Given a $ 1 \times 25$ rectangle divided into $ 25$ "boxes" ($ 1 \times 1$), is it possible to write integers $ 1$ to $ 25$ so that the sum of any two adjacent "boxes" is a perfect square?

Using digits $ 1, 2, 3, 4, 5, 6$, without repetition, $ 3$ two-digit numbers are formed. The numbers are then added together. Through this procedure, how many different sums may be obtained?

A sack contains blue and red marbles*. Consider the following game: marbles are taken out of the sack, one-by-one, until there is an equal number of blue and red marbles; once the number of blue marbles equals the number of red marbles, the game is over. In an instance of this game, it is observed that, at the end, $ 10$ marbles were taken out of the bag, and no $ 3$ consecutive marbles were all of the same color. Prove that, in said instance of the game, the fifth and sixth marbles were of different color. *The original problem involved "stones."

If the sides of a triangle have lengths $ a, b, c$, such that $ a + b - c = 2$, and $ 2ab - c^{2} = 4$, prove that the triangle is equilateral.

Consider a triangle $ ABC$, with $ \angle A = 90^{\circ}$, and $ AC > AB$. Let $ D$ be a point in $ AC$ such that $ \angle ACB = \angle ABD$. Draw an altitude $ DE$ in triangle $ BCD$. If $ AC = BD + DE$, find $ \angle ABC$ and $ \angle ACB$.

Let $n$ be a natural composite number. Prove that there are integers $a_1, a_2,. . . , a_k$ all greater than $ 1$, such that $$a_1 + a_2 +... + a_k = n \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}\right)$$