LESSON AIME Format & Scoring

Fifteen problems, three hours, and every answer an integer from 000 to 999.

The AIME removes the two things that make the AMC forgiving. There are no answer choices to work backwards from, and there is no credit for leaving a problem alone. What it gives you in exchange is time — three hours for fifteen problems, four times the AMC's pace — and a guarantee that every answer is a whole number under a thousand, which turns out to shape the problems more than anything else about the format.

1. The Exam at a Glance

Problems15
Time3 hours, which is 180 minutes
Every answeran integer from 000 to 999
Scoring1 point per correct answer; nothing for a wrong one and nothing for a blank
Maximum15
Getting therea strong AMC result — roughly the top 2.5% of AMC 10 takers or the top 5% of AMC 12 takers

Three hours over fifteen problems is $180 / 15 = 12$ minutes each — four times what the AMC allows. That is not generosity; it is a statement about the problems. An AIME problem is expected to take real work, and finishing one in three minutes usually means you have misread it.

Fifteen independent outcomes. There is no partial credit and no scaling. Your score is the number of problems whose exact integer you produced. That makes it unusually easy to set a goal and unusually unforgiving of arithmetic slips in the last step.

2. Why Every Answer Is an Integer

A contest that accepts only integers between 0 and 999 cannot ask "find the area" if the area is $\sqrt{3} + 2$. So it does not ask that. It asks for something derived from the answer that is such an integer, and a handful of devices do almost all of this work:

Reduce, then add

A probability comes out as $m/n$ in lowest terms and the problem asks for $m + n$. The reduction is part of the question, not a formality.

Name the pieces

A length comes out as $a\sqrt{b} + c$ in simplified form and the problem asks for $a + b + c$.

Take a remainder

The quantity itself is enormous, so the problem asks for its remainder on division by 1000 — which is exactly the last three digits.

Use the range as a check

If your final value is negative, or larger than 999, or not a whole number, you have not finished the problem. The format hands you a free error detector; use it every time.

Worked example — the reduction is the trap

Problem. A probability works out to $\frac{27}{125}$, and the problem asks for $m + n$ where $\frac{m}{n}$ is that probability in lowest terms. What do you record?

Step 1 — check it is reduced. $27 = 3^3$ and $125 = 5^3$, so $\gcd(27, 125) = 1$. It is already in lowest terms.

Step 2 — add. $m + n = 27 + 125 = 152$, which is comfortably inside 000 to 999.

Why this is worth a worked example. Had the same probability arrived as $\frac{54}{250}$, adding straight away gives $304$ — a perfectly reasonable-looking three-digit answer that is wrong. Nothing about $304$ warns you. Reducing first is the only thing that does.

Try it — reduce before you add

A probability comes out as `\frac{54}{250}`. The problem asks for `m+n` where `\frac{m}{n}` is that probability in lowest terms. What do you record?

3. The Scoring Rule, and What It Actually Implies

$\text{AIME score} = \text{number of correct answers, out of } 15$

That is the whole rule. A wrong answer scores 0. A blank scores 0. They are the same number, and that single fact settles a question students often get backwards.

Writing an answer can never cost you. Because a blank and a wrong answer are worth the same, filling in your best integer is never worse than leaving the box empty — and it is occasionally better. There is no such thing as an AIME answer you should withhold.

What is true is that a blind guess is nearly worthless. One integer out of a thousand is right about one time in a thousand, so guessing on every one of the fifteen problems is worth about

$15 \cdot \tfrac{1}{1000} = 0.015 \text{ points}$

a fiftieth of one problem. So the rule is not "guess" and it is not "do not guess" — it is about when:

  • Never spend minutes on a blind guess. Those minutes are worth far more spent on a problem you can actually finish.
  • Never leave a box empty at the end. The last seconds cost you nothing and an empty box is the one answer that certainly scores 0.
  • Partial work often shrinks the range. If you have got as far as "the answer is a remainder mod 1000 of something ending in 24", you are no longer guessing among a thousand values.

The contrast with the AMC is exact and it is worth holding both in mind: there, a blank pays 1.5, so a guess genuinely risks something. Here it risks nothing.

What an AIME point is worth. Selection for the proof-based olympiads runs on a combined qualifying index:

$\text{index} = (\text{AMC score}) + 20 \cdot (\text{AIME score})$

Your AMC 12 index is the route to the USAMO; your AMC 10 index is the route to the USAJMO. The qualifying index is published by the MAA after each cycle rather than announced in advance, so this course states no target number for it — but the structure of the formula is worth a great deal on its own.

One AIME answer moves the index by 20. On the AMC, turning a blank into a correct answer moves your score by $4.5$. So a single extra AIME problem is worth about four and a half extra AMC questions. Once you have qualified, the AIME is where preparation pays.
The multiplier changed, and most advice has not. Through the 2024–25 cycle the index used a factor of 10; from the 2025–26 cycle it is 20. Guidance written before that change understates the AIME's weight by half. If a source does not say which cycle it describes, do not use its arithmetic.

Try it — the current index

A student scores 114 on the AMC 12 and 9 on the AIME. Using `\text{index} = \text{AMC} + 20 \cdot \text{AIME}`, what is their index?

4. Twelve Minutes a Problem, Spent Unevenly

The problems climb: the early ones are meant to be reachable and the last ones are meant to be hard, which is exactly why an even twelve minutes each is the wrong plan. Twelve minutes on a problem you can do in six is six minutes taken from a problem that needed twenty.

Worked example — banking time early

Problem. You intend to attempt 12 of the 15 problems. If you spread the 180 minutes evenly over just those twelve, what do you get — and what happens if the first six take only 9 minutes each?

Step 1 — the even split. $180 / 12 = 15$ minutes per attempted problem.

Step 2 — finish the first six under budget. Six problems at 9 minutes instead of 15 frees $6 \cdot (15 - 9) = 36$ minutes.

Step 3 — hand it to the other six. Thirty-six minutes over six problems is 6 minutes each, so those six go from 15 to $15 + 6 = 21$ minutes.

Check the account. $6 \cdot 9 + 6 \cdot 21 = 54 + 126 = 180$. Nothing was invented; the time was moved. The time-management lesson pushes this further.

Where AIME scores are actually made. Not in heroics on problem 15, but in finishing the early problems quickly and correctly. Speed there is worth minutes later; a slip there costs a whole point with no partial credit to soften it.

Try it — the last two minutes

Two minutes remain and one problem is unsolved. What does the AIME scoring rule say about writing down your best integer?

Lesson complete

No choices to work backwards from and no credit for silence: the AIME asks for an exact integer and pays only for exactness.

15 problems, 180 minutes, every answer an integer from 000 to 999, 1 point each, maximum $15$.
$180/15 = 12$ minutes a problem — four times the AMC's pace, because the problems need it.
A wrong answer and a blank both score $0$, so filling in a box can never cost you points.
A blind guess is worth about $1/1000$; do not spend minutes on one, but never leave a box empty.
Reduce before you add: $54/250$ gives $152$, not $304$, and the wrong answer looks perfectly plausible.

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