Ace the AMC explores the topics and techniques needed to excel in mathematical problem-solving, especially for exams like the American Mathematics Competition (AMC).

Through carefully crafted problems and guided explanations, you'll improve your skills in the core topics on the AMC: **Algebra**, **Geometry**, **Number Theory**, and **Discrete Math**. This quiz introduces problems from all of these topics.

**Algebra**

\[ \large x+\frac{1}{x} = 5, \qquad x^2 + \frac{1}{x^2} = \, ? \]

**Geometry**

If the angle of each pencil tip in diagram is \( 30 ^ \circ, \) then what is \(x?\)

In this flat diagram, each identically shaped pencil is formed by a rectangle attached to an isosceles triangle.

**Discrete Math**

For the above arrangement, is it possible to successively flip over pairs of adjacent coins so that all of the coins have their gold side facing up?

"Flip" indicates switching from gold to silver or back again, not a random flip like in probability. "Pairs of adjacent coins" means only two coins next to each other may be flipped at a time.

In the previous problem, it is possible to "play" with the coins and determine that it seems impossible to get all gold.

However, one of the skills which you'll develop in Ace the AMC is the ability to generalize from patterns with small cases.

In particular, note that **flipping pairs of coins does not change the parity** (even/odd) of the total number of gold coins. In this case, since there was initially an odd number of gold coins, it is impossible to ever have four gold coins.

In the next problem, you'll apply this observation to a more general case.

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