ZA-SAMO 2018

One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is.

[list=a] [] Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end. [] Is it possible that precisely one quarter of the glasses contain green liquid at the end? [/list]

In a triangle $ABC$ , $AB = AC$ , and $D$ is on $BC$ . A point $E$ is chosen on $AC$ , and a point $F$ is chosen on $AB$ , such that $DE = DC$ and $DF = DB$ . It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$ . Determine that value of $\frac{AB}{BC}$ .

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$ , and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$ .

Let $ABC$ be a triangle with circumradius $R$ , and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$ . Let $P$ be an arbitrary point in the same plane as $ABC$ . The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation: $$\operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).$$

Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$ .

Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$ . Construct an $n \times 3$ table where the entries of the $k$ -th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$ . Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.

[list=a] [] Prove that at least three entries remain at the end of the procedure. [] Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain. [/list]