MathDB

2

Part of 2005 National High School Mathematics League

Problems(2)

3D Geometry

Source: 2005 National High School Mathematics League, Exam One, Problem 2

3/18/2020
Four points in space A,B,C,DA,B,C,D satisfy that ∣AB∣=3,∣BC∣=7,∣CD∣=11,∣DA∣=9|AB|=3,|BC|=7,|CD|=11,|DA|=9, then the number of values of AC→⋅BD→\overrightarrow{AC}\cdot\overrightarrow{BD} is (A)\text{(A)} Only one. (B)\text{(B)} Two. (C)\text{(C)} Three. (D)\text{(D)} Infinitely many.
geometry3D geometry
Inequality

Source: 2005 National High School Mathematics League, Exam Two, Problem 2

3/20/2020
Positive numbers a,b,c,x,y,za, b, c, x, y, z satisfy that cy+bz=acy + bz = a, az+cx=baz + cx = b, and bx+ay=cbx + ay = c. Find the minimum value of the function f(x,y,z)=x2x+1+y2y+1+z2z+1f(x,y,z) =\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}.
inequalities