Real numbers $a_1$, $a_2$, …, $a_n$ are fixed. For each $1 \le i \le n$ we let $d_i = \max\{a_j : 1 \le j \le i\} - \min\{a_j : i \le j \le n\}$ and let $d = \max \{d_i : 1 \le i \le n\}$.
(a) Prove that for any real numbers $x_1 \le \dots \le x_n$ we have $$
\max \left\{ \left\lvert x_i - a_i \right\rvert :
1 \le i \le n \right\}
\ge \frac12 d.
$$
(b) Moreover, show that there exists some choice of $x_1 \le \dots \le x_n$ which achieves equality.
Consider five points $A$, $B$, $C$, $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.
In a mathematical competition some competitors are (mutual) friends. Call a group of competitors a *clique* if each two of them are friends. The size of a clique is its number of members; any group of fewer than two competitors is also a clique. Given that the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.
In triangle $ABC$ the bisector of $\angle BCA$ meets the circumcircle again at $R$, the perpendicular bisector of $\overline{BC}$ at $P$, and the perpendicular bisector of $\overline{AC}$ at $Q$. The midpoint of $\overline{BC}$ is $K$ and the midpoint of $\overline{AC}$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.
Let $n$ be a positive integer. Consider $$
S = \left\{ (x,y,z) \mid
x,y,z \in \{ 0, 1, \dots, n\}, \;
x+y+z \gt 0 \right\}
$$ as a set of $(n+1)^3-1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.