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墨滴

Opsimath

2021/09/02  阅读:34  主题:默认主题

Mathematics Interview Questions (IV)

16. Sketch .

Solution. From question 1, . is clearly positive within its domain, . Hence, if and only if , i.e., . When , , so is the minimum point. Also,

using L'Hôpital's rule and the continuity of . So the graph begins at, but not passes through, the point . You can view here for a sample.

17. Prove is a multiple of 3.

Solution 1. , so , and , i.e., .

Solution 2. Proof by induction. When , which is a multiple of 3. Suppose for all , is divisible by 3, then is also divisible by 3.

18. How many ways there are of getting from one vertex of a cube to the opposite vertex without going over the same edge twice?

Solution. 30 ways.

The most direct route is via 3 edges: 3 choices for the first edge, 2 for the second, 1 for the third = 6 routes.

The next most direct route is via 5 edges. These routes can be derived by swapping out the first move of a 3-edge route for the 3-edge "long way round" move to the same vertex. So that's 6 routes over 5 edges.

The longest route is via 7 edges. In this case both the first and second move in a 3-edge route are swapped out for 3-edge long-way-round routes. Each starting move can be swapped out in 3 ways. That gives 18 routes over 7 edges.

19. What shape there would be if the cube was cut in half from diagonally opposite vertices?

Solution. A parallelogram.

20. Draw .

Solution.

which means the graph begins at, but not passes through, the point . Also, , so if and only if , i.e., . When , , so is the minimum point. You can view here for a sample.

Opsimath

2021/09/02  阅读:34  主题:默认主题

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Opsimath