8/3/2019 JQ - Prelims

1/2

Junior Quiz Prelims

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

The first 9 questions each weigh 5 marks each, and working is re-

quired for full credit. Write your answers neatly in separate answer

sheets. The last 3 questions weigh 8 marks each. You have 60 minutes

to attempts this paper. There are two pages to the question paper.

Questions

Q 1. Evaluate 2007! + 2004!

2006! + 2005!

Q 2. Two reals x and y are such that x y = 4 and x3 y

3

= 28. Compute xy.

Q 3. Let a and b be integral solutions to 17a + 6b = 13. What is the smallestpossible positive value for a b?

Q 4. You have a staircase of 10 stairs. In one step, you can either climb onestair, or skip one stair and climb two. What is the number of ways you canclimb the staircase?

Q 5. Find the smallest number that gives remainder 1 on division by 2, 2 ondivision by 3, 3 on division by 4,. . . upto 9 on division by 10.

Q 6. A and B are playing a game where there is a pile of 100 coins, verticallystacked, on a table. A plays the first move, and B plays the next, and so on.In any one move, the player is allowed to take out any (integral) number ofcoins greater than 0, and less than 9 (i.e. 1 8). The aim of the game, is tobe the last player picking a coin off the table. Which player has a winningstrategy, and what is it?

1

8/3/2019 JQ - Prelims

2/2

Q 7. Evaluate the sum

1

21 + 1

+1

22 + 1

+1

23 + 1

+ . . . +1

2100 + 1

Q 8. The Reuleaux triangle is a constant width curve based on an equilateraltriangle (shown in the figure below). All points on a side are equidistantfrom the opposite vertex. Find the area of the grey shaded region, in termsof the side length of the triangle a.

Q 9. A and B play a game of placing coins on a circular table. The circle hasradius 37 and the coins have radius 3. When a coin is placed on the table,it should not overlap with any other coin already on the table, although itmay be tangentially touching it. The game ends when there is no possibility

of placing another coin without overlap, on the table. The If A starts first,who has the winning strategy, and what is it?

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 thisinfamous theorem?

Q 12. The great mathematician Isaac Newton once said I am ashamed to tell youto how many figures I carried these computations, having no other businessat the time. What was he talking about?

2