MathDB

8

Part of 2013 IFYM, Sozopol

Problems(3)

Problem 8 of Second round

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/22/2020
Let KK be a point on the angle bisector, such that ∠BKL=∠KBL=30∘\angle BKL=\angle KBL=30^\circ. The lines ABAB and CKCK intersect in point MM and lines ACAC and BKBK intersect in point NN. Determine ∠AMN\angle AMN.
geometry
Problem 8 of Third round

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/24/2020
The irrational numbers α,β,γ,δ\alpha ,\beta ,\gamma ,\delta are such that ∀\forall n∈Nn\in \mathbb{N} : [nα].[nβ]=[nγ].[nδ][n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]. Is it true that the sets {α,β}\{ \alpha ,\beta \} and {γ,δ}\{ \gamma ,\delta \} are equal?
number theorySetsequation
Korea Third Round (FKMO) 2012 #1

Source: FKMO 2012

3/25/2012
Let x,y,z x, y, z be positive real numbers. Prove that 2x2+xy(y+zx+z)2+2y2+yz(z+xy+x)2+2z2+zx(x+yz+y)2≥1 \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1
inequalities unsolvedinequalities