Course II · Open courseware
AMC 10: From Problem 1 to the AIME
From five answer choices that give one away to the problems that decide an AIME ticket: the AMC 10 taught by the idea that opens each problem, on real contest problems credited to the MAA.
7 units · 42 chapters · free
01
Playing the Contest
3 chapters- 01Reading the Answer ChoicesEvery AMC problem hands you five candidate answers, and often only one survives a ten-second check.
- 02Make the Problem SmallerWhen a problem is too big to hold in your head, a smaller version of it often shows you the shape of the answer.
- 03Deduction PuzzlesSome AMC problems contain almost no numbers, only statements that cannot all be true at once.
02
Algebra
9 chapters- 04Choosing the UnknownThe hardest step in a word problem is usually the first one: deciding what to call x.
- 05Rates, Ratios, and PercentsSpeed, shared work, discounts, and mixtures are one idea in different settings: an amount per unit.
- 06Averages Are Totals in DisguiseAn average hides a sum, and most average problems open the moment you write that sum down.
- 07Factor Before You ComputeThe AMC is full of numbers too large to multiply out, chosen so that factoring makes them small.
- 08Quadratics and Their RootsYou can know the sum and the product of a quadratic's roots without ever finding the roots.
- 09Arithmetic and Geometric SequencesA list that grows by adding and a list that grows by multiplying behave in completely different ways.
- 10Sequences Built From Their PastSome sequences are defined by their earlier terms, and some long sums cancel down to almost nothing.
- 11Functions, Graphs, and Absolute ValueA graph turns an equation into a picture, and many AMC problems are easier to see than to solve.
- 12Which Idea Opens It? AlgebraInside a chapter about averages you know to use averages; on the contest, nothing tells you.
03
Counting
6 chapters- 13Multiply or Add?Most counting mistakes are not arithmetic errors: they multiply where they should add, or add where they should multiply.
- 14Count What You Don't WantSometimes the things you want are hard to count, but the things you don't want are easy.
- 15Arrangements and OvercountingThe fastest way to count is often to count too many on purpose, then divide by how many times each one was counted.
- 16Choosing, Pascal, and Stars and BarsChoosing a committee and splitting identical candy among friends turn out to be the same kind of count.
- 17Counting Two WaysIf two different counts describe the same set, they must be equal, and that equation is often the whole solution.
- 18Too Few BoxesSome facts are forced simply because there are more things than places to put them.
04
Probability
4 chapters- 19Fix the Sample Space FirstMost probability mistakes come from counting all the outcomes wrong, not the favorable ones.
- 20Probability One Step at a TimeWhen a random process unfolds in stages, you can follow it one step at a time instead of counting everything at the end.
- 21Geometric Probability and Expected ValueA random point turns probability into a question about area, and a random payout turns it into a question about averages.
- 22Which Idea Opens It? Counting and ProbabilityCounting problems rarely announce which method they need, and the wrong method can take ten times as long.
05
Number Theory
8 chapters- 23Factor Into Primes FirstWhen a number problem looks stuck, writing every number as a product of primes almost always gets it moving.
- 24Divisibility, GCD, and LCMTwo numbers share exactly the primes they share, and that simple fact settles most gcd and lcm questions.
- 25Odd and EvenKnowing only whether numbers are odd or even is often enough to rule out almost every possibility.
- 26Remainders and CyclesEnormous powers and processes repeated a hundred times become small once you notice that they repeat.
- 27Digits, Bases, and DecimalsA number's digits are a disguised polynomial, and that view turns digit puzzles into algebra.
- 28Equations in Whole NumbersAn equation that must have whole-number solutions can often be factored until the answers are just factor pairs.
- 29Whole Numbers Are InformationThe word 'integer' in a problem is a clue: it cuts an equation with endless solutions down to a few.
- 30Which Idea Opens It? Number TheoryPrimes, remainders, parity, and factoring can all look like the right tool; usually only one of them is fast.
06
Geometry
10 chapters- 31Angles, Polygons, and the Triangle InequalityAngles add up in predictable ways, and side lengths obey one inequality that rules out more than you expect.
- 32Right TrianglesRight triangles hide inside most AMC geometry problems, waiting for one well-placed segment.
- 33Similar TrianglesTwo triangles with the same angles are scaled copies, and one scale factor unlocks every length at once.
- 34Area by Pieces and RatiosMost shaded-region problems never need a formula you don't know, only a smarter way to cut the region.
- 35Circles: Angles and MeasureAn angle drawn in a circle is tied to the arc it cuts off, and that tie solves most circle angle problems.
- 36Tangent Lines and CirclesA tangent line meets its radius at a right angle, and that one fact turns circle problems into triangle problems.
- 37Coordinates and Lattice PointsPut a figure on a grid and geometry becomes algebra you already know how to do.
- 38Moves and ShapesReflecting or rotating part of a figure can turn a hard length into one you can see.
- 39Thinking in Three DimensionsMost 3D problems on the AMC become 2D problems once you find the right slice or unfold the right net.
- 40Which Idea Opens It? GeometryThe hardest part of an AMC geometry problem is usually deciding which segment to draw first.
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