Opsimath
2021/09/13 阅读：15 主题：默认主题
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 .
Opsimath
2021/09/13 阅读：15 主题：默认主题
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