location:

 PROBLEMS FOR OCTOBER

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

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 Û b = f(a) .

178.
Suppose that n is a positive integer and that x1, x2, ¼, xn are positive real numbers such that x1 + x2 + ¼+ xn = n. Prove that

 nå i=1 Ö[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 + ¼+ 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, ¼, an from the sides. Prove that

 a nå i=1 1ai > 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 ³ 1 .

183.
Simplify the expression

æ
Ö

 1 + _____Ö1 - x2

( (1 + x)   ____
Ö1 + x

- (1 - x)   ____
Ö1 - x

)

 x (2 + _____Ö1 - x2 )
,
where 0 < |x | < 1.

184.
Using complex numbers, or otherwise, evaluate

 sin10° sin50° sin70° .

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