Some number theory (and miscellaneous) coaching for RMO and INMO: tutorial (problem set) III

Continuing this series of slightly vexing questions, we present below:

1. Prove the inequality \frac{A+a+B+b}{A+a+B+b+c+r} + \frac{B+b+C+c}{B+b+C+c+a+r} > \frac{C+c+A+a}{C+c+A+a++b+r}, where all the variables are positive numbers.

2. A sequence of numbers: Find a sequence of numbers x_{0}, x_{1}, x_{2}, \ldots whose elements are positive and such that a_{0}=1 and a_{n} - a_{n+1}=a_{n+2} for n=0, 1, 2, \ldots. Show that there is only one such sequence.

3. Points in a plane: Consider several points lying in a plane. We connect each point to the nearest point by a straight line. Since we assume all distances to be different, there is no doubt as to which point is the nearest one. Prove that the resulting figure does not containing any closed polygons or intersecting segments.

4. Examination of an angle: Let x_{1}, x_{2}, \ldots, x_{n} be positive numbers. We choose in a plane a ray OX, and we lay off it on a segment OP_{1}=x_{1}. Then, we draw a segment P_{1}P_{2}=x_{2} perpendicular to OP_{1} and next a segment P_{2}P_{3}=x_{3} perpendicular to OP_{2}. We continue in this way up to P_{n-1}P_{n}=x_{n}. The right angles are directed in such a way that their left arms pass through O. We can consider the ray OX to rotate around O from the initial point through points P_{1}, P_{2}, \ldots, P_{n} (the final position being P_{n}). In doing so, it sweeps out a certain angle. Prove that for given numbers x_{i}, this angle is smallest when the numbers x_{i}, that is, x_{1} \geq x_{2} \geq \ldots x_{n} decrease; and the angle is largest when these numbers increase.

5. Area of a triangle: Prove, without the help of trigonometry, that in a triangle with one angle A = 60 \deg the area S of the triangle is given by the formula S = \frac{\sqrt{3}}{4}(a^{2}-(b-c)^{2}) and if A = 120\deg, then S = \frac{\sqrt{3}}{12}(a^{2}-(b-c)^{2}).

More later, cheers, hope you all enjoy. Partial attempts also deserve some credit.

Nalin Pithwa.

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