MathDB

2

Part of 2002 National High School Mathematics League

Problems(2)

The Minumum Value

Source: 2002 National High School Mathematics League, Exam One, Problem 2

3/12/2020
Real numbers x,yx,y satisfy that (x+5)2+(y−12)2=142(x+5)^2+(y-12)^2=14^2, then the minumum value of x2+y2x^2+y^2 is (A)2(B)1(C)3(D)2\text{(A)}2\qquad\text{(B)}1\qquad\text{(C)}\sqrt3\qquad\text{(D)}\sqrt2\qquad
Cubic Equation

Source: 2002 National High School Mathematics League, Exam Two, Problem 2

3/16/2020
For real numbers a,b,ca,b,c and positive number λ\lambda such that three real roots x1,x2,x3x_1,x_2,x_3 of f(x)=x3+ax2+bx+cf(x)=x^3+ax^2+bx+c satisfying: (1)x2−x1=λ(1) x_2-x_1=\lambda; (2)x3>12(x1+x2)(2) x_3>\frac{1}{2}(x_1+x_2). Find the maximum value of 2a3+27c−9abλ3\frac{2a^3+27c-9ab}{\lambda^3}