AMC 10 Math Competition Prep: Algebra to AIME

The AMC 10 gives 6 points for a correct answer, 1.5 for a blank and 0 for a wrong answer. Because a random guess among five choices is worth only 1.2 points on average, a blank beats a wild guess, but once you can rule out two choices a guess is worth 2 points and beats the blank. The common trap is treating the contest like a school test where every question must be attempted.

Many early AMC 10 questions are word problems that become one or two linear equations. In rate and work problems the key move is to add rates (jobs per hour), never times. The trap is averaging or adding hours directly, which gives a number that looks reasonable and is wrong.

For a quadratic x^2 + bx + c = 0, Vieta's formulas say the two roots add to -b and multiply to c. This lets you find things like the sum of the squares of the roots without ever solving the equation, which saves time and avoids messy square roots. The trap is solving with the quadratic formula when the question only asks about a symmetric expression of the roots.

Absolute value |x - a| is simply the distance between x and a on the number line. Thinking in distances turns many absolute value questions into a picture you can read at a glance. The trap is splitting into cases without a plan and losing track of which case applies where.

Every whole number above 1 breaks into primes in exactly one way, and that prime factorisation answers most divisor questions. If n = 2^a x 3^b x 5^c, then n has (a + 1)(b + 1)(c + 1) divisors, because each divisor picks how many of each prime to use. The trap is listing divisors by hand and missing some.

Modular arithmetic means working only with remainders, which makes huge numbers small. It also solves equations where the answers must be whole numbers: looking at an equation's remainders tells you which values are even possible. The trap is forgetting the problem's conditions, such as positive integers only, when counting solutions.

Counting is about choosing a method that cannot miss or double count. Sometimes the easiest way to count what you want is to count everything, then subtract what you do not want: that is complementary counting. The trap is counting the cases directly and either missing some or counting a number twice.

Stars and bars counts ways to share identical objects among people: picture the objects as stars and the dividers between people as bars. Expected value is the long-run average of a random outcome, the same idea that sits behind the AMC 10 blank-versus-guess rule. The trap in stars and bars is forgetting a minimum, such as each person getting at least two.

Two triangles are similar when their angles match, and then all their sides are in the same ratio. In a right triangle, the altitude to the hypotenuse creates two smaller triangles similar to the original, which gives quick formulas for the pieces. The trap is matching the wrong sides when writing a ratio, so always label corresponding vertices in the same order.

An inscribed angle is half the arc it cuts off, and a tangent always meets the radius at a right angle. Power of a point says that for any point and any line through it that meets the circle twice, the product of the two distances is the same, and for a tangent it equals the tangent length squared. The trap is measuring a secant from the circle instead of from the outside point.

Coordinates turn a geometry picture into algebra you can calculate. The shoelace formula gives the area of any polygon straight from its vertex coordinates, with no heights needed. The trap is listing the vertices out of order around the shape, which gives a wrong area.

Solid geometry on the AMC 10 is about boxes, cubes, cylinders, cones, spheres and pyramids. Often the cleanest route is to name the dimensions as unknowns and write equations, rather than reaching for a formula. The trap is mixing up surface area and volume, or forgetting that a box has three pairs of faces.

The first ten questions on the AMC 10 are usually quicker, and every one is worth the same 6 points as the hardest, so accuracy there matters more than speed. On the hardest questions a student should decide quickly whether to work, guess after eliminating choices, or leave a blank. The trap is spending fifteen minutes on one late question and rushing the easier ones.

A real AMC 10 mixes every topic, so students must recognise quickly which tool a problem needs. The error log turns every mistake into a lesson: each entry records the question type, why it went wrong and the fix. The trap is reviewing only the questions that were wrong and ignoring the ones that were right by luck.

From the 2027 AIME, the MAA uses a fixed qualifying score: 100 or more on the AMC 10, plus other eligibility rules. The AIME is a different kind of test, with 15 questions whose answers are whole numbers from 000 to 999 and no answer choices to check against. The trap is relying on answer choices to guide you, a habit that does not work when there are none.

The last weeks are about sharp, calm practice, not cramming new topics. Students also decide, with the teacher, whether their next contest should be the AMC 10 again or the AMC 12, which adds trigonometry, advanced algebra and advanced geometry. The trap is switching to the AMC 12 before those topics are secure.