IMC 2000 · Problem 4

Suppose the graph of a polynomial of degree $6$ is tangent to a straight line at $3$ points $A_1$, $A_2$, $A_3$, where $A_2$ lies between $A_1$ and $A_3$.

a) Prove that if the lengths of the segments $A_1A_2$ and $A_2A_3$ are equal, then the areas of the figures bounded by these segments and the graph of the polynomial are equal as well.

b) Let $k = \dfrac{A_2A_3}{A_1A_2}$, and let $K$ be the ratio of the areas of the appropriate figures. Prove that

$$\frac{2}{7} k^5 < K < \frac{7}{2} k^5.$$