1983 Vietnam National Olympiad

Part of Vietnam National Olympiad

Subcontests

(3)

2

Locus of points whose area of pedal triangle is constant

ABC

and a positive number

k

are given. Find the locus of a point

M

inside the triangle such that the projections of

M

on the sides of

\Delta ABC

form a triangle of area

k

Tetrahedron with two opposite sides equal.

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

2

Two trigonometric inequalities.

(a)

\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}

0 \le t \le\frac{\pi}{2}.

(b)

y, 0 < y < \pi

1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}

Finding the larger of two algebraic expressions

S_n

T_n

is larger, where

S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}

2

Easy number theory: 2^b-1 divides 2^a+1

Are there positive integers

a, b

b \ge 2

2^a + 1

is divisible by

2^b - 1

Expressing 1 as sum of 6 and 9 reciprocals of odd integers.

Show that it is possible to express

1

6

, and as a sum of

9

reciprocals of odd positive integers. Generalize the problem.