MathDB

3

Part of 1995 National High School Mathematics League

Problems(2)

Are You an Excellent Boy?

Source: 1995 National High School Mathematics League, Exam One, Problem 3

3/2/2020
If a person A is taller or heavier than another peoson B, then we note that A is not worse than B. In 100 persons, if someone is not worse than other 99 people, we call him excellent boy. What's the maximum value of the number of excellent boys? (A)1(B)2(C)50(D)100\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}50\qquad\text{(D)}100
Inscribed Circle

Source: 1995 National High School Mathematics League, Exam Two, Problem 3

3/3/2020
Inscribed Circle of rhombus ABCDABCD touches AB,BC,CD,DAAB,BC,CD,DA at E,F,G,HE,F,G,H. l1,l2l_1,l_2 are two lines that are tangent to the circle. l1∩AB=M,l1∩BC=N,l2∩CD=P,l2∩DA=Ql_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q. Prove that MQ/ ⁣/NPMQ/\! /NP.
geometryrhombus