MathDB

1

Part of 1994 National High School Mathematics League

Problems(2)

Inequality

Source: 1994 National High School Mathematics League, Exam One, Problem 1

2/28/2020
a,b,ca,b,c are real numbers. The sufficient and necessary condition of ∀x∈R,asin⁡x+bcos⁡x+c>0\forall x\in\mathbb{R},a\sin x+b\cos x+c>0 is (A)\text{(A)} a=b=0,c>0a=b=0,c>0 (B)\text{(B)} a2+b2=c\sqrt{a^2+b^2}=c (C)\text{(C)} a2+b2<c\sqrt{a^2+b^2}<c (D)\text{(D)} a2+b2>c\sqrt{a^2+b^2}>c
inequalities
Complex Number

Source: 1994 National High School Mathematics League, Exam Two, Problem 1

3/2/2020
In the equation x2+z1x+z2+m=0x^2+z_1x+z_2+m=0, z1,z2,mz_1,z_2,m are complex numbers, and z12−4z2=16+20iz_1^2-4z_2=16+20\text{i}. Two roots of the equations are α,β\alpha,\beta. If ∣α−β∣=27|\alpha-\beta|=2\sqrt7, find the maximum and minumum value of ∣m∣|m|.
complex numbers