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Canadian Mathematical Society
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PROBLEMS FOR OCTOBER

Please send your solution to

Dr. Mihai Rosu
135 Fenelon Drive, #205
Toronto, ON M3A 3K7

It is very important that the front page contain your complete mailing address and your email address. The deadline for this set is November 15, 2002.

Notes. A function f : A B is a bijection iff it is one-one and onto; this means that, if f(u) = f(v), then u = v, and, if w is some element of B, then A contains an element t for which f(t) = w. Such a function has an inverse f-1 which is determined by the condition


f-1(b) = a b = f(a) .


178.
Suppose that n is a positive integer and that x1, x2, , xn are positive real numbers such that x1 + x2 + + xn = n. Prove that


n

i=1 
[n ]axi + b a + b + n - 1
for every pair a, b or real numbers with all axi + b nonnegative. Describe the situation when equality occurs.

179.
Determine the units digit of the numbers a2, b2 and ab (in base 10 numeration), where


a = 22002 + 32002 + 42002 + 52002
and


b = 31 + 32 + 33 + + 32002 .

180.
Consider the function f that takes the set of complex numbers into itself defined by f(z) = 3z + |z |. Prove that f is a bijection and find its inverse.

181.
Consider a regular polygon with n sides, each of length a, and an interior point located at distances a1, a2, , an from the sides. Prove that


a n

i=1 
1
ai
> 2p .

182.
Let ABC be an equilateral triangle with each side of unit length. Let M be an interior point in the equilateral triangle ABC with each side of unit length. Prove that


MA.MB + MB.MC + MC.MA 1 .

183.
Simplify the expression


  


1 +   _____
1 - x2
 
 
( (1 + x)   ____
1 + x
 
- (1 - x)   ____
1 - x
 
)

x (2 +   _____
1 - x2
 
)
 ,
where 0 < |x | < 1.

184.
Using complex numbers, or otherwise, evaluate


sin10 sin50 sin70 .


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