MathDB

6

Part of 2018 South East Mathematical Olympiad

Problems(2)

Another Chinese projective cookie

Source: 2018 China Southeast MO Grade 10 P6

7/31/2018
In the isosceles triangle ABCABC with AB=ACAB=AC, the center of ⊙O\odot O is the midpoint of the side BCBC, and AB,ACAB,AC are tangent to the circle at points E,FE,F respectively. Point GG is on ⊙O\odot O with ∠AGE=90∘\angle AGE = 90^{\circ}. A tangent line of ⊙O\odot O passes through GG, and meets ACAC at KK. Prove that line BKBK bisects EFEF.
geometry
"n-interesting" meeting

Source: 2018 China Southeast MO Grade 11 P6

7/31/2018
Assume integer m≥2.m \geq 2. There are 3m3m people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "nn-interesting", only if there exists n(n≤3m−1)n(n\leq 3m-1) people of them, the time everyone of whom shakes hands with other 3m−13m-1 people is exactly 1,2,⋯ ,n,1,2,\cdots,n, respectively. If in any "nn-interesting" meeting, there exists 33 people of them who shake hands with each other, find the minimum value of n.n.
combinatorics