MathDB

Problem 7

Part of 1999 Ukraine National Mathematical Olympiad

Problems(3)

constant quantity of chords inside circle

Source: Ukraine 1999 Grade 9 P7

5/10/2021
Let MM be a fixed point inside a given circle. Two perpendicular chords ACAC and BDBD are drawn through MM, and KK and LL are the midpoints of ABAB and CDCD, respectively. Prove that the quantity AB2+CD2−2KL2AB^2+CD^2-2KL^2 is independent of the chords ACAC and BDBD.
geometry
Ukraine 1999

Source:

12/25/2014
If x1x_1, x2x_2, ..., x6x_6 ∈\in [0,1][0,1], prove that the cyclic sum of
x13x25+x35+x45+x55+x65+5\frac{x_1^3}{x_2^5+x_3^5+x_4^5+x_5^5+x_6^5+5} is less than 35\frac{3}{5}.
inequalities unsolvedinequalities
bounding period of trig function

Source: Ukraine 1999 Grade 11 P7

5/11/2021
Suppose that the function f(x)=tan⁡(a1x+1)+…+tan⁡(a10x+1)f(x)=\tan(a_1x+1)+\ldots+\tan(a_{10}x+1) has the period T>0T>0, where a1,…,a10a_1,\ldots,a_{10} are positive numbers. Prove that T≥π10min⁡{1a1,…,1a10}.T\ge\frac\pi{10}\min\left\{\frac1{a_1},\ldots,\frac1{a_{10}}\right\}.
trigonometryalgebrafunction