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| US triumphs in ‘hardest ever’ maths Olympiad | |
| (about 5 hours later) | |
| This was the final question at the International Mathematical Olympiad (IMO), an annual “maths World Cup” for secondary school-age students, held this year in Thailand, which ended on Wednesday with the US as winners: | This was the final question at the International Mathematical Olympiad (IMO), an annual “maths World Cup” for secondary school-age students, held this year in Thailand, which ended on Wednesday with the US as winners: |
| Don’t panic if it left you bewildered. Some of the world’s best mathematical brains have been having trouble too. The British IMO team’s leader, Dr Geoff Smith of Bath University, said it was the hardest paper in the history of the IMO, which was first held in 1959. | |
| The threshold score for gold medals – which changes yearly depending on how well contestants perform – was set at 26 out of 42 points, the lowest ever. Snatching five gold medals, the US beat the usual winners, China. The British team of six students, including 16-year-old “mathlete” Joe Benton, came 22nd out of 104 competing countries. | The threshold score for gold medals – which changes yearly depending on how well contestants perform – was set at 26 out of 42 points, the lowest ever. Snatching five gold medals, the US beat the usual winners, China. The British team of six students, including 16-year-old “mathlete” Joe Benton, came 22nd out of 104 competing countries. |
| Related: UK maths prodigy sets out to prove his worth at international Olympiad | Related: UK maths prodigy sets out to prove his worth at international Olympiad |
| Smith said his team was “pleased as punch” after taking four silver medals. Team member Warren Li was one point off a gold. | Smith said his team was “pleased as punch” after taking four silver medals. Team member Warren Li was one point off a gold. |
| Smith noted that France had finished in 14th place. “Almost always, the UK finishes above France. This year, malheureusement, the situation is reversed. I congratulated the French leader while mentioning that he was the new Napoleon.” | |
| The examination is held over two consecutive days and contestants have four and a half hours to solve three problems per day, which can include geometry, number theory and algebra. You don’t need knowledge of higher mathematics such as calculus, but the questions are designed to be extremely difficult. No calculators are allowed. | |
| On Wednesday more than 570 teenagers from as far afield as Afghanistan and Ecuador stood around in groups relaxing after the results had been announced. During their trip, some had taken elephant rides or hikes in the mountains around Chiang Mai. | On Wednesday more than 570 teenagers from as far afield as Afghanistan and Ecuador stood around in groups relaxing after the results had been announced. During their trip, some had taken elephant rides or hikes in the mountains around Chiang Mai. |
| Michael Kural, 17, from Connecticut in the US, said he had spent June at a camp with his five team members aged 16 to 18. “It was a definitely a lot harder than we’ve been used to,” the teenager said of the contest, as his teammates played on their phones and discussed the event over steamed rice and spicy pork curry. “I think a lot of teams weren’t really used to that.” | Michael Kural, 17, from Connecticut in the US, said he had spent June at a camp with his five team members aged 16 to 18. “It was a definitely a lot harder than we’ve been used to,” the teenager said of the contest, as his teammates played on their phones and discussed the event over steamed rice and spicy pork curry. “I think a lot of teams weren’t really used to that.” |
| He said they had an edge because their coach had trained them with especially hard test papers. “We managed to get far ahead on the first day.” | |
| Kada Williams, 16, from Hungary, was disappointed with his team’s 20th place. “We expected to do considerably better,” he said. “Based on these low thresholds for gold medals, everyone did not do too well, which made us feel somewhat better about ourselves.” | Kada Williams, 16, from Hungary, was disappointed with his team’s 20th place. “We expected to do considerably better,” he said. “Based on these low thresholds for gold medals, everyone did not do too well, which made us feel somewhat better about ourselves.” |
| • If the above question is too technical, try this from the 2011 IMO, which is easier to understand, if not to solve: | • If the above question is too technical, try this from the 2011 IMO, which is easier to understand, if not to solve: |
| Suppose that you mark a finite collection of points on an infinite plane in such a way that you cannot draw a straight line through any three marked points. We define a windmill to be the following process: draw an infinite straight line on the plane through exactly one of the marked points. Then rotate the line clockwise using the chosen marked point as a pivot until the moving line hits another marked point. At that instant, the new marked point takes over as a pivot, and the line continues to rotate clockwise. This process continues, with new pivots taking over from time to time. | Suppose that you mark a finite collection of points on an infinite plane in such a way that you cannot draw a straight line through any three marked points. We define a windmill to be the following process: draw an infinite straight line on the plane through exactly one of the marked points. Then rotate the line clockwise using the chosen marked point as a pivot until the moving line hits another marked point. At that instant, the new marked point takes over as a pivot, and the line continues to rotate clockwise. This process continues, with new pivots taking over from time to time. |
| Show that it is possible to select one of the marked points, and choose a starting line through that point at a particular angle, so that the resulting windmill uses every marked point as a pivot on infinitely many occasions. | Show that it is possible to select one of the marked points, and choose a starting line through that point at a particular angle, so that the resulting windmill uses every marked point as a pivot on infinitely many occasions. |