Olympic Rings Challenge

1. The Answers Senior Challenge ‘12

2. Ring Totals • Place the numbers 1 to 9 into the Olympic Rings, one in each white space, so that there is the same total inside each ring. Calculate each of the possible totals that satisfy this requirement.

3. Ring Total = 11

4. Ring Total = 13

5. Ring Total = 13 again

6. Ring Total = 14

7. Metal Medals Gold costs twice as much as Silver, which costs 1 ½ times as much as Bronze. Gold weighs 1 ¼ times as much as Silver, which weighs 1 ¾ as much as Bronze. For the boxing medals, twice as many Bronze medals are required as Gold and Silver medals. There are 10 boxing events. A Silver medal weighs 40g and costs £120. How much will all of the boxing medals weigh and cost?

8. Metal Medals - Weight Gold weighs 1¼ × 40 = 50g Bronze weighs 40 ÷ 1¾ = 160/7 ≈ 22.86g So total weight = 10 × 50 + 10 × 40 + 20 × 160/7 = 500 + 400 + 3200 /7 = 1357.14g

9. Metal Medals – cost per medal Gold costs 2 × 120 = £240 Bronze costs 120 ÷ 1½ = £80 So total cost = 10 × 240 + 10 × 120 + 20 × 80 = 2400 + 1200 + 1600 = £5,200

10. Metal Medals – cost per gram 40g of Silver costs £120, so Silver = £3 per gram Gold costs 2 × 3 = £6 per gram, so 50 × 6 = £300 per medal Bronze costs 3 ÷ 1½ = £2 per gram, so 160/7 × 2 ≈ £45.71 per medal So total cost = 10 × 300 + 10 × 120 + 20 × 45.71 = 3000 + 1200 + 914.29 = £5,114.29

11. Four sportswomen meet. Any two of them have something in common: first name, country of origin or the sport they compete in. However, there is nothing that any group of three of them have in common. How is this possible? Common or Uncommon

12. Common or Uncommon Barbara German Swimmer Ann French Swimmer Ann German Rower Barbara French Rower

13. Making Tracks • An 8-lane circular race track has a circumference of 400m in the centre of lane 1. Given that each lane is 122cm wide, how much further back does the runner in lane 1 start than the runner in each of the other lanes?

14. Making Tracks

15. Making Tracks • Each lane will be 2 × π × 1.22m longer, so the runner in lane 1 needs to start 7.67m behind the runner in lane 2.

16. Medals Table Mayhem In a particular Olympic games, the top 4 countries in the medal table were China, USA, Russia and Great Britain, in that order. (The order is determined by the number of Gold medals won.) Each country won more than 10 medals of each type. China won two more Gold than Silver and Bronze combined in their total of 100 medals. USA’s number of Gold medals matched their number of Bronze (a square number) and they won 10 more medals than China in total. Russia and China won the same number of Silver medals and the same number of Bronze medals, the latter being the same number of medals as GBR’s total for Silver and Bronze combined. Russia averaged 24 medals of each type. GBR’s numbers of Gold and Silver were prime, and their total number of medals was 4 fewer than the number of Gold medals won by China. The USA earned twice as many Silver medals as GBR earned Gold medals. The number of Bronze medals for each country is a triangle number. Re-construct the medals table.

17. Medals Table Mayhem • China’s total is 100 • USA got 10 more than China • Russia averaged 24 of each type => total of 72 • China won two more Gold than silver and bronze combined, so Gold = 51

18. Medals Table Mayhem • All Bronze totals are triangle numbers, so 15, 21, 28, 36, 45, • Must be >10, and <50 for all remaining spaces. • USA Bronze is also square, so must be 36, and their Gold is the same

19. Medals Table Mayhem • That leaves 38 for USA Silver. • GBR’s Gold is half the USA’s Silver, so must be 19 • GBR’s total is 4 less than China’s Gold, so must be 47

20. Medals Table Mayhem • That leaves 28 for GBR’s combined Silver and Bronze, which is the same as China’s Bronze. • That leaves 21 for China’s Silver. • Russia and China have the same Silver and Bronze

21. Medals Table Mayhem • That means Russia’s Gold is 23 • The only sum of a prime and a triangle number to 28 is 13 and 15

22. Matchmaking An Olympic handball tournament was organised as follows: each day of 12 days, 5 teams were playing against each other. Any pair having met once could never meet again. The teams were chosen so that, on each day, all ten matched were allowed. Is it possible to do this with 20 teams taking part in the competition?

23. Matchmaking Day 12 Day 11 Day 10 Day 9 Day 8 Day 7 Day 6 Day 5 Day 4 Day 3 Day 2 Day 1 And Team 19 And Team 20

24. Pole Bearers Two men are delivering flagpoles to the Olympic stadium. To enter the stadium, they have to carry each pole along a 3-metre wide corridor with a ceiling 2.5m high throughout. The corridor starts off straight, but then bend through 90 degrees, with the inside wall following an arc of a circle of radius 4m. The Bronze flag pole is 11.5m long, the Silver is 11.75m and the Gold is 12m long. Work out which poles can be carried through the corridor without bending the flag pole!

25. Pole Bearers E B BC = 7m OB = 9.899M OA = 4, so AB = 5.899 AB = AE because ABE is a right angled isosceles triangle A C O D

26. Pole Bearers E B ED = 2 × AE = 11.799m So Bronze and Silver poles will go through flat The corridor is 2.5m high, so we get another right angled triangle... A C O D

27. Pole Bearers Now, by Pythagoras The hypotenuse of this triangle is 12.06m So the Gold pole will go through with 6cm to spare! 2.5m 11.799m