Mathematics Interview Questions (XII)
61. Sketch and find the maximum.
Solution. Consider its derivative , which is greater than 0 when and less than 0 when . Its maximum is therefore at , which is .
The domain of that function is ; also, . For any , ; also, as using L'Hôpital's rule, the -axis is its asymptote.
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62. What's the probability of flipping consecutive heads on a fair coin? What about an even number of consecutive heads?
Solution. The number of heads , so the probability of flipping consecutive heads . For an even number of consecutive heads, we can calculate the probability as
63. Two trains start 30 km apart and travel towards each other. They meet after 20 mins. If the faster train chases the slower train, they meet after 50 mins. How fast are the trains moving?
Solution. Let A be the faster train and B be the slower train. The conditions can be translated as and . Solving gives and .
64. A 10-digit number is made up of only 5's and 0's. It's also divisible by 9. How many possibilities are there for the number?
Solution. Suppose it consists of 5's and 0's. Then the sum of digits is , which should be divisible by 9. Since , can only be 9. There are therefore 9 possible positions for the only 0 (since it cannot be at the first digit), giving 9 possibilities.
65. There is a set of numbers whose sum is equal to the sum of the elements squared. What's bigger, the sum of the cubes or the sum of the fourth powers? (Hint: you only need to consider the case that the sum of the cubes is positive.)
Solution. All the elements should be positive; a counterexample could be made if there exists a negative number, for example, . By supposing all elements be positive, we can conclude that the sum of the fourth powers is greater.
Denoting the th element as , we have that using the AM-GM inequality. Hence, . Since , .
On the other hand, we have that using the AM-GM inequality. Hence, . Since , .