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Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation. Polynomials with integer coefficients and integer values Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–12 3.0

Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.

Problem #PRU-60845

Problems Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–14 3.0

Let the number \(\alpha\) be given by the decimal:

a) \(0.101001000100001000001 \dots\);

b) \(0.123456789101112131415 \dots\).

Will this number be rational?

Problem #PRU-60863

Problems Calculus Real numbers Rational and irrational numbers Algebra and arithmetic Surds. Rational powers. Square roots

12–16 2.0

Prove that the number \(\sqrt {2} + \sqrt {3} + \sqrt {5} + \sqrt {7} + \sqrt {11} + \sqrt {13} + \sqrt {17}\) is irrational.