Inequalities (Olympiad)2 questions1 PYQ

Inequalities (Olympiad) — JEE Maths Practice Questions & Solutions

2 questions on Inequalities (Olympiad) with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

mediumPYQ · RMO 2015
The all real numbers x,yx, y such that x2+2y2+12≤x(2y+1)x^2 + 2y^2 + \dfrac{1}{2} \le x(2y + 1).
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hard
If a,b,ca,b,c are positive real numbers with a+b+c=1a+b+c=1 such that
7+2b1+a+7+2c1+b+7+2a1+c≥P4,\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\ge\frac{P}{4},
 then find the maximum possible value of PP for which the inequality above is always true.
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