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Found: 105

Problem #PRU-116589

Problems Number Theory Divisibility Divisibility of a number. General properties Algebra Sequences Recurrent relations Linear recurrent relations Calculus Number sequences Number sequences (other)

The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and

for all \(n \geq 2\).

Prove that all members of the sequence are integers.

Problem #PRU-116627

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Integer and fractional parts. Archimedean property

Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).

Problem #PRU-35105

Problems Algebra Exponential functions and logarithms Calculus Real numbers Rational and irrational numbers Surds. Rational powers

Are there any irrational numbers \(a\) and \(b\) such that the degree of \(a^b\) is a rational number?

Problem #PRU-35379

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

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

Problem #WSP-000238

Problems Calculus Real numbers Countable and uncountable subsets

We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.

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