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Pair of similar inequalities with square root implies sum greater than 1

Source: Mexico National Olympiad Mock Exam (OMMock) 2020 P1

November 8, 2020

algebraInequalitysquare rootinequalities

Problem Statement

Let

a

b

c

and

d

positive real numbers with

a > c

and

b < d

. Assume that

a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d

Prove that

a + b + c + d > 1

.
Proposed by Victor Domínguez