MathDB

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Part of 2005 Ukraine National Mathematical Olympiad

Problems(4)

find all pairs of real numbers

Source: Ukraine 2005 grade 8

7/24/2009
Find all pairs of real numbers (x,y) (x,y) such that: \frac{x\minus{}2}{y}\plus{}\frac{5}{xy}\equal{}\frac{4\minus{}y}{x}\minus{}\frac{|y\minus{}2x|}{xy}.
algebra unsolvedalgebra
draw the locus

Source: Ukraine 2005 grade 9

7/26/2009
Draw the locus of points M(x,y) M(x,y) in the Cartesian plane xOy xOy satisfying (x^2\minus{}1)(|y|\minus{}1) \ge 0.
geometry unsolvedgeometry
trigonometry

Source: Ukraine 2005 grade 10

7/26/2009
Let 0≤α,β,γ≤π2 0 \le \alpha, \beta, \gamma \le \frac{\pi}{2} satisfy the conditions \sin \alpha\plus{}\sin \beta\plus{}\sin \gamma\equal{}1, \sin \alpha \cos 2\alpha\plus{}\sin \beta\cos 2\beta\plus{}\sin \gamma \cos 2\gamma\equal{}\minus{}1. Find all possible values of \sin^2 \alpha\plus{}\sin^2 \beta\plus{}\sin^2 \gamma.
trigonometry
equation

Source: Ukraine 2005 grade 11

7/28/2009
Solve in R \mathbb{R} the equation \left| x \minus{}\frac{\pi}{6} \right|\plus{}\left| x\plus{}\frac{\pi}{3} \right|\equal{}\arcsin \frac{x^3\minus{}x\plus{}2}{2}.
algebra unsolvedalgebra