Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side.

Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i = 1}^{2n - 1} (a_{i + 1} - a_i)^2 = 1$. Find the greatest possible value of $ (a_{n + 1} + a_{n + 2} + \ldots + a_{2n}) - (a_1 + a_2 + \ldots + a_n)$.

Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n - 1$ that can be divided by $ d$.

Given that the sum of the distances from point $ P$ in the interior of a convex quadrilateral $ ABCD$ to the sides $ AB, BC, CD, DA$ is a constant, prove that $ ABCD$ is a parallelogram.

The sequence $ \{a_n\}$ satisfies $ a_0 = 0, a_{n + 1} = ka_n + \sqrt {(k^2 - 1)a_n^2 + 1}, n = 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n = 0, 1, 2, \ldots$.

A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i = 1}^5 \frac {1}{1 + x_i} = 1$. Prove that $ \sum_{i = 1}^5 \frac {x_i}{4 + x_i^2} \leq 1$.

$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.