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Part of 2020 OMMock - Mexico National Olympiad Mock Exam

Problems(1)

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

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

11/8/2020
Let aa, bb, cc and dd positive real numbers with a>ca > c and b<db < d. Assume that
a+b≥c+danda+b≤c+da + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d
Prove that a+b+c+d>1a + b + c + d > 1.
Proposed by Victor Domínguez
algebraInequalitysquare rootinequalities