Question

Rectangle $ABCD$ has sides $\overline{AB}$ of length $4$ and $\overline{CB}$ of length $3.$ Divide $\overline{AB}$ into $168$ congruent segments with points $A=P_0,$ $P_1,$ $\ldots,$ $P_{168}=B,$ and divide $\overline{CB}$ into $168$ congruent segments with points $C=Q_0,$ $Q_1,$ $\ldots,$ $Q_{168}=B.$ For $1\leq k\leq167,$ draw the segments $\overline{P_kQ_k}.$ Repeat this construction on the sides $\overline{AD}$ and $\overline{CD},$ and then draw the diagonal $\overline{AC}.$ Find the sum of the lengths of the $335$ parallel segments drawn.

Answer

Answer: 840

Concepts:similarity arithmetic sequence rectangle

Difficulty rating: 1830

Small Hint:

Each segment $\overline{P_kQ_k}$ is parallel to the $3$ - $4$ - $5$ diagonal and is a fixed fraction of its length

Big Hint:

The two side constructions give two identical arithmetic sums; remember to include $\overline{AC}$

Solution:

Put $A=(0,0),$ $B=(4,0),$ and $C=(4,3).$ Then $\begin{aligned}P_k&=\left(\frac{4k}{168},0\right),\\Q_k&=\left(4,3-\frac{3k}{168}\right),\end{aligned}$ so the $3$ - $4$ - $5$ ratio gives $P_kQ_k=5\left(1-\frac{k}{168}\right).$ Hence one construction has total length $5\sum_{k=1}^{167}\left(1-\frac{k}{168}\right)=\frac{5\cdot167}{2}.$ The construction on the other two sides has the same total, and $AC=5.$ Thus the requested sum is $5(167)+5=840.$

1991 AIME Problem 2

Answer: 840

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years