129A - Cookies

We can compute the sum of all the integers. Then, we enumerate each element and test whether their difference is even or not.

129B - Students and Shoelaces

A straightforward implementation problem. We can use BFS to find out all the nodes whose degree is exactly one. The answer is just the total number of BFS.

129C - Statues

Note that at the eighth second, all the positions are definitely safe. Therefore, it is sufficient to check whether we can survive within the first eight seconds. One feasible solution is to record and update all the safe positions at each second. If we can survive after eight seconds, it means that we can always reach the right upper corner; otherwise not.

129E - Games with Rectangle

In fact we might obtain some more clear observation if we consider the length and width of each rectangular in an independent perspective. For instance, for the length, the rectangulars can only have positions at (1, 2, 3,... n-1). As there are exactly k rectangulars, we should select 2k positions, which gives C_n - 1^2k different patterns. Similarly, for the width we have C_m - 1^2k patterns as well, and thus the final answer should be C_n - 1^2k × C_m - 1^2k. C_p^q can be calculated by using Pascal triangle.