Opsimath

2021/09/02阅读：96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,

### 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*.

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