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Problem-Solving Workshop UWI, Mona Campus March 5, 2009 1. Find the missing number in the pattern below:

1, 3, 6, 11, 18,_-_, 2. Find the next

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numbers in the sequence 22, 4, 26, 8, 34, 7, 41, 5,

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A· teacher wants to give each student in her .~lass some tountez:s for an activity. She _wants everyone to get an equal number of counters. If she splits the class into groups of 2, 3, or -4 students, she will have 1 counter left over after passing them all out. If she uses groups of 5, she will have none left over. If she has fewer than 100 counters, how many could she possibly have?

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What is the smallest number which has remainder 1 when divided by 2,3,4,5,

and 6?

(b) -What is the smallest number which is divisible by 7 and has remainder 1 when divided by 2, 3, 4, 5, and 6? 5. Find a number x that fits all the following conditions: x

+ 2,

remainder x + 3, remainder x + 4, remainder x + 5, remainder

1 2 3 4

x + 6, x + 7, x + 8, x + 9,

remainder remainder remainder remainder

5 6 7 8

:Is there· more than one such number? If so, what is the smallest number meeting these conditions? 6. Which digit is in the hundreds place of the sum .of the following expression: 7 + 77

+ 777 + 7777 + ... + 77777777777777777777

7. How much money would you make in one week if you made $5 every time the hands of a clock formed a 90° angle?

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The diagram below sh6ws' a. starting number and the result when operation "H" is applied. For instance, if you;i,tartvvi~h 36 anq ,apply the I'-t operation, the result is 8. Three more examples are ,given.belOw.J:"nt'in the blanks for the missing examples . ~." ~.,

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36H

. 53H7;

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11 h4 1;

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H 14;

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49H __

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What rule are you using to determine these values? 9. A certain town has one-way streets arranged as shown below. Cars enter the town center at point A marked in the diagra.m below. As they come to each intersection, half .thecars take each available path (shown by the arrows). If t4ereis qnly one path available ",:all.cars'proce~along that path. If 416 cars enter the town centre at point A and proceed ,,':~ording to these rules, how many will arrive at point C?

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c 10. A 3 x 3 magic square is one in which the numbers 1, 2, ... , 9 are arranged so that each row, column, and diagonal sum is the same. Find all possible 3 x3 'ma;gicsquar~. 11. How many circles of radius

't can fit fnside (without

overlapping) a circle of radius 2.4?

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