Pmwc Problems

7/25/2019 Pmwc Problems

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INDIVIDUAL CONTEST PROBLEMS

Po Leung Kuk 11th Primary Mathematics World ContestEnglish Version

1. A regular hexagon is given. The vertices of the rectangle lie on themidoints of the sides of the hexagon. What is the ratio of the area of therectangle to the area of the hexagon!

". A multilication of a three#digit num$er $y a t%o#digit num$er has the formas sho%n $elo%. &sing only the digits "' (' ) or *' fill in all $oxes tocomlete the correct multilication.

3. +o% many different %ays are there to form a three#digit even num$erchoosing the digits from ,' 1' "' (' - or ) %ithout reetition!

4. or ho% many %hole num$ers $et%een 1,, and /// does the roduct ofthe ones digit and tens digit e0ual the hundreds digit!

5. n a survey of 1,, ersons' it %as found that "2 read maga3ine A' (,

read maga3ine 4' -" read maga3ine C' 2 read maga3ines A and 4' 1,read maga3ines A and C' ) read maga3ines 4 and C and ( read all threemaga3ines. +o% many eole do not read any of these maga3ines!

6. A school has to $uy at least 111 ens. The ens are sold in acks of )%hich cost 56 er ack or acks of * %hich cost 5* er ack. What is thelo%est cost at %hich the school can $uy the ens!

7. +o% many digits does the roduct 16") × (2

" have!

8. 7n a %ooden rod' there are markings for three different scales. The first

set of markings divides the rod into 1, e0ual arts8 the second set of

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markings divides the rod into 1" e0ual arts8 the third set of markingsdivides the rod into 1) e0ual arts. f one cuts the rod at each marking'ho% many ieces does one get!

9. There are ten identical candies in a 9ar. Al$ert can only eat 1 or " of thesecandies at a time. +e does this until there is no more candy left. n ho%many different %ays can he do this!

10. The entrance fee to a museum is 5) er adult and 5- er child. or anygrou of five eole' the entrance fee is 51/. T%o adults %ho ay the fullentrance fee may take a child for free. Three adults and fourteen childrencome to visit the museum. What is the least amount they need to sendon the entrance fee!

11. A' B' C ' D' A:C ' B:C ' A:D' B:D reresent the eight different naturalnum$ers 1 to 2. f A is the largest num$er amongst A' B' C and D' %hat is A!

12. A nine#digit num$er abcdefghi is such that its digits are all distinct andnon#3ero. The t%o#digit num$er ab is divisi$le $y "' the three#digitnum$er abc is divisi$le $y (' the four#digit num$er abcd is divisi$le $y-' and so on so that the nine#digit num$er abcdefghi is divisi$le $y /.ind this nine#digit num$er.

13. n ho% many %ays can seven students A' B' C ' D' E ' F and G line u inone ro% if students B and C are al%ays next to each other!

14. A 1,,1#digit num$er $egins %ith 6. The num$er formed $y any t%oad9acent digits is divisi$le $y 1* or "(. Write do%n the last six digits.

15. The attern $elo% is formed $y dra%ing semi#circles inside s0uares. Theradii of three tyes of semi#circles are - cm' " cm and 1 cm resectively.What is the total area of the shaded regions! ;Take π < (.1-=.

2

8 cm



TEAM CONTEST PROBLEMS

Po Leung Kuk 11th Primary Mathematics World ContestEnglish Version

Question 1

The diagram $elo% is the street ma of a small to%n. There is a very strangetraffic rule. >o turns are allo%ed at any intersection unless it is imossi$le todrive straight on. Then $oth left turns and right turns' if ossi$le' are allo%ed.?ntering the to%n from oint ?' it is ossi$le to exit from any other ointexcet one. Which exit is imossi$le!

Question !There are 1, hats. ?ach hat is a different colour. T%o hats are cotton ;5(,each=' five are leather ;5), each= and three are %ool ;51, each=. +o% many%ays are there to $uy ) hats such that the total cost is more than 51,1 $utless than 51-/!

Question "

7n a 51×

$oard are four counters %hich are %hite on one side and $lack onthe other side. A counter can only change osition $y 9uming over at leastone other counter and landing on the emty sace. When a counter has $een 9umed over' it is flied over' $ut the 9uming counter itself is not flied. Theconfiguration in the diagram $elo% on the left must $e changed to that on theright in six 9ums. @ecord each 9um $y indicating the initial osition of the 9uming counter. ive one ossi$le solution and its corresonding 6#digitnum$er.

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Question # At a certain school' four students W ' X ' Y and Z %ere redicting their grades$efore the final examination.

W saidB We %ill all get different grades.f get an AD' then Y %ill get a ED.

X saidB f Y gets a CD' then W %ill get a ED.W %ill get a $etter grade than Z .

Y saidB f X does not get an AD' then W %ill get a CD.f get a 4D' then Z %ill not get a ED.

Z saidB f Y gets an AD' then %ill get a 4D.f X does not get a 4D' %ill not either.

After the final examination %as graded' each of the students got his gradeas redicted. What grade did each student get!

Ans%erB $ % & '

Question ( A circle of radius 1 cm rolls along the inside lines of the icture. The sidelength of each small s0uare in the icture is 1 cm. What is the area in s0uarecentimetres that the circle covers %hen it rolls along the inside lines once!;Take )14.3=π

Question )n ABC ' E is the midoint of BC . F is on AE %here AE < ( AF . BF meets AC at D as sho%n in the figure. f the area of ABC < -2' find the area of F AFD.

Question *

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Consecutive counting num$ers are groued as follo%sB;1=' ;"' (=' ;-' )' 6=' ;*' 2' /' 1,=' . . .

There is one num$er in the first grou' t%o num$ers in the second grou' andthree num$ers in the third grou' etc. What is the sum of all num$ers in the",,*th grou!

Question + A rectangle of area (-)6 cm" lies on the grid lines of a larger grid %hich isformed $y s0uares of side 1 cm as sho%n $elo%B

We call the oints %here the grid lines meet Goints of intersectionH or examle' the diagonal of a " cm × - cm rectangle asses through ( oints of intersection.

What is the greatest ossi$le num$er of oints of intersection %hich adiagonal of the rectangle of area (-)6 cm" can ass!

Question ,There are ", iles of stones. ?ach has 1,, stones. Choose one of the t%entyiles' take one stone from each of the remaining 1/ iles and ut them onto

the chosen ile. This is called an oeration. n su$se0uent oerations' youmay choose any ile amongst the t%enty iles' and reeat the a$ove rocess. After less than ), oerations' there are 66 stones in one of the iles. Thenum$er of stones in another ile is $et%een 1*, and ",, ;inclusive=. What isthe exact num$er of stones in this ile!

Question 1- A alindromic num$er is a %hole num$er that is the same %hen %rittenfor%ards or $ack%ards ;for examle' 11)11' """""' 1,,,1=. ind the ratio' inroer fraction form' of the num$er of all five#digit alindromic num$ers %hichare multiles of eleven to the num$er of all five#digit alindromic num$ers.

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A>IW?@I 7 T+? C7>T?IT P@74L?MIPo Leung Kuk 11th Primary Mathematics World Contest

+ong Kong' 1( J 12 uly ",,*

Individual Test Team Contest

1. 1:2 1. C

2. 775 x 33 2. 35

3. 52 3. 152415

4. 23 4. W B, X A. Y D, Z C

5. 20 5. 52.99

6. 112 6. 1.6

7. 34 7. 4042148175

8. 28 8. 25

9. 89 9. 186

10. 64 10. 41/45011. 6

12. 381654729

13. 1440

14. 692346

15. 38.88

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Pmwc Problems

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