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# Problems of the Week

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# 2017-05-22 Intermediate

If $$y^2 -4y = x^2 - 4x,$$ then does it necessarily follow that $$y= x?$$

How many ways are there to paint a $$3\times3$$ grid with three colors such that each row and column has all three distinct colors?

Note: If you rotate the above illustration by $$90^\circ, 180^\circ$$ or $$270^\circ$$, they each count as different ways to paint the $$3\times3$$.

In the diagram below, $$ABGF$$ is a trapezoid and $$D$$ is the intersection point of its diagonals.

If $$CE$$ is drawn such that it passes through $$D$$ and is parallel to $$AB,$$ is it true that $$CD = DE?$$

In a small, remote country, there are only two types of bills--the $X bill and the$Y bill.

With combinations of these bills, almost all positive integer prices can be paid exactly. Only fifteen positive integer prices can't be paid exactly, one of which is \$18.

What is $$X+Y?$$

All the resistors in the above circuit have equal resistance $$R = \SI[per-mode=symbol]{10}{\ohm}$$. Calculate the equivalent resistance $$R_e$$ between points $$A$$ and $$B$$ in ohms. Submit your answer to the nearest integer.

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