MathDB

1

Part of 2016 Taiwan TST Round 3

Problems(4)

Inequality

Source: 2016 Taiwan TST Round 3

7/23/2016
Let x,y,zx,y,z be positive real numbers satisfying x+y+z=1x+y+z=1. Find the smallest kk such that x2y21−z+y2z21−x+z2x21−y≤k−3xyz\frac{x^2y^2}{1-z}+\frac{y^2z^2}{1-x}+\frac{z^2x^2}{1-y}\leq k-3xyz.
inequalities
Odd Coefficient

Source: 2016 Taiwan TST Round 3

7/21/2016
Let nn be a positive integer. Find the number of odd coefficients of the polynomial (x2−x+1)n(x^2-x+1)^n.
algebranumber theorypolynomial
300 degree

Source: 2016 Taiwan TST Round 3

7/25/2016
Let λ\lambda be a positive real number satisfying λ=λ2/3+1\lambda=\lambda^{2/3}+1. Show that there exists a positive integer MM such that ∣M−λ300∣<4−100|M-\lambda^{300}|<4^{-100}.
Proposed by Evan Chen
number theory
Altitude and Midpoints

Source: 2016 Taiwan TST Round 3

7/25/2016
Let ABCABC be an acute-angled triangle, with ∠B≠∠C\angle B \neq \angle C . Let MM be the midpoint of side BCBC, and E,FE,F be the feet of the altitude from B,CB,C respectively. Denote by K,LK,L the midpoints of segments ME,MFME,MF, respectively. Suppose TT is a point on the line KLKL such that AT//BCAT//BC. Prove that TA=TMTA=TM .
geometry