isi-entrance 2014 Q1

isi-entrance · India · solved Inequalities Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.)

If $a, b, c, d$ are real numbers such that $a - b^2 \geq 1/4$, $b - c^2 \geq 1/4$, $c - d^2 \geq 1/4$, $d - a^2 \geq 1/4$, find the number of solutions $(a, b, c, d)$.
(A) 0 (B) 1 (C) 2 (D) Infinitely many

(B) Only one solution: $a = b = c = d = 1/2$.

If $a, b, c, d$ are real numbers such that $a - b^2 \geq 1/4$, $b - c^2 \geq 1/4$, $c - d^2 \geq 1/4$, $d - a^2 \geq 1/4$, find the number of solutions $(a, b, c, d)$.

(A) 0 \quad (B) 1 \quad (C) 2 \quad (D) Infinitely many

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