Mathematical Relay

Peter McNamara

April 2005

J1 Altitude AD of triangle ABC right angled at A cuts BC into lengths BD = 1, DC = 4. What is AD?

J2 How many ways can an IMO team of 6 students be chosen from a class of 8 boys and 6 girls if it is to contain at least 3 boys and at least 2 girls?

J4 How many integers less than 2005 can be represented in the form 6m + 8n where m and n are positive integers.

J3 Find the number of ways of arranging two bishops, two rooks, two knights, one king and one queen on a 1 × 8 chessboard coloured in the usual way, such that the two bishops are on opposite coloured squares. (Pieces of the same type are considered indistinguishable.)

J12 Norm has an automatic card shuffler. This shuffler only works on a pack of exactly thirteen cards, and performs the same shuffle each time (so if the first card gets sent to the sixth card, for example, then the first card will always get sent to the sixth card). Ross takes a deck of cards A, 2, 3, …, J, Q, K and sends them through the card shuffler twice. The cards end up in the following order:

Q K 7 2 6 8 9 A 10 4 5 3 J

What was the order of the cards after one shuffle?

S10 An emergency service is to be set up in a national park. Ranger stations are to be connected by phone lines. Each station must be able to connect to every other station, either by a direct connection or else through at most one other station. Each station can be connected to at most three other stations. What is the largest number of stations which can be connected in this way?

S4 Suppose that P QS is a triangle and R is a point on QS such that P Q · RS = QS · P R. Furthermore suppose that ∠P QR = 37^{o} and ∠P RQ = 47^{o}. Find ∠RP S.

S14 P QRS is a quadrilateral with right angles at P and S. T is a point on P R such that ST ⊥ P R. It is also true that ∠T QR = ∠P ST = 30^{o}. If P S = 1, what are all possible lengths of P Q?

S7 Let O be a vertex of a twelve dimensional unit cube. Find the number of vertices A of this cube for which |OA| < 2.

S2 What is the largest number of knights that can be placed on a standard chessboard such that no two can capture each other (in one move)?

S9 Let f : ℤ⟶ℤ be a function satisfying the following equation.

f(n + 2) = f(n + 1) − f(n)

What is the maximum number of different values of f(n) that can be attained as n ranges over all integers?

S12 Let ABC be a triangle for which ∠B = 30^{0} and ∠C = 45^{o}. Let Γ be a circle through A, tangent to BC and to the circumcircle of ABC. Let X be the point of tangency of Γ and BC. What is the value of the ratio \frac{BX}{XC}?

S3 Find all pairs of integers (F, V) for which there exists a polyhedron with F faces, V vertices and eight edges.

J10 Ross is counting his marbles. He notices that the number of marbles he has is a prime that is less than the number of days in a leap year. Furthermore, he notices the following interesting facts about this number.

(a) Its first power is congruent to 1 mod 3.

(b) Its third power is congruent to 1 mod 5.

(c) Its fifth power is congruent to 1 mod 7.

How many marbles does Ross have?

J9 Find the number of positive integers, strictly less than 2005, which are divisible by none of 33, 5 and 6.

J5 Nick is rowing a boat upstream. He rows at a speed of 17 metres per second relative to the current, which flows at a rate of 5 metres per second. At precisely 12 noon, Nick’s hat blows off, and is carried downwind by the current. At 1pm, Nick realises that he has lost his hat, abruptly turns around, and begins rowing downstream. At what time will Nick reach his hat?

J7 Find the area of a cyclic quadrilateral with sides of length (in order) 1,7,8,4.

S15 Find all composite r which can be written as the sum of two nonzero squares, and is also the remainder upon dividing some prime number by 210.

J14 Suppose 0 ≤ m ≤ n are integers. Find all such pairs (m, n) that additionally satisfy the following equation:

\frac{m^{2} + mn + n^{2}}{m + n} = \frac{49}{4}.

S11 Let n be a positive integer, and G a graph with the following properties

(a) G contains n vertices.

(b) G contains at least one edge.

(c) There exists two vertices of G, without an edge between them.

(d) Every vertex of G has the same degree.

(e) If there are three vertices A, B, C of G such that there is an edge between A and B, and an edge between A and C, then there is an edge between B and C.

Find the number of integers n ≤ 100 for which there exists such a graph G.

J6 Suppose that the incircle of triangle ABC touches the sides AB, BC and CA at points F, E and D respectively. Suppose that D, E and F divide the circumference of the incircle in the ratio DE : EF : F D = 16 : 11 : 9. What is the size of ∠BAC?

J8 (General Knowledge - 5 marks) Everybody has heard of the Möbius strip, named after one August Ferdinand Möbius. But the question is, in what century did Möbius live the majority of his life in?

S13 What is the last nonzero digit of 99!, when written in decimal notation?

S5 In an n × n chessboard, aligned with the co-ordinate axes, how many pairs of (1 × 1) squares are there with one lying above and to the right of the other?

J11 Two rods of length 1 are divided in to 400 and 501 parts respectively. They are then placed side by side so that their ends coincide. Find the minimum distance apart of the division marks.

S8 (General Knowledge - 5 marks) How may children did Euler have? [hint: it is prime]

S1 David is older than Stewart. Ivan comes along and says: ”Hey guys, check this out, if p(x) is the monic quadratic polynomial, whose roots are David and Stewart’s age (which are assumed to be non-negative integers), then p(1) is prime!

How old is Stewart?

J13 Let AB be a chord of a circle with centre O. C is a point on AB, and the radius through C intersects the circle at D. AC = 3, BC = 5 and CD = 1. What is the radius of the circle?

S6 What is the maximum number of terms of a geometric progression with common ratio greater than one, whose entries are all integers from 25 to 250 inclusive?

J15 Alice: I am insane.

Bob: I am pure.

Charlie: I am applied

Dorothy: I am sane.

Alice: Charlie is pure.

Bob: Dorothy is insane.

Charlie: Bob is applied.

Dorothy: Charlie is sane.

Describe Alice’s sanity and purity, given that pure mathematicians tell the truth about their beliefs, applied mathematicians lie about their beliefs, sane mathematicians’ beliefs are correct and insane mathematicians’ beliefs are incorrect.