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2017-05-22 Intermediate

         

If y24y=x24x,y^2 -4y = x^2 - 4x, then does it necessarily follow that y=x? y= x?

How many ways are there to paint a 3×33\times3 grid with three colors such that each row and column has all three distinct colors?

Note: If you rotate the above illustration by 90,18090^\circ, 180^\circ or 270270^\circ, they each count as different ways to paint the 3×33\times3.

In the diagram below, ABGFABGF is a trapezoid and DD is the intersection point of its diagonals.

If CECE is drawn such that it passes through DD and is parallel to AB,AB, is it true that CD=DE?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?X+Y?

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

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