The 21st Canadian Mathematical Olympiads
1989年第二十一届加拿大数学奥林匹克 |
- How many permutations of 1, 2, 3, ... , n have each number larger than all the preceding numbers or smaller than all the preceding numbers ?
- Each vertex of a right angle triangle of area 1 is reflected in the opposite side. What is the area of the triangle formed by the three reflected points ?
- Tranform a number by taking the sum of its digits. Start with 19891989 and make four transformations. What is the result ?
- There are five ladders. There are also some ropes. Each rope attaches a rung of one ladder to a rung of another ladder. No ladder has two ropes attached to the same rung. A monkey starts at the bottom of each ladder and climbs. Each time it reaches a rope, it traverses the rope to the other ladder and continues climbing up the other ladder. Show that each monkey eventually reaches the top of a different ladder .
- For every permutation a1, a2, ... , an of 1, 2, 4, 8, ... , 2n-1 form the product of all n partial sums a1 + a2 + ... + ak. Find the sum of the inverses of all these products .
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