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Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Set theory and logic Logic and set theory

12–14 2.0

Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.

Problem #PRU-61353

Problems Algebra and arithmetic 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-64548

Problems Calculus Functions of one variable. Continuity Monotonicity, boundedness Algebra and arithmetic Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

3.0

Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).