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Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities Probability and statistics Probability theory Discrete distribution Number inequalities. Number comparison

14–16 4.0

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

Problem #PRU-61353

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

11–14 2.0

Prove that $\sqrt{\frac{a^{2} + b^{2}}{2}} \geq \frac{a + b}{2}$ .

Problem #PRU-61392

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

13–15 3.0

We are given rational positive numbers $p, q$ where $1 / p + 1 / q = 1$ . Prove that for positive $a$ and $b$ , the following inequality holds: $a b \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$ .