MathDB

2

Part of 1999 Estonia National Olympiad

Problems(4)

x^2 + 6x + 4a = 0 and x^2 + 2bx - 12 = 0 , have a common solution

Source: 1999 Estonia National Olympiad Final Round grade 9 p2

3/13/2020
It is known that the quadratic equations x2+6x+4a=0x^2 + 6x + 4a = 0 and x2+2bx−12=0x^2 + 2bx - 12 = 0 have a common solution. Prove that then there is a common solution to the quadratic equations x2+9x+9a=0x^2 + 9x + 9a = 0 and x2+3bx−27=0x^2 + 3bx - 27 = 0.
trinomialquadratic polynomialalgebra
sum f(i/2000)+ f(2000/i)+ f(2000/2000) , when i=1,...,1999

Source: 1999 Estonia National Olympiad Final Round grade 11 p2

3/11/2020
Find the value of the expression f(12000)+f(22000)+...+f(19992000)+f(20002000)+f(20001999)+...+f(20001)f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right) assuming f(x)=x21+x2f(x) =\frac{x^2}{1 + x^2} .
Sumalgebra
x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0 , |x_1|=2|x_2|

Source: 1999 Estonia National Olympiad Final Round grade 10 p2

3/11/2020
Find all values of aa such that absolute value of one of the roots of the equation x2+(a−2)x−2a2+5a−3=0x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0 is twice of absolute value of the other root.
algebraparametertrinomial
\int_{-1}^{1} ln (x +\sqrt{1 + x^2} )

Source: 1999 Estonia National Olympiad Final Round grade 12 p2

3/11/2020
Find the value of the integral ∫−11ln(x+1+x2)dx\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx.
calculusintegrationIntegral