51. Sketch and on the same axis.

Solution. They are all odd functions and increasing on . is the point of inflection for both of them. Notice that, when or , ; when or , . You can view here for a sample.

52. What would the two sides of a rectangle ( and ) be to maximise the area if , where is a constant?

Solution. From the AM-GM inequality, the area with equality when . Since , when the area is maximised, , .

53. Can 1000003 be written as the sum of 2 square numbers?

Solution. No. Suppose there exists such that . Consider mod 4 on both sides, since square numbers appear. Since , but , so there is no solution to this equation.

54. Show that when you square an odd number, you always get one more than a multiple of 8.

Solution. All the odd numbers have the form where . Squaring gives . If is even, then is even, so is also even. If is odd, then is also odd, so is still even. This means we can write in the form where .

55. Prove that equals infinity.

Solution.

which diverges, so the original series (commonly called the harmonic series) also diverges.

56. Prove that for , when , .

Solution. Taking natural logarithms on both sides gives , i.e., . Suppose . Using quotient rule, which is less than 0 when . Therefore, when , . This gives , completing the proof.

57. Prove that is irrational.

Solution. Proof by contradiction. Assume that is rational, so it can be written as where and . Squaring both sides gives , so . This means must be a multiple of 3, so is also a multiple of 3. (Otherwise, if is not a multiple of 3, then is not a multiple of 3.) We can then write as for some . Therefore, , i.e., . This means must be a multiple of 3, so is also a multiple of 3, by the same logic above. and are both even, but ; contradiction.

58. What are the possible unit digits for perfect squares?

Solution. For , since , we can see that .

59. What are the possible remainders when a cube is divided by 9?

Solution. For , since , we can see that .

60. Prove that 801,279,386,104 can't be written as the sum of 3 cubes.

Solution. Suppose there exists such that . The sum of digits of 801,279,386,104 is 4, so . However, since , , so there is no solution to this equation.