AMC 12 in one month: a plan made of what you cut

A month is not a shorter version of a year. It is a different problem. Given ten months you can raise your ceiling, and the plan is a syllabus. Given four weeks you can only raise your floor, and the plan is a list of things you are going to stop doing.

So nothing below tells you to learn a topic. Every section is a cut, with the arithmetic that justifies it.

Size the month before you plan it

Thirty days sounds generous until you convert it into hours. Forty-five minutes a day, every day, for four weeks is 21 hours. That is the entire budget, and every decision here is measured against it.

Now price one full practice paper properly. Seventy-five minutes to sit it, and, if the review is honest, meaning you re-solve every question you got wrong and write down the step where you went wrong, roughly two hours after it. Call the pair 3 hours 45 minutes. Four papers is 15 hours, which is over 70% of everything you have.

That is the first cut, and it is the one people resist: two full mocks in the month, not four. Two costs a little over a third of the budget and still delivers what a mock is for, which is calibration under a clock. It is not what makes you better, and the arithmetic says you cannot afford to pretend otherwise.

Cut the last five questions

The paper runs 25 questions in 75 minutes, roughly in order of difficulty, and the final five are a different exam wearing the same cover. In a month they are not yours. Decide that now rather than in the room.

Two numbers make the decision easy. First, the ceiling: answer the first twenty correctly, leave the last five blank at 1.5 points each, and you have $6 \times 20 + 1.5 \times 5 = 127.5$ points, far above the range where qualification is usually settled. Nothing about your target requires you to read questions 21 to 25 at all. The companion guide on target scores works the floor out in full, and the scoring rule is the same on both papers.

Second, the clock. Twenty-five questions in 75 minutes is 3 minutes each. Twenty questions in 75 minutes is 3.75 minutes each, a 25% raise on every question you actually intend to answer. The five you gave up were never going to be solved in three minutes anyway, so the trade is free.

Give them a glance in the last two minutes, mark anything where you can genuinely eliminate choices, and otherwise let them go.

Cut your errors before you cut your ignorance

Inside the first twenty, two different things cost you points, and they are not worth the same.

A question you answered wrongly scores 0, so fixing it is worth the full 6 points. A question you left blank already banked 1.5, so solving it is worth 4.5. Fixing an error pays $6 / 4.5 = 4/3$ of what extending your range pays, and it is the cheaper of the two to buy, because the work is re-deriving something you already know rather than learning something you do not.

Put numbers on a realistic student. You attempt the first twenty, get 15 right and 5 wrong, and leave the last five blank: $6 \times 15 + 1.5 \times 5 = 97.5$.

• Fix three of your five errors, changing nothing else: 18 right, 2 wrong, 5 blank, which is 115.5.
• Instead, keep all five errors and solve three of the hard five: 18 right, 5 wrong, 2 blank, which is 111.

Same three extra correct answers, 4.5 points apart, and the first route is the one you can actually complete in four weeks. So the second cut is this: until your error rate on the first twenty is near zero, no time goes to hard problems.

That requires knowing what your errors are, which means keeping a written log from day one. One line per mistake: the question, the step that broke, and whether it broke because of arithmetic, a misread, or a genuine gap. After a week the log stops being a list and starts being a diagnosis, and almost everybody who keeps one is surprised by the split.

Cut every topic that cannot pay inside four weeks

New material has a long lag. A technique met for the first time three weeks before an exam is a technique you will half-remember under pressure, which is worse than not having it. The things worth adding now are not topics; they are habits that shorten problems you can already do.

Three of them pay almost immediately.

1. Use the answer choices. Five options is information, and this is a multiple-choice paper, not a proof. Test a candidate, check parity or size, rule out two options and reconsider. Working backwards from an option you can test is often faster than solving forwards, and on this paper being fast is the same thing as being right more often.
2. Count constructively, then check the overcount. The commonest wrong answer in a counting question is a small integer multiple of the right one. A convex 12-gon has $12 \times 9 = 108$ if you count nine diagonals from each of twelve vertices, and 54 once you notice every diagonal was counted from both ends. Exactly 108 is twice 54, and that factor of 2 is the whole error. Constructive counting is mostly the discipline of asking what you counted twice.
3. Check arithmetic mod 9. It costs ten seconds. Digit sums give $4321 \equiv 1$ and $1234 \equiv 1$, so their product has to be $1$ mod 9, and indeed $4321 \times 1234 = 5332114$ has digit sum 19. A slip to 5332414 has digit sum 22 and fails the test on the spot. Modular thinking is blind to a transposition, though, since 5332141 keeps the same digit sum, so use it for the errors it catches and do not trust it past them.

Notice what all three have in common: none of them is new mathematics. They are ways of finishing faster and being wrong less, which is precisely the resource a month can buy.

The four weeks

• Week 1. Sit one mock, cold, timed. It is a measurement, so do not prepare for it. Then start the error log and work only in the band where you are sometimes right and sometimes wrong.
• Weeks 2 and 3. No full papers at all. Sets of eight to ten timed questions from your band, reviewed the same day, with every mistake logged and every logged mistake re-solved from a blank page later in the week.
• Week 4. The second mock early in the week, with the triage rule rehearsed: first pass, twenty questions, no bargaining. Spend the rest of the week on your own log, and stop introducing anything new.
• The last three days. No new material and no new sources. Re-read the log, sleep properly, and let the plan be over.

The thing that makes this study plan work in one month is that it is small enough to actually finish. A plan that is too big does not get done at 80%; it gets abandoned in week two and replaced by whatever is nearest.

If you want a measurement of your band instead of a guess, the diagnostic is the quickest route to one. The daily problem posts a new one every day in three tiers and needs no account, which is the cheapest way to keep a triage habit warm between sessions, and the technique pages and problem archive are what you work through once the log tells you where to aim.