Week 21 · Part 2 — Speed Drill
30 Q1-Q5 problems · 60 seconds each · Target ≥ 25/30
Final Sprint · Week 21 · Part 2

Speed Drill — 30 × 60s

Lock in your Q1-Q5 reflexes. 30 problems back-to-back. 60 seconds each. No second chances — auto-advance at zero.
30
Problems
60s
Per Problem
25+
Target Score
2
Rounds

Why this drill?

Q1-Q5 on AIMO are the guaranteed points. Strong students miss them not because the maths is hard, but because they rush, misread, or freeze. This drill forges the reflex: see → solve → write → next, in under a minute, every time.

1 · Timer is brutal

60 seconds. When it hits zero, the page auto-advances and locks your answer as a timeout. No going back.

2 · Integer answers only

Every problem in this drill has a verified integer answer (0-999 typical). Type the number, press Enter, move on.

3 · Skip is allowed

If you have no idea in 10 seconds, skip. Banking 30 seconds on a hard one is worse than locking 4 easy ones.

4 · Two rounds

Round 1 = diagnostic. After results, Round 2 reshuffles same problems for improvement focus. Track your delta.

Scoring & breakdown

After 30 problems you get:

Composition of the 30

Q1-Q2 · easy · 12 problems

2-mark warmups. These should be reflex answers — single concept, one or two steps. If you miss more than 1 of these, your fundamentals need a Week 1-5 review.

Q3-Q4 · medium · 12 problems

3-4 mark working problems. These need a strategy choice plus 2-4 steps of execution. Target: 10/12. This is where speed gains live.

Q5 · harder · 6 problems

4-mark bridge problems between Easy and Hard halves. Often a clever observation unlocks a 30-second solution. Target: 4/6.

Category mix

Problems are drawn from Number Theory, Algebra, Geometry, and Combinatorics in roughly even proportion. The mix is randomised per drill so two runs aren't identical even on Round 1.

The 60-second decision tree

The single biggest gain from this drill is learning when to not spend a minute. Use this tree on every problem:

  1. 0-10s · Read & recognise. Read the whole problem once. What category is it? Have you seen something like it before? If yes — go. If no — go to step 2.
  2. 10-20s · Pick a tool. NT → modular arithmetic / factorisation / digit-sum. ALG → substitute / factorise / Vieta's. GEO → coordinates / similar triangles / power of a point. COMB → casework / complement / pigeonhole.
  3. 20-40s · Execute. One direction only. If your tool isn't working after 20 seconds of honest work, abandon — don't switch tools mid-problem.
  4. 40-55s · Verify or skip. If you have an answer, sanity-check it once (units, sign, parity). If you don't, type your best integer guess and move on.
  5. 55-60s · Lock it. Press Save. Don't second-guess. The next problem might be a free 4 marks waiting for you.

Common time-traps to watch for

The "almost there" trap

You've spent 50 seconds, the answer is "just one more step away." It almost never is. If you're not done at 50s, lock your best guess and move. The opportunity cost is the next problem you won't get to.

The arithmetic trap

You picked the right method but botched 14 × 17. Slow your final arithmetic step by 5 seconds — that's the highest-leverage time in the whole minute.

The over-reading trap

Reading the problem three times "to be safe" costs you 25 seconds. Read once with focus. If you misread, the auto-advance will punish you exactly once and you'll stop doing it.

The category-mismatch trap

You're attacking a NT problem with algebra, or a geo problem with brute coordinates. If 20 seconds in you've made zero progress, the most likely cause is wrong-tool. Restart the diagnosis, don't push harder.

Round 1 vs Round 2 — what to track

Round 1 is your diagnostic photo. Don't game it. Don't pre-study the bank. The number that comes out is your honest current state.

Round 2 (same 30 problems, reshuffled) measures two things:

The target Round-2 delta is +3 to +5. Anything less means you didn't actually re-do the misses; anything more means Round 1 was nerves, not knowledge.

Category cheat sheet — recall before you start

Skim this list once before pressing Start. The point isn't to memorise it now — it's to tag the categories so your brain can route a problem in 3 seconds instead of 15.

Number Theory · NT

Digits & digit sums

If the problem mentions digits, think: sum of digits mod 9 = number mod 9. Last-digit problems usually reduce to mod 10. Trailing zeros of $n!$ = number of 5s in its factorisation.

Divisibility & factors

Number of divisors of $n = p_1^{a_1} p_2^{a_2} \cdots$ is $(a_1+1)(a_2+1)\cdots$. Sum of divisors is multiplicative. For "smallest $n$ such that…" problems, factorise the target first.

Modular arithmetic

Squares mod 4 are only 0 or 1. Squares mod 8 are 0, 1, 4. Squares mod 9 are 0, 1, 4, 7. Fermat's little theorem: $a^{p-1} \equiv 1 \pmod{p}$ if $\gcd(a,p)=1$.

GCD & LCM

$\gcd(a,b) \cdot \text{lcm}(a,b) = ab$. Bezout: $\gcd(a,b) = ax+by$ for some integers $x,y$. Euclidean algorithm runs in $\log$ time — use it on any "find gcd" problem instantly.

Algebra · ALG

Vieta's & symmetric sums

For $x^2 + bx + c = 0$: sum of roots $= -b$, product $= c$. For cubic $x^3 + px^2 + qx + r$: sum $= -p$, sum-of-products-pairs $= q$, product $= -r$. Many AIMO Q3-Q5 algebra problems collapse the moment you write Vieta's.

Substitution tricks

Symmetric in $a$ and $b$? Try $s = a+b$, $p = ab$. Reciprocal-symmetric in $x$? Divide by $x^k$ and let $y = x + 1/x$. Square root in a denominator? Rationalise immediately.

AM-GM & Cauchy

AM-GM: $\frac{a+b}{2} \ge \sqrt{ab}$, equality at $a=b$. Useful for "minimum of $a+b$ given $ab = k$" or vice versa. Cauchy-Schwarz: $(a_1^2+a_2^2)(b_1^2+b_2^2) \ge (a_1 b_1 + a_2 b_2)^2$.

Telescoping sums

If the problem is a sum with $\frac{1}{k(k+1)}$ style terms, partial-fraction it: $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$. Almost every "find the sum to 99 terms" problem telescopes.

Geometry · GEO

Angle chasing

Sum of angles in a triangle is 180°. Exterior angle = sum of two remote interior. Cyclic quadrilateral: opposite angles sum to 180°. If you see a circle, look for inscribed angles — half the central angle subtending the same arc.

Similar triangles

If two angles match, the triangles are similar; ratios of all sides are equal. The biggest source of Q3-Q5 geometry shortcuts is spotting two similar triangles sharing an angle or a side.

Area tools

Heron's: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is the semi-perimeter. For a triangle with two sides $a, b$ and included angle $C$: $A = \frac{1}{2} ab \sin C$. Coordinates: shoelace formula for any polygon.

Power of a point

For a point $P$ and a circle, the product of signed lengths along any line through $P$ is constant. Great for chord/secant problems. Equivalent to similar triangles in disguise — but faster.

Combinatorics · COMB

Casework vs complement

If the problem asks "how many ways such that X", try the complement: count "ways such that not-X" and subtract from total. Often the complement is one clean count, while the direct version is five messy cases.

Stars and bars

Number of ways to put $n$ identical balls into $k$ distinct boxes = $\binom{n+k-1}{k-1}$. With "at least one per box" constraint: $\binom{n-1}{k-1}$. Memorise both; they cover most COMB Q3-Q4.

Pigeonhole

If $n+1$ pigeons are in $n$ holes, some hole has at least 2. Generalised: with $kn+1$ pigeons in $n$ holes, some hole has $\ge k+1$. The trigger phrase is "prove that at least…" or "must there exist…".

Probability shortcut

$P(\text{event}) = \frac{\text{favourable}}{\text{total}}$ — but in olympiad problems, both numerator and denominator are usually $\binom{n}{k}$ expressions. Reduce the fraction before multiplying out — the cancellations are the whole point.

Pre-flight checklist (30 seconds)

Before you press Start, do this once:

  1. Pen and paper ready. The drill is on screen, but your scratch work is on paper. Don't try to do this in your head.
  2. Phone away. Notification at 28s of a 60s window = lost problem.
  3. Calculator off. AIMO is no-calculator. Practising with one builds the wrong reflexes.
  4. Water within reach. 30 minutes of intense focus dehydrates you faster than you think.
  5. One deep breath. Tension on Q1 ripples to Q30. Start calm.

What "good" looks like at each level

Round-1 scoreVerdictNext step
27-30Elite reflexSpend the rest of Week 21 on Q6-Q10 (Part 3 Hard Half).
25-26Target hitRe-do misses cold tonight; run Round 2 tomorrow morning for stickiness.
20-24Strong but leakyPattern-spot the misses by category. The category with worst accuracy is your week's drill focus.
15-19Foundations wobblyRe-do all misses slowly (no clock). Re-run drill in 48 hours.
< 15Slow downDrop back to Week 1-5 fundamentals. Come back to this drill when you can do AIMO Q1-Q2 untimed at 95%+.

Three worked walkthroughs — what a 60-second solve actually looks like

Before you start, watch what the "60-second decision tree" looks like in practice on three sample problems. These aren't in the drill — they're illustrative.

Walkthrough 1 · NT · easy (Q1-Q2 style)

Problem

Find the smallest positive integer $n$ such that $n^2 + n$ is divisible by 12.

60-second solve

0-5s · Read. Category: NT, divisibility. Recognise: $n^2 + n = n(n+1)$ is a product of consecutive integers.

5-20s · Pick the tool. $12 = 4 \times 3$. Product of two consecutive integers is always even (one of them is even). For divisibility by 4, need one of $n, n+1$ to be divisible by 4. For divisibility by 3, need one of $n, n+1$ to be a multiple of 3.

20-35s · Execute. Try $n=1$: $1 \cdot 2 = 2$, no. $n=2$: $2 \cdot 3 = 6$, no. $n=3$: $3 \cdot 4 = 12$, yes! Divisible by 12.

35-40s · Verify. $12 / 12 = 1$. Check. Answer: $n = 3$.

Total: 40s. Margin: 20s. Lock and move on. Don't waste the buffer.

Walkthrough 2 · GEO · medium (Q3-Q4 style)

Problem

In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Find the area.

60-second solve

0-5s · Read. Category: GEO, area of triangle with three sides given. Trigger: Heron's formula.

5-15s · Pick the tool. Semi-perimeter $s = (13 + 14 + 15) / 2 = 21$.

15-40s · Execute. $A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}$. Group cleverly: $21 \cdot 6 = 126$ and $8 \cdot 7 = 56$. So $A = \sqrt{126 \cdot 56}$. Better: $21 \cdot 8 \cdot 7 \cdot 6 = (21 \cdot 7) \cdot (8 \cdot 6) = 147 \cdot 48$. Better still: $\sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{(3 \cdot 7) \cdot (2^3) \cdot 7 \cdot (2 \cdot 3)} = \sqrt{2^4 \cdot 3^2 \cdot 7^2} = 4 \cdot 3 \cdot 7 = 84$.

40-50s · Verify. 13-14-15 is a famous triangle. Area 84. Check.

Total: 50s. Margin: 10s. Lock. The "verify" step was 5 seconds well-spent — Heron's is arithmetic-heavy.

Walkthrough 3 · ALG · harder (Q5 style)

Problem

Let $x, y$ be positive real numbers with $x + y = 10$ and $xy = 21$. Find $x^3 + y^3$.

60-second solve

0-5s · Read. Category: ALG, symmetric in $x$ and $y$. Trigger: Newton's identities / symmetric-sum identity for $x^3 + y^3$.

5-15s · Pick the tool. Identity: $x^3 + y^3 = (x+y)^3 - 3xy(x+y)$.

15-35s · Execute. $(x+y)^3 = 10^3 = 1000$. $3xy(x+y) = 3 \cdot 21 \cdot 10 = 630$. So $x^3 + y^3 = 1000 - 630 = 370$.

35-45s · Verify. Solve quadratic: $t^2 - 10t + 21 = 0 \Rightarrow t = 3, 7$. So $\{x, y\} = \{3, 7\}$. Then $3^3 + 7^3 = 27 + 343 = 370$. Check.

Total: 45s. Margin: 15s. The verification by solving the quadratic was insurance — on a Q5 you can afford 10 seconds to double-check.

The single most important habit

If you only take one thing from this drill brief, take this:

Write the answer in the box at 40 seconds, even if you're not sure. Use the last 20 seconds to either improve it or move on. Never enter the final 5 seconds with an empty box.

This one rule alone typically lifts a student's score by 2-4 problems out of 30, because timeouts with an empty box are zero, whereas timeouts with even a roughly-guessed answer have a non-zero chance of being right.

FAQ

What if I finish in 20 seconds — should I move on early?

Yes. Press Save & Next. The timer is a maximum, not a target. Banking 40 seconds gives you a buffer for a harder problem later. Many top scorers finish the easy 12 in under 4 minutes total, leaving ~26 minutes for the medium and hard problems.

What if I'm completely stuck at 30 seconds?

Skip. Don't burn the full minute. Skipping costs you potential marks on that one problem, but burning the minute costs you marks on a problem you might've solved instead.

What if the integer answer is huge — like 12345?

It can be. AIMO Q1-Q5 answers are typically 0-999, but some go higher. The input field accepts up to 6 digits. Don't worry about the size; type what you get.

What if I make an arithmetic mistake?

You'll see it in the re-do list. Tag it as "(c) arithmetic slip" during your 20-minute review. If you have more than 3 of these in one drill, your scratch-work discipline needs work — write bigger, write neater, verify products before moving on.

How often should I run this drill?

During final-sprint weeks (Weeks 20-22): 3-4 times per week, ideally Mon/Wed/Fri/Sun. Track the trend, not any single score. Look for the Round-1 baseline rising week-on-week — that's the only metric that matters.

Mental math micro-drills (do these once before pressing Start)

The single largest source of avoidable misses on Q1-Q5 is slow mental arithmetic. Run through these 30 prompts quickly — they'll warm up the same circuits the drill will hammer. Aim for under 2 seconds each. If any one takes more than 5 seconds, circle it for slow practice this week.

Multiplication reflexes

PromptAnswerWhy it matters
$13 \times 17$221$(15-2)(15+2) = 225-4$ trick — every AIMO student should own this.
$14 \times 16$224Same trick: $(15-1)(15+1) = 225 - 1$.
$24 \times 25$600$25 \cdot n = n \cdot 100 / 4$. So $24 \cdot 25 = 2400/4 = 600$.
$19 \times 21$399$(20-1)(20+1) = 400 - 1$.
$27 \times 33$891$(30-3)(30+3) = 900 - 9$.
$11 \times 13$143Memorise this — it shows up in $11 \cdot 13 = 143$ factorisations constantly.
$12 \times 17$204$12 \cdot 17 = 12 \cdot 16 + 12 = 192 + 12 = 204$.
$15 \times 18$270$15 \cdot 18 = 15 \cdot 20 - 15 \cdot 2 = 300 - 30 = 270$.

Squares and cubes to memorise

SquaresCubes
$11^2 = 121$$2^3 = 8$, $3^3 = 27$
$12^2 = 144$$4^3 = 64$, $5^3 = 125$
$13^2 = 169$$6^3 = 216$, $7^3 = 343$
$14^2 = 196$$8^3 = 512$, $9^3 = 729$
$15^2 = 225$$10^3 = 1000$, $11^3 = 1331$
$16^2 = 256$$12^3 = 1728$, $13^3 = 2197$
$17^2 = 289$$2^4 = 16$, $2^5 = 32$
$18^2 = 324$$2^{10} = 1024$, $2^{16} = 65536$
$19^2 = 361$$3^4 = 81$, $3^5 = 243$
$20^2 = 400$, $25^2 = 625$$5^4 = 625$, $6^4 = 1296$

Primes to know cold (under 100)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

If a number under 100 isn't in this list, it's composite. Test for primality of $n$ up to 200 by checking divisibility against primes up to $\sqrt{n}$ — for $n < 200$, that means primes up to 14, so just 2, 3, 5, 7, 11, 13. Six divisions max.

Famous Pythagorean triples

If a problem gives two sides of a right triangle, the third is often in this list. Recognising a triple saves 20 seconds of computation.

TripleMultiples that appear
3-4-56-8-10, 9-12-15, 12-16-20, 15-20-25
5-12-1310-24-26, 15-36-39
8-15-1716-30-34
7-24-2514-48-50
20-21-29
9-40-41

Common AIMO problem patterns by question slot

Across 30+ years of AIMO papers, each question slot has a characteristic flavour. Pattern-spotting these saves you 5-10 seconds of "what kind of problem is this?" thinking at the top of each minute.

Q1 (2 marks) — usually a single-step computation

Q1 is the warm-up. It tests one concept, executed cleanly. Examples of typical Q1 templates:

Speed target: under 30 seconds.

Q2 (2 marks) — single concept with one twist

Q2 is Q1 with one extra layer of substitution or reframing. Examples:

Speed target: under 40 seconds.

Q3 (3 marks) — two concepts combined

Q3 is where the drill starts biting. You need to spot two things: the right setup, and the right execution path. Typical templates:

Speed target: under 50 seconds.

Q4 (3-4 marks) — multi-step execution

Q4 requires three or more steps from setup to answer. You should still be at the recognise → execute level, but the execution itself is longer. Examples:

Speed target: full 60 seconds is reasonable. Don't panic if you use the whole minute.

Q5 (4 marks) — bridge problem

Q5 is the boundary between the "easy half" and the "hard half" of AIMO. Sometimes it's a clean Q3-style problem with one harder twist; sometimes it's almost Q6-difficulty. The 60-second window is tight. If at 30 seconds you're nowhere, skip and come back if time permits in a real exam. In this drill, you can't come back — so make your best guess at 55s.

What the timer ring tells you

The yellow countdown ring at the top-left of each problem isn't decorative — use it as a decision signal.

Ring at 75%+ (60-45s left)

You're in the comfort zone. Read carefully, plan deliberately. If the problem feels easy, don't rush — careful is faster than fast-and-wrong.

Ring at 50% (30s left)

Decision point. If you have a clear path, execute. If you don't have a clear path, you have 25 seconds — pick a tool and commit, or skip now.

Ring at 25% (15s left)

Lock something in the box. Even a rough estimate beats an empty box. Your job now is "least bad answer", not "perfect answer".

Ring turns red (10s left)

Press Save now if there's anything in the box. The auto-advance will trigger your locked value as a timeout — fine, but voluntary save is cleaner.

Six sample problems with timed solutions

These six examples cover the range of difficulty and category you'll see in the drill. Each is presented with a full 60-second walkthrough so you can calibrate your own pace before pressing Start.

Sample 1 · Q1 · NT

Problem

What is the units digit of $2024^{2024}$?

Solution (target 30s)

Units digit of $2024^k$ depends only on units digit of $4^k$. The cycle of units digits of $4^k$ is $4, 6, 4, 6, \ldots$ — period 2. Since 2024 is even, units digit is $6$.

Answer: 6.

Sample 2 · Q2 · ALG

Problem

If $x + \frac{1}{x} = 5$, find $x^2 + \frac{1}{x^2}$.

Solution (target 30s)

Square both sides: $\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 25$. So $x^2 + \frac{1}{x^2} = 25 - 2 = 23$.

Answer: 23.

Sample 3 · Q3 · COMB

Problem

How many three-digit positive integers have all distinct digits?

Solution (target 45s)

First digit: 9 choices (1-9). Second digit: 9 choices (0-9 minus first). Third digit: 8 choices. Total: $9 \cdot 9 \cdot 8 = 648$.

Answer: 648.

Sample 4 · Q4 · GEO

Problem

A circle is inscribed in a right triangle with legs 6 and 8. Find the radius of the inscribed circle.

Solution (target 50s)

Hypotenuse: $\sqrt{36+64} = 10$ (Pythagorean triple 6-8-10). For a right triangle with legs $a, b$ and hypotenuse $c$, inradius $r = \frac{a+b-c}{2} = \frac{6+8-10}{2} = 2$.

Verification by area: $A = \frac{1}{2} \cdot 6 \cdot 8 = 24$. Semi-perimeter $s = (6+8+10)/2 = 12$. $r = A/s = 24/12 = 2$. Check.

Answer: 2.

Sample 5 · Q4 · NT

Problem

How many positive integer divisors does $720$ have?

Solution (target 45s)

Factorise: $720 = 2^4 \cdot 3^2 \cdot 5$. Number of divisors: $(4+1)(2+1)(1+1) = 5 \cdot 3 \cdot 2 = 30$.

Answer: 30.

Sample 6 · Q5 · ALG

Problem

Find the sum of all positive integer values of $n$ such that $\frac{n+12}{n-3}$ is a positive integer.

Solution (target 60s)

Rewrite: $\frac{n+12}{n-3} = \frac{(n-3) + 15}{n-3} = 1 + \frac{15}{n-3}$. For the expression to be a positive integer, $n-3$ must be a positive divisor of 15.

Positive divisors of 15: $1, 3, 5, 15$. So $n - 3 \in \{1, 3, 5, 15\}$, giving $n \in \{4, 6, 8, 18\}$.

Sum: $4 + 6 + 8 + 18 = 36$.

Answer: 36.

One last thing — your physical setup

This isn't a software drill. It's a habits drill that uses software as the metronome. Before you press Start, set up your physical environment like an exam:

If even one of these is off, fix it before starting. The 30 minutes you'll spend on the drill is too valuable to waste on a setup you wouldn't accept on exam day.

After the drill — the 20-minute review

Don't close this page when results show. Spend exactly 20 minutes on this protocol:

  1. 5 min · Misses by category. Look at the per-category table. Which category had the worst accuracy? Write it at the top of your scratch paper.
  2. 10 min · Cold re-do of misses. Re-do every missed problem on paper, without a clock. Don't peek at the answer until you've genuinely tried for 2-3 minutes per problem.
  3. 3 min · Tag the cause. For each miss, tag it: (a) didn't know the method, (b) knew method but slow, (c) arithmetic slip, (d) misread the problem. The tag tells you what to fix.
  4. 2 min · Errorbook check. Your misses are auto-saved to localStorage under aimoErrors_W21_SpeedDrill with their tags ready for the Week 21 Part 5 errorbook review.
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Round 1 Complete

Speed Drill Results

0/30
0% correct · 0s total time

By category

CategoryScoreAccuracyAvg time
Avg sec / correct
Timeouts
Skipped

Re-do list — problems you missed

Round 2 reshuffles the same 30 problems for an improvement attempt.