Opsimath

2021/09/09阅读：84主题：默认主题

# Mathematics Interview Questions (X)

### 46. Integrate .

*Solution*. Let
, then
. As a result,
.

### 47. Draw , then draw , then draw a graph of the number of solutions of against for , and then find the value of where there is only 1 solution.

*Solution*. The sketch is trivial.

From the graph, we can see that there are infinite solutions when . Suppose then . When they are tangent to each other at some point, say , and . As a result, and . We can then conclude that there is only 1 solution when or using the sketch. Otherwise, there are 2 solutions. The "graph of the number of solutions" is trivial and will be left to the reader.

### 48. I got a Rubik's cube, held it by two diagonally opposite vertices, and rotated it till it reached the same position. By how many degrees did it take a turn?

*Solution*.
using intuition.

### 49. Solve for all natural numbers and where .

*Solution*. Obviously
. We can prove that only 2 sets of solutions exist for all
:
and
.

WLOG, suppose . Taking natural logarithms on both sides gives , i.e., since . Suppose . Using quotient rule, which is 0 when only. Therefore, iff . can therefore only be 1 or 2; if , then must also be 1, contradicting . When , , so . Since is increasing when and decreasing when , must be greater than . Trial and error gives , and by monotonicity, no more solutions exist. The equation is symmetric about , so and are the only solutions.

### 50. There is a game with 2 players (A&B) who take turns to roll a die and have to roll a six to win. What is the probability of person A winning?

*Solution 1*. Assume A throws first. Let the probability be
. Then
because either A wins on the first roll, or it's as if B started first and we want the probability he doesn't win. Solving gives
.

*Solution 2*. Assume A throws first. The probability of A wins on the first roll is
. The probability of A wins on the second roll is
because both A and B have to fail on the first roll. Repeat the calculation, and we end up getting a geometric series
.

作者介绍