Puzzle Hut

The Mathematical Crusade 2011The Mathematical Society, Delhi Public School, R.K. Puram

Instructions

1. There are six questions in the entire paper, with one question on each page,each with a weightage of 10 marks. Page 7, with question 6, is markedwith “—End—”. Ensure that your question paper is complete – it is yourresponsibility.

2. Complete solutions (not referring to any obscure/highly advanced theory)are required for full marks, in any of the questions.

3. Partial marking will be given for any relevant working submitted for anyquestion.

4. You are free to use extra, separate answer sheets (which you should ask usfor) to elaborate your solutions. You can also choose to not use the questionpapers themselves for the solutions at all, and do all the working in thesheets.

5. You are allowed (, and advised) to tear the question paper so that differentteam members can work on different questions.

6. You are free (, and again, advised) to work on these questions in whateverfree time you have while participating in any other event, or while anotherevent is taking place.

7. It is your responsibility to ensure you submit all the sheets which contain anyof your answers or working during collection of answer sheets, for inclusionof evaluation of all your efforts on the questions. Each of those sheets shouldalso include your school details clearly.

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Questions

Question 1. Suppose a and b are real numbers, such that the roots of the cubic equationax3 − x2 + bx− 1 = 0 are all positive real numbers. Prove that:

(a) 0 < 3ab ≤ 1; and

(b) b ≥√

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Question 2. A log of wood, 1 unit in length, is cut into three pieces. What is the proba-bility that these pieces can be used to form the sides of a triangle?

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Question 3. A and B are playing a game, and they initially have a and b units of money,respectively. The game involves a coin toss at every move. If the coin comesup heads, A gives one coin to B. If the coin comes up tails, B gives one cointo A. The game ends when somebody runs out of money, and is declared theloser. What is the probability that A loses? (Give the answer in terms of aand b)

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Question 4. There are 10 people in a room. For any one person, every other person iseither an acquaintance or not. Prove that at least two people have the samenumber of acquaintances.

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Question 5. Prove that, for a given n

1 + x +x2

2!+ . . . +

x2n

(2n)!=

2n∑i=0

xi

i!> 0

for all x ∈ R.

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Question 6. In a regular nonagon (nine-sided polygon) ABCDEFGHI, prove that AF =AB + AC.

*End*

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