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Found: 3

Problems Methods Algebraic methods Counting in two ways Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.

Problem #PRU-32884

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-98170

Problems Methods Extremal principle Extremal principle (other) Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

10–14 4.0

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?