OMM 1997

Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$ , and $R$ on side $AB$ , such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$ . Prove that points $P,G,K$ are collinear.

The numbers $1$ through $16$ are to be written in the cells of a $4\times 4$ board. (a) Prove that this can be done in such a way that any two numbers in cells that share a side differ by at most $4$ . (b) Prove that this cannot be done in such a way that any two numbers in cells that share a side differ by at most $3$ .

What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ . Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$ , and $AP$ and $BQ$ meet at $C'$ , such that $AB' = B'C', BC' =C'A'$ , and $CA'= A'B'$ . Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$ .

Prove that number $1$ has infinitely many representations of the form $$1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}$$ , where $n$ and $a_i$ are positive integers with $5 < a_1 < a_2 < ... < a_n$ .