CMO 2018

Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying $$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$ are both integers that are relatively prime to $p$ . If $A_n$ is finite, let $f(n)$ denote $|A_n|$ .

a) Prove that $A_n$ is finite if and only if $n \not = 2$ .

b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that $$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$

Let $n$ and $k$ be positive integers and let $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points.

Prove that there exists at least $k$ unit cubes of length $1$ , all of whose vertices are colored red.

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$ , one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$ .

$ABCD$ is a cyclic quadrilateral whose diagonals intersect at $P$ . The circumcircle of $\triangle APD$ meets segment $AB$ at points $A$ and $E$ . The circumcircle of $\triangle BPC$ meets segment $AB$ at points $B$ and $F$ . Let $I$ and $J$ be the incenters of $\triangle ADE$ and $\triangle BCF$ , respectively. Segments $IJ$ and $AC$ meet at $K$ . Prove that the points $A,I,K,E$ are cyclic.

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$ .

Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

China Mathematical Olympiad 2018 Q6 Given the positive integer $n ,k$ $(n>k)$ and $a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ , $|\mathbb{I} |=k$ ,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$ Attachments: