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The Mathematical Relay Juniors

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

Instructions

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

2. All answers shall be evaluated only objectively. Marks shall be given onlyfor the correct answer written in the provided space. There is no negativemarking.

3. You are supposed to return the answers in the question paper, along withyour rough sheets, when you are leaving. Do your rough work in separatesheets. There should only be the answer in the box provided for the same.

4. You are not permitted to leave before 35 minutes of the commencement ofthe event. You will be required to submit your work after 65 minutes. You

are free to submit your answers anywhere between that span of time. Theentire event is 100 minutes long, and the earlier you leave, the more timeyour seniors will have to attempt their questions (the Common and Seniorspapers).

5. Tip: There are a lot of questions, especially in comparison to the time youhave. If a question seems too tough to manage, consider abandoning it.

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Questions

Question 1. How many subsets A of the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 have the property that

no two elements of A sum to 11?

Question 2. How many positive integers divide 10! 1?

Question 3. Two circles have radii 15 and 95. If the two external tangents to the circlesintersect at 60 degrees, how far apart are the centers of the circles?

Question 4. The rodent control task force went into the woods one day and caught 200rabbits and 18 squirrels. The next day they went into the woods and caught3 fewer rabbits and two more squirrels than the day before. Each day theywent into the woods and caught 3 fewer rabbits and two more squirrels thanthe day before. This continued through the day when they caught moresquirrels than rabbits. Up through that day how many rabbits did theycatch in all?

1n!, read as n factorial, is defined as n! = n (n 1) (n 2) . . . 2 1

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Question 5. Paul starts with the number 19. In one step, he can add 1 to his number,divide his number by 2, or divide his number by 3. What is the minimumnumber of steps Paul needs to get to 1?

Question 6. The sum of nine consecutive odd numbers is 2007. Find the greatest of thesenine numbers.

Question 7. Five guys are eating hamburgers. Each one puts a top half and a bottomhalf of a hamburger bun on the grill. When the buns are toasted, each guyrandomly takes two pieces of bread off the grill. What is the probability thateach guy gets a top half and a bottom half?

Question 8. And it came to pass that Jeb owned over a thousand chickens. So Jebcounted his chickens. And Jeb reported the count to Hannah. And Hannahreported the count to Joshua. And Joshua reported the count to Caleb.And Caleb reported the count to Rachel. But as fate would have it, Jebhad over-counted his chickens by nine chickens. Then Hannah interchangedthe last two digits of the count before reporting it to Joshua. And Joshuainterchanged the first and the third digits of the number reported to him

before reporting it to Caleb. Then Caleb doubled the number reported tohim before reporting it to Rachel. Now it so happens that the count reportedto Rachel was the correct number of chickens that Jeb owned. How manychickens was that?

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Question 9. There is a coin of radius 5 cm, and a smaller coin of radius 3 cm that isrotating around it, such that there is no slipping. How much distance iscovered by centre of the smaller coin, when it comes back to the starting

point?

Question 10. Find all possible quadruples (a,b,c,d) of positive integers such that ab+cd =a + b + c + d + 3.

*End*

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