MathDB

9

Part of 2013 Hanoi Open Mathematics Competitions

Problems(2)

x+y<= 1 and 2/xy +1/(x^2+y^2)=10 and x,y>0 (HOMC 2013 J Q9)

Source:

7/28/2019
Solve the following system in positive numbers {x+y≤12xy+1x2+y2=10\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}
Systemalgebra
P(t) = t^3 + at^2 + bt + c, P(2013) >1/64 (HOMC 2013 Q9)

Source:

8/1/2019
A given polynomial P(t)=t3+at2+bt+cP(t) = t^3 + at^2 + bt + c has 33 distinct real roots. If the equation (x2+x+2013)3+a(x2+x+2013)2+b(x2+x+2013)+c=0(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0 has no real roots, prove that P(2013)>164P(2013) >\frac{1}{64}
polynomialcubic equation