ukmt maths competition 2016.pdf · crossnumber response sheet team maths challenge 2016 national...

CrossnumberTeam Maths Challenge

2016National Final

UKMT

UKM

TUKMT

1 2 3 4

5 6

7 8 9

10 11 12

13 14 15 16

17 18 19

20 21 22

23

24 25

Across1. 7 Across minus 11 Down (3)

3. The square root of233 + 243 + 253 (3)

5. A palindromic number whosedigits have a sum of 33 (5)

7. The difference between twoconsecutive Fibonaccinumbers (3)

8. A number less than 3 Down (3)

10. A multiple of 18 Down (4)

13. The largest two-digit factor of2 Down (2)

15. 43 plus the square root of12 Down (2)

17. 4 + 10 × 122 (4)

20. 21 Down minus a cube (3)

22. nn + 1 for some n (3)

23. A factor of 5 Across (5)

24. The mean of 14 Down and1 Across (3)

25. 4994 minus 19 Down (3)

© UKMT 2016

CrossnumberTeam Maths Challenge

2016National Final

UKMT

UKM

TUKMT

1 2 3 4

5 6

7 8 9

10 11 12

13 14 15 16

17 18 19

20 21 22

23

24 25

Down1. A palindromic number whose

digits have a sum of 41 andwhich consists of two differentdigits (5)

2. 88 × 77 (4)

3. A number greater than8 Across (3)

4. A square that is8 Across × 3 − 102 (3)

6. A Fibonacci number that is amultiple of 13 (3)

9. 15 Across × 11 + 11 Down (3)

11. A multiple of 19 (3)

12. A square (3)

14. A multiple of 13 Across (3)

16. 55 + 45 + 75 + 45 + 85 (5)

18. A factor of 10 Across (3)

19. The product of its digits is432 (4)

20. The mean of 20 Down and21 Down (3)

21. A power of 2 (3)

© UKMT 2016

Crossnumber response sheet Team Maths Challenge 2016 National Final

Team number School name

1 2 3 4

5 6

7 8 9

10 11 12

13 14 15 16

17 18 19

20 21 22

23

24 25

Rowtotals

/6

/7

/7

/6

/6

/6

/7

/7

/6

Correct digit: place a tick in the dotted circle.Incorrect digit: cross out the answer, write in the correct digit,

and place a cross in the dotted circle.Row totals: enter the number of ticks in each row.

Final score /58

© UKMT 2016

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A1Place the numbers A, B and C in decreasing order of size.

A = 78

B = 0.86C= 87 %

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A2What is the next year with digits having the same median andrange as the digits of this year, 2016?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A3A wheel makes 840 revolutions in one hour.

How long does it take in minutes, at the same rate, to complete28 revolutions?

Answer: minutes

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A4Fifty people were stopped in the street and asked if they watchedTV or listened to the radio that day.

Eight said they did neither, forty-one watched TV and six listenedto the radio.

How many watched TV and also listened to the radio?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A5A large tin has 100 sweets in it. The first person in a long queuetakes 1, the second 2, the third 3, with each person taking onemore sweet than the person before them.

How many sweets are left when the next person cannot taketheir number of sweets?

Answer: sweets

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A6The sign in a railway carriage lists the stops: Havant, Cosham,Portchester, Fareham, Swanwick and Southampton.

What is the median number of letters in the station names?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A7a, b, c, and d represent the numbers 2, 3, 4 and 5, but notnecessarily in that order.

What is the smallest fraction with numerator (a + d) anddenominator (b + c)?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A8The fair offers a special ticket of £12 for an evening pass allowingunlimited rides. Alternatively each ride will cost £1 per person.

Four friends, Bert, Carol, Del and Eve each buy the specialticket. Carol and Eve go on 18 rides each and Bert and Del goon 9 and 14 respectively.

Altogether how much did they save by buying the special tickets?

Answer: £

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A9The points A (2,1), B (5,3), C (3,6) and D form a square.

The square is reflected in the line x = 4.

What are the coordinates of the image of point D?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A10A hiker practising compass work walks a distance of 24 metreson a bearing of 050° then 32 metres on a bearing of 140°.

Finally she returns in a straight line to her starting point.

What total distance has she walked?

Answer: metres

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A11The trapezium PQRS is plotted on a graph.

PQ is parallel to SR.

The trapezium has one line of symmetry,

P = (0,0)Q = (3,0)R = (5,5)

What is the area of the trapezium?

Answer: square units

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A12An official document needed the date written in the formdd/mm/yyyy.

On the 26 January this year I noticed that 4 digits were eachused twice.

How many days later is the last date for which this happensagain this year?

Answer: days

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A13Ann buys wool in a shop which charges her £6 for 10 balls. Suegets her wool in a shop where £7.80 is the cost of 12 balls.

They use the same number of balls of wool to knit identicaljumpers.

One of them spent 20p less than the other on the wool that sheused.

How many balls of wool did they use between them?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A14A tap is dripping constantly at 12 ml per 30 seconds. Thiscontinues until detected three hours later and stopped.

How much water was wasted in the three hours?

Answer: litres

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

A15A book containing 390 puzzles is printed with two puzzles oneach page. It took Nick 17 minutes, on average, to complete onepage.

How long, in hours and minutes, did it take Nick to completethe puzzle book?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B1A rectangle with length 20 cm and width 16 cm has the sameperimeter as a square.

What is the area of the square?

Answer: cm2

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B2A big box contains 100 sweets.

The first person takes one sweet, the next two, and so on, eachtaking twice as many as the one before.

How many sweets are left when the next person cannot take herfull share?

Answer: sweets

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B3a, b, c, and d represent the numbers 2, 3, 4 and 5, but notnecessarily in that order.

What is the largest fraction with numerator (a + b) anddenominator (c + d)?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B4On the first day of each month Joshua receives £7.60 pocketmoney. On his birthday, on 29 February, he is given a 5%increase.

How much will he receive this year in pocket money?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B5An office joker arranges for the office clock to run backwardsfrom midnight on 31 March.

What time does the clock show when the manager enters theoffice at 08.37 on April Fool’s Day (1 April)?

Use a 24-hour clock.

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B6Thirty-five students were asked to sign up for at least one ofthe three sports at the club. 24 signed up for squash and 16for badminton. All the badminton players also signed up forsquash. One squash player signed up for table tennis. None ofthe students signed up for all three.

How many students signed up for table tennis?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B7A square ABCD of side 3 cm is rotated90° clockwise about the point A, then90° clockwise about the point C, then 90°clockwise about the point B, and finally90° clockwise about the point D.

How far is the vertex A from its originalposition?

A B

CD

Answer: cm

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B8The trapezium ABCD has area 16 cm2.

AB is parallel to DC.

A = (1, 1)B = (3, 1)C = (6, 5)

What are the coordinates of D?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B9A passenger pays for his £3.72 bus ticket and receives somechange from the driver. No paper money is used by either.

What is the smallest possible number of coins used in thetransaction?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B10When 63 is divided by a positive integer x the remainder is 3.

What is the total of all the possible values of x?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B11A new ride at the fair is a roundabout in the shape of a pentagon,with identical seats at the vertices.

In how many different ways can five friends sit around theroundabout?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B12The swimming teacher has four sessions on Monday withbetween 6 and 9 students per session. She charges £18 perperson for each session.

What is the difference between the least and the most she wouldreceive for 10 Mondays?

Answer: £

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B13The sign in a railway carriage lists the stops: Barnham,Chichester, Bosham, Southbourne, Emsworth and Havant.

What is the median number of letters in the station names?

Answer:

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B14Kenneth began drawing a net of a squarebased pyramid. He got as far as drawinga rectangle x by 2x and dividing it intotwo identical squares, one of which is thebase of the pyramid. One triangular facewas drawn by joining the midpoint of theshorter side of the rectangle to the verticesof the base of the pyramid. The otherequal triangular faces were then drawn.

What is the area of the complete net?x

2x

Answer: square units

UKMTUK

MTUKM

T

Team MathsChallenge

2016

National Final

Relay

© UKMT 2016

B15Sara is practising her compass-work before an expedition in theNew Forest.

She walks 84 paces on a bearing of 041°.

She then walks 35 paces on a bearing of 311°.

Finally, she walks 50 paces directly towards her starting point.

How many paces is she away from her starting point?

Answer: paces

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

ßï

Ð¿ ±² ¬¸» ª¿´«» ±ºæ

ïî øïï ïð ç÷ øé ê ë÷ õ ì í ø î ï÷ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò ßí

íøÌ è÷¨ ã îÌ ø¨ õ î÷ ìøî ç¨÷

Ð¿ ±² ¬¸» ª¿´«» ±º ¨ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò ßî

· Ì ò

Ì¸» ³»¿² ±º ¬¸» ¿³» »¬ ±º ·²¬»¹»® · Ìî

¹®»¿¬»® ¬¸¿²

¬¸» ³»¼·¿²ò

Ð¿ ±² ¬¸» ª¿´«» ±º ¬¸» ´¿®¹»¬ °±·¾´» ·²¬»¹»® ·² ¬¸»

»¬ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò ßì

ß °±¬ ±º °¿·²¬ ½±²¬¿·² »²±«¹¸ °¿·²¬ ¬± ½±ª»® ¿² ¿®»¿

±º ïèðð ½³îò

É®·¬» ¼±©² ¸±© ³¿²§ °±¬ ±º °¿·²¬ ¿®» ²»»¼»¼ ¬±

½±ª»® ¬¸» «®º¿½» ±º ¿ ½«¾±·¼ ©·¬¸ ¼·³»²·±² Ìî ½³

¾§ Ìí

½³ ¾§ Ìì

½³ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Þï

ëð Ç»¿® ç °«°·´ ©»®» ¿µ»¼ ©¸»¬¸»® ¬¸»§ ¸¿ª» ¿

¾·½§½´» ±® ¿ µ¿¬»¾±¿®¼ò

îë ¿·¼ ¬¸»§ ¸¿ª» ¿ ¾·½§½´»ô

îð ¿·¼ ¬¸»§ ¸¿ª» ¿ µ¿¬»¾±¿®¼ô ¿²¼

¬©·½» ¿ ³¿²§ ¿·¼ ¬¸»§ ¸¿ª» ²»·¬¸»® ¿ ¿·¼ ¬¸»§

¸¿ª» ¾±¬¸ò

Ð¿ ±² ¬¸» ²«³¾»® ±º °»±°´» ©¸± ¿·¼ ¬¸»§ ¸¿ª» ¿

¾·½§½´» ¾«¬ ²±¬ ¿ µ¿¬»¾±¿®¼ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Þí

Ì¸» ¼·¿¹®¿³ ¸±© ¬®·¿²¹´» ßÝÜò Ì¸» °±·²¬ Þ ´·»

±² ßÝ ± ¬¸¿¬ ßÞ ã ßÜ ¿²¼ ÜÞ ã ÞÝò ß²¹´»

ÜßÞ ã èÌpò

ß ÞÝ

Ü

èÌp

Ð¿ ±² ¬¸» ·¦» ·² ¼»¹®»» ±º ¿²¹´» ÞÝÜò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Þî

Ô»©· ¸¿ ¿ ¾¿¹ ½±²¬¿·²·²¹ øÌ õ ï÷ ½±«²¬»®ô è ±º

©¸·½¸ ¿®» ®»¼ò Ø» ¬¿µ» ±«¬ ¬©± ½±«²¬»®ô ±²» ¿º¬»®

¬¸» ±¬¸»®ô ©·¬¸±«¬ ®»°´¿½»³»²¬ò

Ì¸» °®±¾¿¾·´·¬§ ¬¸¿¬ ¾±¬¸ ±º Ô»©·ù ½±«²¬»® ¿®» ®»¼

·°¯ô ¿ º®¿½¬·±² ·² ·¬ ´±©»¬ ¬»®³ò

Ð¿ ±² ¬¸» ª¿´«» ±º ¯ í°ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Þì

ß² ¿®·¬¸³»¬·½ »¯«»²½» ¬¿®¬ ïíêô ïîèô ïîðô ïïîò ò ò

É®·¬» ¼±©² ¬¸» ª¿´«» ±º ¬¸» øïðÌ ÷¬¸ ¬»®³ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ýï

îíì ìë

êêé

è ã¿

¾¿

¾· ¿ º®¿½¬·±² ·² ·¬ ´±©»¬ ¬»®³ò

Ð¿ ±² ¬¸» ª¿´«» ±º î¾ ¿ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Ýí

Ûª¿² ¸¿ ¬©± ¸¿¼» ±º °«®°´» °¿·²¬æ

Ð¿·²¬ ß ½±²¬¿·² ®»¼ ¿²¼ ¾´«» °¿·²¬ ·² ¬¸» ®¿¬·± ï æ íò

Ð¿·²¬ Þ ½±²¬¿·² ®»¼ ¿²¼ ¾´«» ·² ¬¸» ®¿¬·± î æ íò

Ø» ³¿µ» ¿ ²»© ¸¿¼»ô Ûª¿²»½»²½»ô ¾§ ³·¨·²¹

°¿·²¬ ß ¿²¼ Þ ¬±¹»¬¸»® ·² ¬¸» ®¿¬·± ï æ øÌ ï÷ò

Ì¸» ®¿¬·± ±º ®»¼ ¬± ¾´«» °¿·²¬ ·² Ûª¿²»½»²½» · ¿ æ ¾ô

©®·¬¬»² ·² ·¬ ´±©»¬ ¬»®³ò

Ð¿ ±² ¬¸» ª¿´«» ±º ¾ ¿ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Ýî

Ì¸» ¸¿°» ¾»´±© ½±²·¬ ±º ïî ½«¾» ±º ·¼» ïîÌ ½³ò

Ì¸» ¬±¬¿´ «®º¿½» ¿®»¿ · ß ½³îò

Ð¿ ±² ¬¸» ª¿´«» ±º ßîò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Ýì

Ì·¿ù ¿¹» · Ì ¬·³» ¸»® ¿¹» ·² Ì §»¿®ù ¬·³» ³·²« Ì

¬·³» ¸»® ¿¹» Ì §»¿® ¿¹±ò

É®·¬» ¼±©² Ì·¿ù ¿¹»ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Üï

Ì¸» °®·³» º¿½¬±®·¿¬·±² ±º çççððð ½¿² ¾» ©®·¬¬»² ¿

î¿ í¾ ë½ ¼ô

©¸»®» ¿ô ¾ ¿²¼ ½ ¿®» ©¸±´» ²«³¾»®ô ¿²¼ ¼ · ¿ °®·³»

²«³¾»®ò

Ð¿ ±² ¬¸» ª¿´«» ±º ¼ ¿¾½ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Üí

Ö»²²·º»® ¸¿ ¾«·´¬ ±³» °§®¿³·¼ ©·¬¸ Ì ó·¼»¼

°±´§¹±² ¿ ¬¸»·® ¾¿»ô ¿²¼ ±³» °®·³ ©·¬¸ Ìó·¼»¼

°±´§¹±² ¿ ¬¸»·® ¾¿»ò

×² ¬±¬¿´ ¬¸»®» ¿®» éð »¼¹» ¿²¼ ìì ª»®¬·½» ±² ¬¸»

íÜó¸¿°»ò

Ð¿ ±² ¬¸» ¬±¬¿´ ²«³¾»® ±º íÜó¸¿°» Ö»²²·º»® ¸¿

¾«·´¬ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Üî

Ûª·» ¬®¿ª»´ ¿ ¼·¬¿²½» ±º Ì µ³ ¿¬ ¿² ¿ª»®¿¹» °»»¼

±º ¨ µ³ °»® ¸±«®ò Í¸» ¬¸»² ¬®¿ª»´ ¿ º«®¬¸»® íÌî

µ³ ¿¬

¿² ¿ª»®¿¹» °»»¼ ±º ¨î µ³ °»® ¸±«®ò

Ì¸» ©¸±´» ¶±«®²»§ ¬¿µ» è ¸±«®ò

Ð¿ ±² ¬¸» ª¿´«» ±º ¨ò

ËÕÓÌ

Ì»¿³ Ó¿¬¸

Ý¸¿´´»²¹»

îðïê

Ò¿¬·±²¿´ Ú·²¿´

Í¸«¬¬´»

w ËÕÓÌ îðïê

Ì · ¬¸» ²«³¾»® §±« ©·´´ ®»½»·ª»ò Üì

ß ¾´«» °¿²¬¸»® ®«² îðû ³±®» ´±©´§ ¬¸¿² ¿ ¾´¿½µ

°¿²¬¸»®ô ¾«¬ ìÌû ³±®» ¯«·½µ´§ ¬¸¿² ¿ °·²µ °¿²¬¸»®ò

ß ¾´¿½µ °¿²¬¸»® ®«² Þû ³±®» ¯«·½µ´§ ¬¸¿² ¿ °·²µ

°¿²¬¸»®ò

É®·¬» ¼±©² ¬¸» ª¿´«» ±º Þò

Í¸«¬¬´» ®»°±²» ¸»»¬ Ì»¿³ Ó¿¬¸ Ý¸¿´´»²¹» îðïê Ò¿¬·±²¿´ Ú·²¿´

Ì»¿³ ²«³¾»® Í½¸±±´ ²¿³»

ßï

ð ï í

ßî

ð ï í

ßí

ð ï í

ßì

ð ï í

Þ±²« í

ß ¬±¬¿´ ñïë

Þï

ð ï í

Þî

ð ï í

Þí

ð ï í

Þì

ð ï í

Þ±²« í

Þ ¬±¬¿´ ñïë

Ýï

ð ï í

Ýî

ð ï í

Ýí

ð ï í

Ýì

ð ï í

Þ±²« í

Ý ¬±¬¿´ ñïë

Üï

ð ï í

Üî

ð ï í

Üí

ð ï í

Üì

ð ï í

Þ±²« í

Ü ¬±¬¿´ ñïë

Ý·®½´» ¬¸» ³¿®µ ¿©¿®¼»¼ º±® »¿½¸ ¯«»¬·±² ¿²¼ ½®± ±«¬ ¬¸» ±¬¸»®ò

ß¬ ¬¸» »²¼ ±º ¬¸» ®±«²¼ô »·¬¸»® ½·®½´» ¬¸» ¾±²« ³¿®µ ±® ½®± ·¬ ±«¬ò

Ú·²¿´ ½±®» ñêð

w ËÕÓÌ îðïê

PosterCompetition

Team Maths Challenge2016

National Final UKMT

UKM

TUKMT

Instructions

In addition to four rounds similar to those at the regional finals (Group Circus, Crossnumber,Shuttle and Relay), at the National Final there will be a Poster Competition, with a chance to winthe Jacqui Lewis Trophy.

All teams are required to submit a poster. The poster competition will be judged separately andwill not affect the Team Maths Challenge score, but forms an integral part of the National Final.

After the competition some posters may be retained by the UKMT in order to be reproduced forpromotional purposes.

On the day, teams will have 50 minutes to create a poster on a sheet of A1 paper (landscape),which will be provided. Sheets of A4 paper will also be available.

The subject of the poster will be Folding (see overleaf). Teams must carry out research into thistopic in the weeks leading up to the final.

Teams may create materials beforehand, but such prepared work must be on sheetsno larger than A4 and must be assembled to create the poster on the day.

A team which arrives with a poster already assembled will be disqualified.

The materials of the poster must not extend beyond the edge of the A1 paper.

The judges will not touch the poster, so all information must be clearly visible.

Your team number (assigned to you on arrival) must be clearly visible in the bottomright-hand corner of the poster. There must be nothing else on the poster to identifythe team.

Reference books may not be used at the competition, and large extracts copieddirectly from books or the internet will not receive much credit.

Teams must bring with them any drawing equipment they think they will need.

Glue sticks and scissors will be provided.

The content of each poster is limited only by the imagination of the team members. However, onthe day each team will be presented with three questions on the subject—the answers to thesequestions must be incorporated into the structure of the poster. Teams may be asked to provideproofs, and some ingenuity may be involved.

Posters will be judged on the following criteria:

General mathematical content 12 marksImagination and presentation 12 marksAnswers to the questions 24 marks

PosterCompetitionTopic


National Final UKMT

UKM

TUKMT

Folding

When paper-folding is mentioned, your thoughts may immediately turn to origami and foldingobjects like birds or flowers. This artistic side of paper-folding is well-known, but it is less wellknown that there are mathematical aspects too, and many of them. For example, it is possible tofold some simple mathematical shapes. Note that, here and elsewhere, we use the expression‘fold [something]’ to mean ‘create [something] by folding a piece of paper along straight lines’.

Starting from a rectangular (or square) piece of paperwhat simple mathematical shapes is it possible to fold?

One natural question to ask is “what lengths can be folded?”

What is Haga’s theorem?

Start with a square sheet of paper with sides of length 1.

Given some length x < 1, show how to calculate the value ofz in the diagram alongside in terms of x.

x

z

1

Can any rational length (less than one) be folded?

Anyone familiar with classical ‘straightedge and compass’ constructions may be surprised to learnthat everything that can be constructed in the classical way may also be achieved by folding paper,indeed, even more is possible by paper-folding than by classical constructions. For example, agiven angle may be trisected, something that is not possible with straightedge and compass.

What constructions are possible by foldingbut impossible using ‘straightedge and compass’?

PosterCompetitionQuestions


National Final UKMT

UKM

TUKMT

Folding

Question 1

A square sheet of paper is folded once along a straight line.

For each of the following shapes, either show how to make it, or prove that it cannot be made:

(a) a rectangle that is not a square;(b) an isosceles triangle;(c) a right-angled triangle;(d) a rhombus that is not a square.

Question 2

The diagram shows a square sheet of paper ABCD with sidesof length 6 cm. The point M is the midpoint of side CD.

The paper is folded along PQ, where P lies on DA and Q lieson BC, so that the vertex A folds to the point M .

What is the length of PQ?

A

BC

D

M

Q

P

6 cm

Question 3

The square sheet of paper S has sides of length 2.

Explain how to fold S along straight lines to make a line segment of length√

3.

Group CircusTeam Maths Challenge

2016National Final

UKMT

UKM

TUKMT

Station 1

A friend has a box with four balls in it - two are red, one is blackand one is white. She takes two balls out at random and shows methat one of them is red.

What is the probability that both of them are red?

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 2

(a) You are provided with a large red square divided into 16congruent small squares.

Cut the large square into three pieces, just one of which is asquare, and rearrange them to make an 8× 2 rectangle, withoutgaps or overlaps. Cuts may only be made along grid lines andyou are allowed to turn pieces over. [3 marks]

(b) You are provided with a large blue square divided into 16congruent small squares.

Cut the large square into five pieces, one of which is a squareand the other four are congruent, and rearrange them to makean 8 × 2 rectangle, without gaps or overlaps. Cuts may only bemade along grid lines and you are not allowed to turn piecesover. [3 marks]

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 3

Three buses, numbered 117, 127 and 137, arrived together at mylocal bus station at 05:43.

The number 117 buses then arrive at this bus station every 18minutes.



What will be the next time that three buses numbered 117, 127 and137 arrive together at my local bus station?

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 4

You are provided with two hexagons each divided into six triangles,like the one shown.

11

(a) Place one of the number cards 1, 3, 5, 7, and 9 in each triangleon one hexagon so that the sum of the numbers in any threeadjacent triangles is prime. The number 11 is already in place.Show your solution to the supervising teacher. [3 marks]

(b) Using the second hexagon find a second solution to the taskgiven in part (a) and show it to your supervising teacher. Yoursecond solution must not be a reflection of the first one.

[3 marks]

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 5

The number of days in a non-leap year is equal to the sum of twosquares.

(a) Find two squares which give one solution. [2 marks]

(b) Find two other squares which give a second solution.[2 marks]

The number of days in a leap year is equal to the sum of two cubesand one square.

(c) Find these three numbers. [2 marks]

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 6

You are provided with the 7 × 7 grid of squares below and ninecards with arrows on them.

A

B

Place the cards so that:

(i) the arrows form a continuous route starting at corner A ofthe grid and ending at corner B;

(ii) all of the cards are used;(iii) the cards do not overlap;(iv) no part of a card lies outside the grid;(v) the edges of each card are parallel to the lines of the grid.

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 7

(a) A rectangle has a perimeter of 38 cm and an area of 48 cm2.

What are the dimensions of the rectangle? [3 marks]

(b) An equilateral triangle has a perimeter of 48 cm. The area ofthe triangle can be written in the form a2

√b cm2, where a and

b are single-digit numbers.

What are the values of a and b? [3 marks]

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 8

How many different answers can be obtained using only the digits1, 2, 3 and 4 once each, in any order, together with a single × sign?

For example, 31 × 24 and 2 × 413 give two of the answers.

The numbers you use as factors in the calculations must all beintegers.

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Station 5 Worksheet

(a) 2 squares:

(b) 2 squares:

(c) 2 cubes, 1 square:

© UKMT 2016

Team Maths Challenge 2016 National Final Group Circus Station 2 Resource


11

1 3 5 7 9

Group Circus response sheet TMC 2016 National Final


Station 1

Complete the worksheet andshow it to the supervisor.

0 6

Station 2

Show your answer(s) to thesupervisor.

0 3 6

Station 3


0 6

Station 4


0 3 6

Station 5


(a) 0 2(b) 0 2(c) 0 2

Station 6


0 6

Station 7


(a) 0 3(b) 0 3

Station 8


0 6

Circle the mark awarded for each question and cross out the others. Final score /48

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT

Supervisor Station 1

0.2 or 15 or equivalent

Ensure that the worksheet and any scrap paper are cleared away before the nextteam arrives.

Marksto award

6 correct solution0 otherwise

ResourcesQuestion paperWorksheetScrap paper

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT


(a)

(b)

Please check carefully as there are other solutions as well as thoseshown above.

Three marks to be awarded in each part for the correct presentationof an 8 × 2 rectangle with no gaps or overlaps and judged by eye.Poor cutting of the pieces should not be penalised.

Ensure any evidence of each team’s solutions and any scrap paper are clearedaway before the next team arrives.

Marksto award

6 correct solution3 one part correct0 otherwise

ResourcesQuestion paperRed and blue square grids on cardScissorsPractice grids on paper

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT


09:55

Ensure that the worksheet and any scrap paper are cleared away before the nextteam arrives.

Marksto award



© UKMT 2016


2016National Final

UKMT

UKM

TUKMT


Two of these solutions - but not two from the same column.

3

91

7

511

3

97

1

511

5

39

7

111

5

39

1

711

5

71

9

311

5

17

9

311

1

79

3

511

7

19

3

511


Marksto award


ResourcesQuestion paperTwo laminated hexagonsTwo sets of laminated number cards

© UKMT 2016


2016National Final

UKMT

UKM

TUKMT


(a) 169 or 132, 196 or 142

(b) 4 or 22, 361 or 192

(c) 125 or 53, 216 or 63, 25 or 52

Answers (a) and (b) are interchangeable. Accept answers such as192 or 53, but do not accept answers such as 19 or 5.

The numbers in each of the three parts can be given in any order.

Ensure that the worksheet and any scrap paper used are cleared away before thenext team arrives.

Marksto award

6 correct solution4 two parts correct2 one part correct0 otherwise


© UKMT 2016


2016National Final

UKMT

UKM

TUKMT


Please check carefully as there are other solutions as well as thatshown above.

Ensure that all nine cards are moved off the grid before the next team arrives.

Marksto award


ResourcesQuestion paper (4 copies)Nine laminated cards (4 sets)

© UKMT 2016

UKMT

UKM

TUKMT


National Final

Supervisor’s booklet

Please ensure that students do not have access to thisbooklet, and take care to hold it so that answers cannotbe seen.

Please ensure that students use blue or black ink towrite their answers; teachers are asked to use red inkfor marking.

Shuttle answers

A1

12

A2

62

A3

120

A4

6

B1

20

B2

9

B3

27

B4

−2016

C1

1

C2

6

C3

2

C4

8

D1

10

D2

5

D3

6

D4

55

On the response sheet: Circle the mark awarded for each question and cross out the others.At the end of the round, either circle the bonus mark or cross it out.

© UKMT 2016 Team Maths Challenge 2016 National Final 2

Shuttle flowchart

The flowchart explains the order in which questions should be marked.

Checkquestion 1

Hand back topair X

Award 3 marks if correct first time,otherwise award 1 mark

Checkquestion 2

Hand back topair Y


Checkquestion 3

Hand back topair X


Checkquestion 4

Hand back topair Y


The response sheet passes in turnfrom pair to pair.Having completed all questions, pair Yfinally hands in the sheet for marking.

The pairs can nowtake the opportunityto amend incorrectand subsequent

answers.

Award a bonus of 3 marks if allcorrect (at the first or laterattempt) before the 6 minute

whistle

Whistlesbonus 6 minutesfinal 8 minutes

Give out thequestions and

responsesheet

StartKey

pair X have questions 1 and 3 in the current roundpair Y have questions 2 and 4

maximumscore 15

Correct Wrong

Correct Wrong

Correct Wrong

Correct Wrong

Already markedcorrect




Crossnumber

1 2 3 4

5 6

7 8 9

10 11 12

13 14 15 16

17 18 19

20 21 22

23

24 25

Across1. 7 Across minus 11 Down (3)3. The square root of 233 + 243 + 253 (3)5. A palindromic number whose digits have a

sum of 33 (5)7. The difference between two consecutive

Fibonacci numbers (3)8. A number less than 3 Down (3)

10. A multiple of 18 Down (4)13. The largest two-digit factor of 2 Down (2)15. 43 plus the square root of 12 Down (2)17. 4 + 10 × 122 (4)20. 21 Down minus a cube (3)22. nn + 1 for some n (3)23. A factor of 5 Across (5)24. The mean of 14 Down and 1 Across (3)25. 4994 minus 19 Down (3)

Down1. A palindromic number whose digits have a

sum of 41 and which consists of twodifferent digits (5)

2. 88 × 77 (4)3. A number greater than 8 Across (3)4. A square that is 8 Across × 3 − 102 (3)6. A Fibonacci number that is a multiple of

13 (3)9. 15 Across × 11 + 11 Down (3)

11. A multiple of 19 (3)12. A square (3)14. A multiple of 13 Across (3)16. 55 + 45 + 75 + 45 + 85 (5)18. A factor of 10 Across (3)19. The product of its digits is 432 (4)20. The mean of 20 Down and 21 Down (3)21. A power of 2 (3)


Crossnumber

18 1

26

32 0

44

857 8

63 8 7 4

79 8 7 7

81

98 1

8106

111 7

124 8

138

148 7 8

156

165

8171

184 4

194 4

201 0

211 4

222 5 7

2232 6 1 2 9 4

248 4 8

256 9 8

Marking Instructions—a reminder

• Pairs may only communicate through the teacher, and only to request that theother pair works on a particular clue.

• When a pair enters an answer in the grid on the response sheet, the teacherchecks each digit of the answer:

– if it is correct, place a tick in the dotted circle and award one mark– if it is wrong, cross it out, write in the correct digit, and place a cross in

the dotted circle– the correct answer is then shown to both pairs so that they are up-to-date.

• A pair may enter just one digit if they wish, rather than a complete answer.

• A pair may sacrifice a square, by guessing, if they wish.


Group Circus answers

Station 1

0.2 or 15 or equivalent

Station 2

(a) (b)

Station 3

09:55

Station 4

Two of these solutions — but the second must not reflect the first.

3

91

7

511

3

97

1

511

5

39

7

111

5

39

1

711

5

71

9

311

5

17

9

311

1

79

3

511

7

19

3

511

Station 5

(a) 169 or 132, 196 or 142 (b) 4 or 22, 361 or 192 (c) 125 or 53, 216 or 63, 25 or 52

Station 6

Station 7

(a) 16 cm × 3 cm (b) a = 8, b = 3

Station 8

36

On the response sheet: Circle the mark awarded for each question and cross out the others.


Relay scoresheet


A1A C B

0 2B1

324cm2 0 2

A22061

0 2B2

37sweets 0 2

A32

minutes 0 2B3

95

0 2A4

50 2

B4£95.00

0 2A5

9sweets 0 2

B515.23

0 2

A67.5 or 71

2 or 1520 2

B612

0 2A7

59

0 2B7

6cm 0 2

A811

£ 0 2B8

(0, 5)0 2

A9(8,4)

0 2B9

50 2

A1096metres 0 2

B10162

0 2

A1125

square units 0 2B11

240 2

A12315

days 0 2B12

£2160£ 0 2

A138

0 2B13

7.5 or 712 or 15

20 2

A144.32

litres 0 2B14

3x2

square units 0 2A1555 hours 15 minutes

0 2B15

41paces 0 2

Correct answers score 2 points: circle 2 or 0 for each questionand cross out the other number.At the end of the round, draw a line under the last questionattempted.

Final score /60


BACK

PAGE

ukmt maths competition 2016.pdf · crossnumber response sheet team maths challenge 2016 national...

Documents