Session 2 Problems

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From 1985

A1. Determine, with proof, the number of ordered triples of sets which satisfy:

(i) and

(ii)

Express the answer in the form 2^a 3^b 5^c 7^d where a, b, c, d are non-negative integers.

A5. Let

For which integers m between 1 and 10 is I_m not equal to 0?

From 1990

A1. Let and for n Ñ 3,

The first few terms are

2, 3, 6, 14, 40, 152, 784, 5168, 40576, 363392

Find, with proof, a formula for T_n of the form A_n + B_n, where {A_n} and {B_n} are well-known sequences.

A2. Is the square root of 2 the limit of a sequence of numbers of the form

(n, m = 0, 1, 2, 3, . . .)?

From 1995

A4. Suppose we have a necklace of n beads. Each bead is labeled with an integer and the sum of all these labels is n ≠ 1. Prove that we can cut the necklace to form a string whose consecutive labels x₁, x₂, x₃, . . . satisfy

B4. Evaluate

Express your answer in the form

From 2000

A1. Let A be a positive real number. What are the possible values of given that x₀, x₁, x₂, x₃ . . . are positive numbers for which ?

A5. Three distinct points with integer coordinates lie in the plane on a circle of radius r > 0. Show that a distance of at least r^1/3 separates two of these points.

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