MathDB

10.2

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

draw cos(x + y)^2 <= cos(x - y)^2 (V Soros Olympiad 1998-99 Round 1 10.2)

Source:

5/25/2024
On the coordinate plane, draw all pointsM(x,y) M(x, y), the coordinates of which satisfy the inequalities cos⁡(x+y)2≤cos⁡(x−y)2,   0≤x3,   0≤y3.\cos(x + y)^2 \le \cos(x - y)^2, \,\,\, 0 \le x^3, \,\,\, 0 \le y^3.
trigonometryanalytic geometry
infinite sum 1/ (2p+1)^2 (V Soros Olympiad 1998-99 Round 2 10.2)

Source:

5/25/2024
In 17481748, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums 1+14+19+116+...1 +\frac14 +\frac19+ \frac{1}{16}+... and 1+19+116+...1 +\frac19+ \frac{1}{16}+... (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals π26\frac{\pi^2}{6}. Given this result, find the value of the second sum.
algebra
|cos3x-tgt+ |cos3x+tgt|=|tg^2t -3|(V Soros Olympiad 1998-99 Round 3 10.2)

Source:

5/26/2024
Solve the equation ∣cos⁡3x−tgt∣+∣cos⁡3x+tgt∣=∣tg2t−3∣. |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.
algebratrigonometry