1989 APMO Problems

Problem 1

Let $x_1, x_2, x_3, \dots , x_n$ be positive real numbers, and let

$S=x_1+x_2+x_3+\cdots +x_n$ .

Prove that

$(1+x_1)(1+x_2)(1+x_3)\cdots (1+x_n)\leq 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots +\dfrac{S^n}{n!}$ .

Solution

Problem 2

Prove that the equation

$6(6a^2+3b^2+c^2)=5n^2$

has no solutions in integers except $a=b=c=n=0$ .

Solution

Problem 3

Let $A_1,A_2,A_3$ be three points in the plane, and for convenience, let $A_4=A_1$ , $A_5=A_2$ . For $n=1, 2,$ and $3$ , suppose that $B_n$ is the midpoint of $A_nA_{n+1}$ , and suppose that $C_n$ is the midpoint of $A_nB_n$ . Suppose that $A_nC_{n+1}$ and $B_nA_{n+2}$ meet at $D_n$ , and that $A_nB_{n+1}$ and $C_nA_{n+2}$ meet at $E_n$ . Calculate the ratio of the area of triangle $D_1D_2D_3$ to the area of triangle $E_1E_2E_3$ .