2017 CHKMO

Q1

A, B and C are three persons among a set P of n (n>3) persons. It is known that A, B and C are friends of one another, and that every one of the three persons has already made friends with more than half the total number of people in P. Given that every three persons who are friends of one another form a friendly group, what is the minimum number of friendly groups that may exist in P?

Q2

Let k be a positive integer. Find the number of non-negative integers n less than or equal to $10^k$ satisfying the following conditions: (i) n is divisible by 3; (ii) Each decimal digit of n is one of the digits 2,0,1 or 7.

Q3

Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.

Q4

Find the smallest possible value of the nonnegative number $\lambda$ such that the inequality $$\frac{a+b}{2}\geq\lambda \sqrt{ab}+(1-\lambda )\sqrt{\frac{a^2+b^2}{2}}$$holds for all positive real numbers $a, b$.