Mathematics Interview Questions (XIII)
66. Draw .
Solution. Since it is an exponential function, it must be greater than zero for all values. Observing that does not change its expression, one concludes that it is symmetric about the -axis. So, we may only consider the part where . The derivative which is less than 0 for , so it is a decreasing function with maximum value 1 at . As , . One more little thing is that, since for small values, near the origin, so it looks like a parabola at a small neighborhood of . You can view here for a sample.
67. Draw .
Solution. Since it is a cosine function, it must lie between for all values. Compared to the usual function, near , it will be "flattened"; far from , it will be "squashed". Taking the derivative and equating it to zero gives , which are local minima and maxima. You can view here for a sample.
68. What are the last two digits of the number which is formed by multiplying all the odd numbers from 1 to 1000000?
Solution. First, note that multiplying from 1 to 99 and from 100k+1 to 100k+99 makes no difference, since . Thus, we may only consider . It contains a factor of 25 so the answer could only be 25 or 75. Note that ends in 25 if and ends in 75 if . We have
69. Prove that has no square values for .
Solution. , and any additional term is . So there can be no larger power than 2 when , as 3 is a factor but 27 is not. There can be no square either for , as and all additional terms have last digit 0, and no square ends in a 3. The remaining finite cases are easily checked.
My number theory sucks and has no idea how to do this by myself. I therefore quoted a MathStackExchange answer. Hope you guys will not blame me.
70. How many zeros are there in ?
Solution. Each pair of 2 and 5 produce a 0. In , there are much more 2's than 5's, so considering 5's only would be acceptable. Each number with a factor of 5 produce one 5, and produce an extra 5, with and giving one more. Accordingly, there are 73+14+2=89 5's; therefore, there are 89 0's at the end of .