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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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