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Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Algebraic equations and systems of equations Systems of linear equations

14–15 3.0

Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).

Problem #PRU-78130

Problems Set theory and logic Algorithm Theory Game Theory Game Theory Algebra Algebraic equations and systems of equations Systems of linear equations

12–14 3.0

There is a system of equations \[\begin{aligned} * x + * y + * z &= 0,\\ * x + * y + * z &= 0,\\ * x + * y + * z &= 0. \end{aligned}\] Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.