8/3/2019 Senior Quiz - Prelims

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Senior Quiz Prelims

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

Complete solutions are required for full credit. Write your answers

neatly in separate answer sheets. The first 9 questions each weigh 10

marks each. The last 3 questions weigh 13 marks each. You have 60

minutes to attempts this paper.

QuestionsQ 1. I have four distinct rings that I want to wear on my right hand hand (five

distinct fingers.) One of these rings is a Canadian ring that must be wornon a finger by itself, the rest I can arrange however I want. If I have two ormore rings on the same finger, then I consider different orders of rings alongthe same finger to be different arrangements. How many different ways canI wear the rings on my fingers?

Q 2. Suppose there is a 100-storey building, and we are given 2 eggs. We need tofind the highest floor from which an egg can be dropped without breaking

it. You also know the following:

An egg that survives a fall can be used again.

A broken egg must be discarded.

The effect of a fall is the same for all eggs.

If an egg breaks when dropped, then it would break if dropped from ahigher window.

If an egg survives a fall then it would survive a shorter fall.

It is not ruled out that the first-floor windows break eggs, nor is itruled out that the 100th-floor windows do not cause an egg to break.(Question on next page)

What is the least number of egg-droppings that is guaranteed to work in allcases?

Q 3. Prove that there exists an injective mapping1 from:

(a) The set of integers, to the set of naturals2.

(b) The set of the positive rationals to the set of naturals.

1An injective mapping, also called one-to-one correspondence is a relation f such that f(x) =f(y) = x = y

2The set of naturals is defined as the set of positive integers (not including zero)

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Q 4. I have a pan balance. The pan balance can only tell me when the weightsplaced on either side are equal. I need to a measure the weight of a bagwhich I know has an integral weight less than a thousand kilograms.Now, Ihave to order standard weights to measure this bag. The standard weightsall have to be distinct and integral. What is the minimum number of weightsthat I require?

Q 5. Prove that n! + 3 can never be a square for n 4.

Q 6. Prove that n

3

n3

is a natural number divisible by 3 for all values of n.

Q 7. A and B are playing a game with an empty 2n2n matrix, and in each move,one player adds one number to a position, as per his desire. Player A wins ifthe matrix formed at the end of the game has a non-zero determinant, whilePlayer B wins if it has determinant zero. Who has the winning strategy andwhat is it?

Q 8. The function f satisfies f(x) + f(2x + y) + 5xy = f(3x y) + 2x2 + 1 for allreal numbers x, y. Determine the value of f(10).

Q 9. Prove the A.M. G.M. inequality for 4 variables, i.e. prove the following:

a + b + c + d

4 (abcd)

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Q 10. At the second International Congress of Mathematicians, in 1900, a renownedmathematician of that time announced a now famous list of unsolved prob-lems in mathematics, some of which remain unresolved or partially resolvedto this day, and greatly shaped the direction of mathematical research wellinto the 20th century. By what name do we call this list?

Q 11. This well known theorem is the undisputed contender for the largest numberof proofs in published mathematical literature. One book about this propo-sition contains 370 proofs by itself. The closest contender to this title isthe Law of Quadratic Reciprocity which itself has 200 proofs. What is this

infamous theorem?Q 12. The great mathematician Isaac Newton once said I am ashamed to tell you

to how many figures I carried these computations, having no other businessat the time. What was he talking about?

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