8/3/2019 Seniors Relay

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Mathematical Relay - Seniors

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

Instructions

1. This question paper has 5 questions each with a total weightage of 10 marks.

2. Working along with the answer is required for full credit. There will bepartial marking for relevant working elucidated.

3. You are free to ask for additional answer sheets to elaborate your answers,or to simply work on the questions.

4. All working on these questions must be submitted by the final time limit of100 minutes after the commencement of the event.

Questions

Q 1. On a 2n 2n board, numbers are written from 1 to 4n2 row wise such thatnumbers from 1 to 2n are written from left to right in the first row and soon. Now the board is coloured in such a way that each row and each columnhas n red squares and n white squares1. Prove that the sum of numbers onthe red squares is equal to that on the white squares.

Q 2. A and B play a game of tossing coins. A wins the game if HTH appears inthree consecutive tosses and B wins if TTH appears2. The game terminatesonce either of the two wins. What is the probability that A wins?

Q 3. Let f be a continuous function that maps [0, 1] to [0, 1]. Prove that, thereexists a c belonging to [0, 1] such that f(c) = c.

Q 4. Prove that there exists a value a R, which is larger than every term ofthe sequence (a

n) = (

2,

22

,

22

2

, . . . n terms), as n tends to infinity.Find the smallest such a.

Q 5. Evaluate the integral 1

0

log(1 + x)

1 + x2dx

1The chess board colouring is only a special case of this2where H stands for heads, and T stands for tails, and HHH implies a sequence of three

consecutive coin tosses in which the coin turns heads up

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