AMC Practice Problems

The following problems were sent to us by the MAA. These example problems are copyrighted by the MAA and are reproduced here under the principle of fair use for the benefit of high school students who visit USM to participate in the competition. You can find more practice problems here. As of 2017, schools that register with USM can participate in a “pre-competition” that makes more practice problems available.

October 2016

Given that \(x\) and \(y\) are distinct nonzero real numbers such that \(x+2x=y+2y\), what is \(xy\)?

(A) 1/4(B) 1/2(C) 1(D) 2(E)4

November 2016

Several problems this month. Solutions are E, D, B.

1. In a chess tournament, each player played a game with every other player. A player earned 1 point for winning a game, 1/2 of a point for a draw, and 0 points for losing a game. At the end of the tournament, several people added the points for all the players, but they got five different results: 2012, 2013, 2014, 2015, and 2016. Given that one of those results was the correct one. Which one was correct?

   (A) 2012(B) 2013(C) 2014(D) 2015(E) 2016

2. The difference between a two-digit number and its reverse is five times the sum of the digits of either number. The sum of the two-digit number and its reverse is

   (A) 44(B) 55(C) 77(D) 99(E) 110

3. A certain convex polyhedron has 14 faces, specifically 11 triangles, 2 quadrilaterals, and 1 pentagon. How many vertices does it have?

   (A) 10(B) 11(C) 12(D) 13(E) 14