
=
-
-
I.
n- 1.
Second Day 8:OO-12: 30
January 16, 2003
>
@.$@Find all ternary positive integer groups (a, m, n>satisfying a 2 and m 2 such that an 203 is a multiple of a" 1. (posed by Chen Yonggao) Solution We will discuss the following three cases for n and m. (i) In the case when n < m , from an 203 a" 1, we have
>
+
202 Therefore ,
>a"
-
an
>an( a
+
-
+ > + 1) > a ( a 1).
2<a<14.
-