MathDB

5

Part of 2017 South East Mathematical Olympiad

Problems(2)

CSMO Grade 10 Problem 5

Source: CSMO Grade 10 Problem 5

8/6/2017
Let ABCDABCD be a cyclic quadrilateral inscribed in circle OO, where AC⊥BDAC\perp BD. M,NM,N are the midpoint of arc ADC,ABCADC,ABC. DODO and ANAN intersect each other at GG, the line passes through GG and parellel to NCNC intersect CDCD at KK. Prove that AK⊥BMAK\perp BM.
geometryperpendicular
CSMO Grade 11 Problem 5

Source: China Southeast Mathematical Olympiad

8/1/2017
Let a,b,ca, b, c be real numbers, a≠0a \neq 0. If the equation 2ax2+bx+c=02ax^2 + bx + c = 0 has real root on the interval [−1,1][-1, 1]. Prove that min⁡{c,a+c+1}≤max⁡{∣b−a+1∣,∣b+a−1∣},\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\}, and determine the necessary and sufficient conditions of a,b,ca, b, c for the equality case to be achieved.
inequalities