rmo aakash lecture 10, 15th oct 2026

Q1.
Let a, b, c ∈ ℝ. Prove the inequalities:

3(ab + bc + ca) ≤ (a + b + c)² ≤ 3(a² + b² + c²)
S1.
First see the RHS one.
Cauchy shwartz.
(1^2 + 1^2 + 1^2)(a^2 + b^2 + c^2) >= (1.a + 1.b + 1.c)^2
Since
(a1.b1 + a2.b2 + a3.b3)^2 <= (a1^2 + a2^2 + a3^2).(b1^2 + b2^2 + b3^2)
And equality happens when:
a1/b1 = a2/b2 = a3/b3
=>
a/1 = b/1 = c/1 => a = b = c

Now the LHS one,
we need to do effectively this:
ab + bc + ca <= a^2 + b^2 + c^2
WLOG a <= b <= c
Which is simple using rearrangement inequality.
And in RE also a = b = c gives the equality.


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