AMC 12 Math Competition Prep: Advanced Algebra to the AIME

The AMC 12 gives 6 points for a right answer, 1.5 for a blank and 0 for a wrong one, so a blind guess is worth only 1.2 points on average and loses to leaving the question empty. Once two choices are ruled out, a guess among three is worth 2 points and beats the blank. The common trap is guessing everything out of habit from tests with no penalty, or leaving everything blank out of fear. This fortnight also sets a diagnostic so each student knows where to start.

Vieta's formulas link the roots of a polynomial to its coefficients, so you can find sums and products of roots without ever solving the equation. That matters because AMC 12 polynomials are often impossible to factor by hand, yet the question only asks about a symmetric expression of the roots. The trap is mixing up signs: for x^3 + ax^2 + bx + c the sum of roots is minus a, not a.

A logarithm answers the question: what power do I raise the base to? On the AMC 12 the key skill is changing every logarithm in a problem to one common base so they can be added like ordinary numbers. The usual trap is treating log_4 x as if it were log_2 x, forgetting that log_4 x is only half of log_2 x.

A telescoping sum is one where each term cancels part of the next, so a long sum collapses to its first and last pieces. The AMC 12 loves sums that look hopeless until one term is rewritten, often by rationalising a denominator or splitting a fraction. The trap is starting to add terms one by one and running out of time.

The law of cosines is the Pythagorean theorem for any triangle: it gives an angle from three sides, or a side from two sides and the angle between them. Once an angle is known, the area is half of two sides times the sine of the angle between them. The trap is using the wrong angle: the angle in the formula must sit opposite the side you single out.

A complex number can be written by its distance from zero and its angle, and in that form multiplying means multiply the distances and add the angles. That turns a huge power like (1 + i)^20 into a quick calculation. The trap is expanding the power by hand, which takes far too long and invites mistakes.

Modular arithmetic means keeping only the remainder, like reading a clock. To find the last two digits of a huge power you only need the remainder when dividing by 100, and powers always fall into a repeating cycle. The trap is trying to compute the power itself instead of finding how long the cycle is.

The Chinese remainder idea says that if you know a number's remainders when divided by numbers that share no factors, the number is fixed up to their product. In practice you solve one condition, list the numbers that satisfy it, and test them against the next condition. The trap is listing all numbers up to the product one by one when stepping through a single condition is far faster.

Recursion means counting the big case by building it from smaller cases you have already counted. For example, a string with no two 1s next to each other either ends in 0 or ends in 01, which links each length to the two before it. The trap is trying to list every case directly, which is slow and almost always misses some.

Conditional probability asks for a chance when you already know something has happened, so the known event becomes the whole new set of possibilities. Expected value is the long-run average, and linearity of expectation lets you add expected values even when events depend on each other. The trap is dividing by all 36 dice outcomes when the question has already told you which smaller set you are in.

Power of a point says that for any line through a point and a circle, the product of the two distances from the point to the circle is always the same number. For a tangent, both distances are the tangent length, so the product is the tangent length squared. The trap is multiplying the inside piece of a secant instead of the whole distance from the point.

Many hard geometry questions become routine once you place the figure on coordinate axes, because lengths, planes and distances turn into short formulas. A cube is ideal for this, since its corners have simple coordinates. The trap is placing the axes awkwardly, which makes every later step messy.

Pacing means deciding in advance how long each part of the paper deserves, so the hard questions at the end get real time. Estimation is a big part of this: often you only need to know which answer is largest, not its exact value. The trap is spending ten minutes on one early question and arriving at the last ten with no time left.

An error log turns each mistake into a rule you will remember: what went wrong, why, and what to check next time. Many AMC 12 points are lost not to hard maths but to traps, such as an equation that looks solvable but has no solutions. The trap is reviewing only the questions you got wrong, not the ones you got right by luck.

From 2027, the AIME is open to students who score 85 or more on the AMC 12, plus other eligibility rules. It has 15 questions with no answer choices, and every answer is an integer from 000 to 999, so you cannot check against choices or guess usefully. The trap is stopping at an answer that is close but not finished, for example forgetting to take the remainder the question asks for.

Because the 2027 AIME has two separate 90-minute parts and no return to Part 1, students must finish and check Part 1 before moving on. Practising under that split teaches you to budget time for 8 questions, then 7. The trap is leaving a Part 1 question half done, expecting to come back to it.