MathDB

1

Part of 2005 National High School Mathematics League

Problems(2)

Inequality

Source: 2005 National High School Mathematics League, Exam One, Problem 1

3/18/2020
The maximum value of kk such that the enequality x−3+6−x≥k\sqrt{x-3}+\sqrt{6-x}\geq k has a real solution is (A)6−3(B)3(C)3+6(D)6\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6
inequalities
Geometry

Source: 2005 National High School Mathematics League, Exam Two, Problem 1

3/20/2020
In △ABC\triangle ABC, AB>ACAB>AC, ll is tangent line of the circumscribed circle of △ABC\triangle ABC that passes AA. The circle with center AA and radius ACAC, intersects segment ABAB at DD, and line ll at E,FE, F (F,BF,B are on the same side). Prove that lines DE,DFDE, DF pass the incenter and an excenter of △ABC\triangle ABC respectively.
geometry