OSN 2017

$ABCD$ is a parallelogram. $g$ is a line passing $A$ . Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$ , and the distance from $D$ to $g$ .

Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a trio if one of the following is true:

• A shakes hands with B, and B shakes hands with C, or
• A doesn't shake hands with B, and B doesn't shake hands with C.

If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.

A positive integer $d$ is special if every integer can be represented as $a^2 + b^2 - dc^2$ for some integers $a, b, c$ .

• Find the smallest positive integer that is not special.
• Prove 2017 is special.

Determine all pairs of distinct real numbers $(x, y)$ such that both of the following are true:

• $x^{100} - y^{100} = 2^{99} (x-y)$
• $x^{200} - y^{200} = 2^{199} (x-y)$

A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$ . Prove that $P(n) = 0$ .

Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$ .

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$ .

A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the $1500 \times 1500$ area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of $k$ in which this is possible.