How hard is the AIME compared to the AMC 10?

The most common way this question gets asked is whether the AIME is "harder" than the AMC 10. That word is doing too much. The AIME is not a stretched AMC 10 with meaner questions, and treating it that way is the mistake that costs students the most time in the month between them.

It is a different paper, in four ways that can each be measured. Once you have measured them the "how much harder" question stops being a feeling and becomes an inventory of four format changes, three of which are actively working against every habit the AMC 10 rewarded.

The four numbers

Both papers are written for the same audience on the same weekend of the year, and after that they diverge. The AMC 10 gives you 25 multiple-choice questions in 75 minutes. The AIME gives you 15 questions, each answered by an integer from 000 to 999, in three hours.

That single sentence contains all four format changes. Pulled out one at a time:

• Time per question. $75 \times 60 = 4500$ seconds over 25 questions is 180 seconds each. $3 \times 60 \times 60 = 10800$ seconds over 15 questions is 720 seconds each. The AIME gives you four times as long per question. That is not a courtesy, it is what the questions cost.
• Answer space. The AMC 10 asks you to pick one of five. The AIME asks you to write an integer from 000 to 999. The space of possible answers grew by a factor of 200, from 5 to 1000, and every strategy that used the answer choices as information vanishes at the same stroke.
• Penalty for a wrong answer. On the AMC 10 a blank scores 1.5 and a wrong scores 0, so a wrong answer is 1.5 points worse than not answering. On the AIME a blank and a wrong both score 0. The penalty is gone, and with it every reason to leave anything blank.
• Score range. The AMC 10 tops out at $6 \times 25 = 150$ points. The AIME tops out at $1 \times 15 = 15$. A single AIME question is worth $150 / 15 = 10$ AMC 10 points of maximum, which is roughly the impact of two AMC 10 questions on the boundary between a good score and a very good one.

That is the whole comparison. Everything below is what each of those four does to how you should sit the paper.

What the extra time buys, and what it does not

Four times as long per question is the change that looks most generous, and it is the one most likely to be misread. The gain is not 540 extra seconds to keep doing what you were doing. It is 540 seconds of headroom for a longer solution path.

The AMC 10 rewarded solving inside 180 seconds. Under that clock, a nine-minute solution was a failure mode, not a plan. The AIME clock is exactly nine minutes longer per question, and the questions are shaped to fill it. A question that could be finished in 180 seconds would be an AMC 10 question, and it would not have been chosen.

So the mistake is thinking the extra time means the AMC 10 pace is now safe. The pace is still tight, because the questions are correspondingly longer. What the time actually buys is the option to sit with a problem long enough to notice the setup, which is exactly what a three-minute clock does not allow. Sitting with the problem is the technique, not a symptom of losing to it.

The free sample lesson is the shortest demonstration of what a longer setup looks like — reading the object before touching the arithmetic — and it works for the same reason. A paper that gives you long questions is a paper that rewards reading the setup before you compute anything at all.

The answer space is the biggest change

Five choices to 1000 is a factor of 200. That is the single largest gap between the two papers, and it is also the one most invisible until you sit an AIME cold.

Every AMC 10 strategy that used the answer choices — testing candidates, ruling out on parity, checking size, comparing to a rough estimate — is worth nothing on the AIME. You cannot try all 1000 candidates in 720 seconds. You cannot even try a hundred. On the AIME you must actually produce the integer, and there is no shortcut through the format.

There is one small mercy. Fewer than 4% of the 1000 possible answers are perfect squares, and even fewer are round numbers like 100 or 500, so an answer that looks unusual is not a mistake to distrust. On the AMC 10 an unusual value was suspicious because the paper chose from five and the setter had to make them all plausible. On the AIME the setter never chose your answer; the mathematics did.

The penalty is gone, and it means one thing

This one is the easiest to state and the easiest to get wrong.

On the AMC 10, a random guess between five options has expected value $6 \times \tfrac{1}{5} = 1.2$, which is less than the 1.5 a blank pays. So a blind guess is strictly worse than a blank. Eliminate one option and the expected value rises to 1.5, a break-even; eliminate two and it is 2, and guessing is clearly right. The paper is built to reward disciplined blanks and punish undisciplined guessing.

On the AIME a blank and a wrong both score zero, so a guess has strictly non-negative expected value. Even a random integer from 000 to 999, with expected value $15 \times \tfrac{1}{1000} = 0.015$, is not worse than a blank; it is just almost identical to one. And any guess where you actually know something — a parity, a size, a residue — is worth more than nothing.

The implication is one sentence: you should write an answer in every box, and you should do it whether or not you finished the question. Leaving a box blank on the AIME is throwing away the free lottery ticket the format is handing you. It is not a large ticket, but it is not zero and there is no reason to refuse it.

The score is coarser, and it changes what a mock tells you

Fifteen points instead of 150 means each AIME question is worth 10 AMC 10 points of ceiling, and that coarseness deserves a moment before you decide anything based on a mock score.

The gap between 5 and 6 out of 15 on the AIME is $\tfrac{6}{15} - \tfrac{5}{15} = \tfrac{1}{15}$ of the paper, or a little under 7 percentage points of the ceiling. The equivalent gap on the AMC 10 is 6 points out of 150, or 4%. One AIME question moves you further than one AMC 10 question does, so a single AIME mock — one paper, one afternoon — is a much noisier estimate of where you actually stand than a single AMC 10 mock.

That has two practical consequences. You need more AIME mocks than AMC 10 mocks to get the same signal about your level, and you should not read too much into any single mock score in either direction. Score six one weekend and eight the next, and the difference is one question you happened to see; it is not a change in ability.

The elimination cliff

Combining the answer-space change and the penalty change gives the sharpest way to state the difference in one sentence: on the AMC 10, being able to eliminate two of five options was worth as much as a real solution; on the AIME, no partial elimination is worth anything at all.

That is the elimination cliff. And it explains something students report often, which is that the AIME feels not four times harder than the AMC 10 but more than that. The AMC 10 had two ways to earn a point — solve the question, or narrow the choices — and the AIME only has one. The paper that took away the shortcut also removed every intermediate credit.

What to do with this

The AMC 10 taught you three habits that will actively hurt you on the AIME: fast solving inside a three-minute clock, using the answer choices as information, and disciplined blanking. All three were correct for the paper they belonged to. None of them is correct for the AIME.

In the month between the two papers, the training that pays is the one that inverts each of those habits.

1. Time discipline in twelve minutes, not three. Sit AIME questions with a twelve-minute clock and let the solution be long. A question you would have abandoned at three minutes on the AMC 10 is a question you might solve at eight on the AIME, and the timer is what teaches your instinct to wait.
2. Compute the answer, do not test it. Everything that ends in "and one of the choices was this" has to be replaced by "and the answer is this." That is a different piece of work.
3. Never leave a box empty. Even at the last minute, even if the guess is a wild integer, write something. The expected cost of the habit is zero and the expected value is positive.

None of the three requires learning new mathematics. They are format habits, and they train fast in the four weeks the calendar usually gives you. If you want to keep pattern recognition warm for both papers, the daily problem posts a new one every day in three tiers and needs no account, the technique pages are the vocabulary the AIME assumes, and the problem archive is the closest bench of longer problems the site keeps.