AMC 10: how many questions do you need for AIME?

"How many do I need?" is the most asked question about this exam, and it is asked as though the answer were a number somebody is refusing to hand over. It is not being withheld. It does not exist yet.

The score you need to qualify for the AIME is a percentile of the people who sat the paper, and the score matching that percentile is worked out after the papers are marked. It moves between the two versions of the same year, and it moves from one year to the next, because the difficulty of the paper and the strength of the field both move. Any fixed number you have been given is a past result being quoted as a rule.

What does not move is the scoring. And the scoring answers a better question, one that is entirely under your control: for a target score you pick yourself, how many questions must you actually get right? That has an exact answer, and it is lower than most people assume.

Count in quarters, not in points

The paper has 25 questions with five answer choices each. A correct answer is worth 6 points, a question left blank is worth 1.5, and a wrong answer is worth 0. The maximum is 150.

Those numbers are awkward to hold in your head with a clock running, and there is no need to. Notice that $1.5 \times 4 = 6$. A blank is worth exactly a quarter of a correct answer, so count in quarters:

• A correct answer is 4.
• A blank is 1.
• A wrong answer is 0.

Write $c$ for the number you get right and $b$ for the number you leave blank. Your quarter total is $4c + b$, out of a maximum of 100, and your score is $1.5(4c + b)$.

That is not a trick for adding up your score faster. It is the model that makes every strategic decision on this paper immediate, because the only choice you ever make in the room is answer or leave it, and those are now worth 4 and 1.

The floor: the fewest right answers that can reach a target

Pick a target score $T$. What is the smallest number of correct answers that could reach it?

Working backwards from the target is the right direction, because the target tells you more than your current score does. The best you can do with $c$ right answers is to leave every remaining question blank, which is $4c + (25 - c) = 3c + 25$ quarters, or $4.5c + 37.5$ points. So $c$ right answers can reach $T$ only if $4.5c + 37.5 \ge T$, which rearranges to $c \ge (T - 37.5)/4.5$.

That is a floor, and a floor on its own proves nothing. The other half is a construction that reaches it, which is bounding and squeezing in its plainest form: show that nothing smaller can work, then exhibit something that does. Both halves, for four targets people commonly aim at:

• Target 90: 12 right is enough, leaving the other 13 blank, for 91.5 points. With 11 right the ceiling is 87.
• Target 100: 14 right is enough, leaving the other 11 blank, for 100.5 points. With 13 right the ceiling is 96, and no arrangement of the remaining questions rescues it.
• Target 110: 17 right is enough, leaving the other 8 blank, for 114 points. With 16 right the ceiling is 109.5.
• Target 120: 19 right is enough, leaving the other 6 blank, for 123 points. With 18 right the ceiling is 118.5.

That 109.5 is worth a moment. A student who answers 16 correctly and leaves every single remaining question blank, which is perfect discipline, finishes half a point under 110. Nothing about that is bad luck. It is just what the rule does.

The second number: what answering everything costs

Every floor above assumes you leave blank what you cannot solve. Most students do not. If you answer all 25 there are no blanks to collect, and you need $6c \ge T$ outright:

• Target 90: 15 right, against 12 with blanks.
• Target 100: 17 right, against 14.
• Target 110: 19 right, against 17.
• Target 120: 20 right, against 19.

The gaps are 3, 3, 2 and 1, and that shrinking pattern is the finding on this page worth carrying into the exam. Blank discipline is worth three questions to somebody aiming at 100 and one question to somebody aiming at 120. If you are near the boundary, how you treat the questions you cannot solve is worth more to you than another technique. If you are aiming high, it is worth almost nothing, and there is no substitute for being right more often.

When a guess beats a blank

Same model, one line of arithmetic. A blank is 1 quarter, guaranteed. A guess is 4 quarters with probability $p$, so it is worth $4p$ quarters, and guessing wins exactly when $p > \tfrac14$.

With five choices and nothing to go on, $p = \tfrac15$, which is below a quarter: a blind guess is strictly worse than a blank. Eliminate one choice and $p = \tfrac14$ exactly, a coin flip you should be indifferent to. Eliminate two and $p = \tfrac13$, and guessing is clearly right.

A corner worth knowing

Since every score is $1.5(4c + b)$ with $c + b \le 25$, not every multiple of 1.5 is a score anybody can get. Above 141 the only attainable scores are 144, 145.5 and 150. There is no 147 and no 148.5, because reaching those quarter totals would need more questions than the paper has.

What this says about practice

The floor result says something plain and slightly uncomfortable: the resource you are short of is right answers, not attempts. A wrong answer scores exactly what never reading the question scores. The paper does not reward coverage at all. It rewards certainty, and it pays a small dividend for knowing when you do not have it.

So the training splits in two, and most people only do the first half:

1. Accuracy inside a band. Find the range of questions where you are usually right but not always, and work until you are always. Moving from 14 right to 17 right is three questions, and they are the three you already nearly solve.
2. Triage in the first pass. Deciding fast which questions you will not answer is a separate skill from solving them, and it is the one that turns the arithmetic above into points.

Triage is trainable, and it trains faster than solving does, because a single question gives you the whole exercise: read it, decide inside twenty seconds whether it is yours, and check afterwards whether the decision was right. You are scoring your judgement, not your algebra. The daily problem posts a new one every day in three tiers and needs no account, which makes it a cheap way to run that drill between sessions. If you want a measurement rather than a feeling about where your band actually sits, the diagnostic is the fastest route to one, and the problem archive and the technique pages are what you work through once you know.