Worksheet1

Problem-Solving Workshop February 26, 2009 1. Find the missing number in the pattern below:

1, 3, 6, 11, 18,:...--, 42 2. Which digit is in the hundreds place of the sum of the following expression: 7

+ 77 + 777 + 7777 + ... + 77777777777777777777

3. How much money would you make in one week if you made $5 every time the hands of a. clock formed a 90° angle? 4.:. A teacher wants to give each student in .her class some count.ers 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? 5. A certain town has streets and avenues forming a 3 x 3 array of blocks as shown below.

Every night, the police chief posts officers at intersections to watch the streets of the 'town. Each officer can watch one block from where he stands in eaCh of the four main directions. a) What is the minimum number of officers required to watch every block? b) Prove that your answer in part (a) above is correct. c) Generalize your conclusi9ns to cases with more streets and more avenues (e.g.; 4 x 4 and 5 x 6 arrays) .. 6. 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 x 3 magic squar~.

Problem-Solving Workshop February 26, 2009 1. Find the missing number in the pattern below:

1, 3, 6, 11, 18,:...--, 42 2. Which digit is in the hundreds place of the sum of the following expression: 7

+ 77 + 777 + 7777 + ... + 77777777777777777777

3. How much money would you make in one week if you made $5 every time the hands of a. clock formed a 90° angle? 4.:. A teacher wants to give each student in her class some count.ers 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? 5. A certain town has streets and avenues forming a 3 x 3 array of blocks as shown below.

Every night, the police chief posts officers at intersections to watch the streets of the ·town. Each officer can watch one block from where he stands in each of the four main directions. a) What is the minimum number of officers required to watch every block? b) Prove that your answer in part (a) above is correct. c) Generalize your conclusi9ns to cases with more streets and more avenues (e.g.; 4 x 4 and 5 x 6 arrays). 6. 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 x 3 magic square$.

7. An acute-angled triangle ABC has orthocentre H. The circle passing throu~ H with centre the midpoint of BC intersects the. line BC at Al and Ai; . Similarly,the; circle passing through H with centre the midpoint 9f CA intersect6 t.he line CA atB1and B2, and the circle passing through H with centre the midpoint of AB ili.tersects :the ,line AB at C1 and C2• Show that At, A2, Bt, B2,Ct, C2, lie on a circle .. .' . 8. Find all functions f : (0, 00) ~ (0, 00) (sp, numbers to the positive real numbers) such th~

f

is a function from the positive real

'i.

+ (f(x»2 + f(Z2)

(J(W)}2 f(y2)

for all-positive real numbersw,

x, y, z

_ w2 - y2

satisfying wx

+ x2 + Z2

= yz.

9(a). Prove that x2

p; - J.r

+I

y2

4 ~•.•

+

I

z2

4~"'>

1

for. all real numbers x, y, z, each different from 1, arid satisfying xyz

= 1.

b. Prove that equality holds~1:)9ve for infinitely many triples of rational numbers x, y, z, each different from 1, and satis,fying xy z -'I.

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