W21 Part 1 β€” Atomic Skill Self-Check

Final Sprint diagnostic across ~80 atomic skills from Weeks 1–20. Rate honestly. Find your weakest 5. Build a personalised drill plan for the next 14 days.

Final Sprint Β· Week 21 Diagnostic Β· No Calculator ~15 min self-rating Auto-saves to browser

How to use this dashboard

Walk through each skill. Click stars to rate yourself honestly. Be brutal β€” this is for you, not a grade.

⭐ = Weak β€” I get stuck, blank, or guess. Re-study needed.
⭐⭐ = Okay β€” I know the idea but slow or error-prone. Drill needed.
⭐⭐⭐ = Strong β€” I can solve in under 90 s with no notes. Maintenance only.
After all 80+ skills rated β†’ scroll to Weakest 5 Skills β†’ that is your Week 21–22 drill priority.
Skills rated: 0 / 0
⭐ 0
⭐⭐ 0
⭐⭐⭐ 0

Why this matters β€” the Final Sprint logic

You've covered ~80 atomic skills across Weeks 1–20. The AIMO paper draws unevenly from this pool: Q1–Q5 (~40% of marks) test the easy half, Q6–Q8 (~30%) test medium synthesis, Q9–Q10 (~30%) test the proof-heavy advanced layer. You don't need all 80 strong. You need zero ⭐ Weak skills in the easy half, and at least 4 of 5 Strong in your top medium areas.

⭐ Weak (1-star)
If still ⭐ on competition day β†’ you will lose 4-7 marks where this skill appears. Re-study before drill.
⭐⭐ Okay (2-star)
You know it but you're slow. Drill 5-10 reps under timer. Speed gain = exam confidence.
⭐⭐⭐ Strong (3-star)
Maintenance only. Touch it once a week in mixed-set drills so you don't lose it.

⚑ 5-Second Checkpoint Tests

Quick gut-check per skill cluster. Read each prompt β€” if you can't answer in 5 seconds, that cluster needs work. These are NOT the rating β€” they're a fast triage before you sit down for the full self-check.

NT I-II-III Β· Number Theory
  1. What is 7³⁷ mod 10?
  2. How many divisors does 720 have?
  3. Last 2 digits of 3¹⁰⁰?
  4. Find x: 7x ≑ 3 (mod 11)
  5. What is Ο†(60)?
Answers: 7 Β· 30 Β· 01 Β· x=2 Β· 16
Algebra I-IV
  1. Factor xy + 3x βˆ’ 2y βˆ’ 6
  2. If x+y=5 and xy=6, find xΒ²+yΒ²
  3. Solve xΒ² βˆ’ 5x + 4 < 0
  4. Simplify √(7+4√3)
  5. If f(x+y)=f(x)+f(y), what is f(0)?
Answers: (xβˆ’2)(y+3) Β· 13 Β· 1<x<4 Β· 2+√3 Β· 0
Geometry I-VI Β· 3D
  1. Triangle 5-12-13: area?
  2. Heron 13-14-15: area?
  3. Stewart's: cevian length when sides 4, 5, 6 split as 2:3
  4. Power of P inside circle r=5, OP=3?
  5. Cube edge 6: space diagonal?
Answers: 30 · 84 · ~3.4 · 16 · 6√3
Counting & Probability
  1. How many ways: arrange MISSISSIPPI letters?
  2. C(10, 3) = ?
  3. P(roll sum 7 on 2 dice)?
  4. |AβˆͺBβˆͺC| if each = 10, pairwise=3, triple=1?
  5. 3rd Catalan number?
Answers: 34650 Β· 120 Β· 6/36 Β· 22 Β· C₃=5
Proof I-II
  1. Induction: what is the base step for 1+2+…+n = n(n+1)/2?
  2. What's the parity invariant when swapping any 2 adjacent items?
  3. Extremal: in pigeonhole 13β†’12, at least how many in one hole?
  4. Contradiction setup for "√2 is irrational"?
  5. Strong induction: when do you need it vs weak?
Answers: n=1, LHS=RHS=1 · parity flips · 2 · assume √2=p/q reduced · when P(k+1) needs P(k) AND earlier
FE & Sequences
  1. If f(xy) = f(x) + f(y), what is f(1)?
  2. Geometric sum 1+r+rΒ²+…+rⁿ when r=1/2, nβ†’βˆž?
  3. Telescope: Σ 1/[n(n+1)] from 1 to ∞?
  4. Cauchy FE solutions if continuous?
  5. aβ‚™ = 2aₙ₋₁ + 3, aβ‚€=1: closed form?
Answers: 0 Β· 2 Β· 1 Β· f(x)=cx Β· aβ‚™ = 4Β·2ⁿ βˆ’ 3

πŸ“š Quick Reference β€” Common AIMO Stumble Points

Examples of where students most often lose marks per skill cluster. If any of these look unfamiliar, mark the corresponding skill as ⭐ Weak immediately.

NT

The "last 3 digits of 7^2024" trap

Step 1: split mod 1000 = mod 8 Γ— mod 125 (CRT). Step 2: 7^? mod 125 needs Ο†(125) = 100. Step 3: 7^? mod 8 needs the cycle 7, 1, 7, 1… Step 4: stitch back. If you blanked on any step β†’ T10-A2 + T10-A5 weak.

Algebra

SFFT instant-recognition test

Given xy + 3x + 2y = 47, what is the first move? Answer: add 6 both sides β†’ (x+2)(y+3) = 53. If you didn't see this in 5 seconds β†’ T4-A1 weak. Drill 10 SFFT problems in 20 min.

Geometry

Stewart's vs Mass Points

Cevian asked: length β†’ Stewart's (T9-A1). Ratio of segments β†’ Mass Points (T9-A2). If you reach for Pythagoras + similar triangles every time β†’ both weak, you're working too hard.

Counting

"How many strings of 8 with no 2 consecutive A's"

Reach for recursion: aβ‚™ = aₙ₋₁ + aβ‚™β‚‹β‚‚ (Fibonacci-style). If you started writing cases β†’ T6-A4 weak. Re-study before competition.

Proof

Q9 partial-credit structure

Marks are awarded for: (1) clear setup with variables defined, (2) the key claim stated explicitly, (3) reasoning chain, (4) conclusion line. Skipping any of these = -1 mark each. Even a wrong proof can score 3/7 if structured.

3D Geo

Space diagonal of a box

3D distance = √(l²+w²+h²). The "British Flag" 3D variant: for any P inside box ABCD-EFGH, PA²+PC²+PG²+PE² = PB²+PD²+PH²+PF². T18-A5 is often tested as Q5-Q7.

Your Weakest 5 Skills

Auto-generated from your ratings. These are your Week 21–22 drill priority. Re-study the linked Part, then re-rate to mark progress.

Rate at least 5 skills to see your priority list. Skills rated ⭐ (weak) appear first, then ⭐⭐ (okay).

πŸ—ΊοΈ Past Paper Skill Map (2021–2024)

Which atomic skills were tested in each AIMO question. Use this to prioritise: skills tested 3+ times across 4 years are high-frequency β€” they almost certainly appear again. Skills tested 0-1 times are low-frequency β€” don't over-invest.

Q# 2021 (skill) 2022 (skill) 2023 (skill) 2024 (skill) Pattern
Q1 T1-A2 digit T2-A1 linear T1-A1 base T2-A4 rate Easy NT or Alg setup
Q2 T5-A1 area T6-A3 comb T4-A1 SFFT T5-A3 Heron Basic geo or count
Q3 T4-A1 SFFT T3-A4 GCD T6-A4 recur T1-A4 divisors Mid NT/Alg trick
Q4 T7-A1 PIE T8-A4 decomp T9-A1 Stewart T7-A2 prob Counting or Geo
Q5 T3-A3 last digit T19-A2 sequence T18-A1 3D T10-A1 cycles NT/Alg with hook
Q6 T13-A4 PoP T14-A1 PIE+ T20-A2 poly T9-A5 6-thms Advanced single-skill
Q7 T14-A2 Catalan T13-A1 60Β° T20-A3 AM-GM T14-A3 graph Synthesis 2-skill
Q8 T12-A4 polygon T20-A1 FE T14-A4 GF T17-A2 power Hardest non-proof
Q9 T11-A1 ind T11-A3 geo proof T15-A1 invariant T15-A2 extremal Proof, partial credit
Q10 T15-A4 syn T15-A3 strong ind T15-A4 syn T11-A2 contra Hardest proof
πŸ”₯ High frequency (3-4 of 4 years)
SFFT (T4-A1), PIE (T7-A1, T14-A1), Heron/area (T5-A3), Last-digit cycles (T3-A3), Mass Points / Stewart's (T9), Power of Point (T13-A4), Induction (T11-A1), Invariants (T15-A1).
⚑ Medium frequency (2 of 4 years)
Recursion (T6-A4), Catalan (T14-A2), 60Β° rotation (T13-A1), AM-GM (T20-A3), Generating functions (T14-A4), 3D coords (T18-A3), FE (T20-A1).
πŸŒ™ Low frequency (0-1 of 4 years)
Descartes (T17-A1), Inversion (T17-A3), Apollonius (T17-A5), Well-ordering (T15-A5), 3D British Flag (T18-A5), advanced FE (T20-A1).

🚨 Top 10 Mark-Losing Patterns

Across the W16 mock papers, these are the recurring errors. If any sound familiar, mark the linked atomic skill ⭐ Weak and re-study.

  1. Forgetting to check edge cases in counting problems
    "How many integers from 1 to 100…" β€” students forget either endpoint. Fix: write the count as N βˆ’ M + 1, NOT N βˆ’ M. Linked skill: T6-A1 + T6-A5.
  2. Mod cycle errors: confusing period with exponent
    7⁡⁰ mod 10: cycle is {7, 9, 3, 1} length 4. Need 50 mod 4 = 2, so answer = 9. Common mistake: using 50 directly. Linked skill: T3-A2 + T10-A1.
  3. SFFT with wrong constant added
    xy + ax + by = c. You must add ab, not a+b or aΓ—c. Fix: always factor (x+b)(y+a) and check by expanding. Linked: T4-A1.
  4. Heron's: forgetting s = (a+b+c)/2 (NOT a+b+c)
    s is the semi-perimeter. For 13-14-15, s = 21, not 42. Triangle area = √(21·8·7·6) = 84. Linked: T5-A3.
  5. Power of a Point: wrong sign for "inside vs outside"
    P inside circle: Power = -PAΒ·PB (negative). P outside: Power = PAΒ·PB (positive, secant) or PTΒ² (tangent). Always check P's position first. Linked: T13-A4 + T17-A2.
  6. Inclusion-Exclusion: missing the triple-overlap term
    |AβˆͺBβˆͺC| = |A|+|B|+|C| βˆ’ |A∩B| βˆ’ |A∩C| βˆ’ |B∩C| + |A∩B∩C|. The final + is easy to skip. Linked: T7-A1 + T14-A1.
  7. Stewart's: confusing m, n, d positions
    man + dad = bmb + cnc. m, n are the cevian-split parts; d is the cevian length; b, c are the OTHER two sides. Draw a labelled diagram before plugging in. Linked: T9-A1.
  8. Induction: inductive step assumes the conclusion
    You must use P(k) to derive P(k+1), not assert P(k+1). Markers penalise circular reasoning -3 marks even with correct algebra. Linked: T11-A1 + T15-A3.
  9. Diophantine: missing negative integer solutions
    3x + 5y = 41 has positive solutions (2,7), (7,4), (12,1) β€” but also (-3, 10), etc. If "positive integers" is specified, exclude. Read the problem twice. Linked: T4-A3.
  10. Probability: not reducing the fraction
    AIMO often asks for p+q where p/q is in lowest terms. Computing 18/30 instead of 3/5 β†’ wrong final answer. Always reduce. Linked: T7-A2 + T7-A3.

Suggested 14-Day Sprint Calendar

Default plan β€” adapt based on your weakest 5. If your weak count > 10, extend the re-study phase to 5 days and trim mock papers.

Day 1
Re-study weak #1
20 drill reps
Day 2
Re-study weak #2
20 drill reps
Day 3
Re-study weak #3
+ #1/#2 review
Day 4
AIMO 2021 paper
Untimed
Day 5
AIMO 2022 paper
Untimed
Day 6
Re-study weak #4-5
Drill mixed
Day 7
Drill ⭐⭐ skills
Speed focus
Day 8
AIMO 2023 paper
Timed Q1-Q5
Day 9
AIMO 2024 paper
Timed Q1-Q8
Day 10
Re-rate all skills
Re-study any new ⭐
Day 11
Mock #1
Full 4-hour timed
Day 12
Mock #1 review
Mark + analyse errors
Day 13
Mock #2
Light review only
Day 14
Rest
Early sleep, light Qs
Re-study & drill Past papers Mock + review Rest day

🎯 Exam Day Checklist

Print this. Tape it to your wall. Read it the morning of the exam.

Night Before

  • Pack: 2 pens (blue/black), 2 pencils, ruler, eraser, water bottle, ID
  • Lay out clothes β€” comfortable, layered (exam rooms can be cold or hot)
  • Re-read the W21 Part 4 formula sheet ONCE β€” no new problems
  • Glance at your Weakest 5 list β€” confirm the priority-1 patterns
  • Eat a normal dinner (no novelty food, no caffeine after 6pm)
  • Bed by 10pm β€” sleep matters more than 1 extra hour of revision

Morning of

  • Wake 3+ hours before the exam β€” brain needs time to boot
  • Normal breakfast: protein + carbs (eggs + toast, oats + milk)
  • 10-minute warm-up: solve ONE easy Q1-Q3 style problem from a past paper
  • Leave home with 20-min buffer for the venue
  • Arrive, find seat, drink water, breathe slow 4-count for 2 min

First 5 minutes

  • Read ALL 10 questions before starting β€” 3 min skim
  • Star Q1-Q5 (do first), circle Q6-Q8 (medium), mark Q9-Q10 (proof, do last)
  • Pick the easiest-looking question β€” get 1 right answer fast for confidence
  • Write your name on every page if instructed

During the exam

  • Show working for EVERY answer β€” even Q1-Q8 get partial credit if you show method
  • Q9-Q10 (proof): write SETUP β†’ CLAIM β†’ PROOF β†’ CONCLUSION β€” even partial structure scores
  • If stuck > 5 min, leave and move on β€” circle the Q# to return
  • Check units, signs, and reduction in final answers
  • If you finish early: re-do the questions you were least confident about, not the easy ones
β›” Anti-cramming warning: If on exam morning you discover a weak skill, DO NOT panic-study it. You won't internalise a new technique in 2 hours. Trust the 14-day sprint. Use that morning slot for warm-up only.

πŸ“Š Your Confidence Map

After rating, this map shows your strength distribution across topic clusters. Aim for β‰₯ 70% green in your top 3 frequency clusters.

Number Theory (W1, W3, W10) Β· 15 skills
Algebra (W2, W4, W19, W20) Β· 20 skills
Geometry (W5, W8, W9, W12, W13, W17, W18) Β· 30 skills
Counting & Probability (W6, W7, W14) Β· 15 skills
Proof (W11, W15) Β· 10 skills
Weak Okay Strong Not rated yet

Common Questions β€” Final Sprint Doubts

"I rated 20+ skills as ⭐ Weak. Am I cooked?"
No β€” but you must triage. Don't try to fix all 20. Pick the 5 that appear most in past papers: SFFT (T4-A1), Heron (T5-A3), PIE (T7-A1), Power of Point (T13-A4), and Last-digit cycles (T3-A3). Get these to ⭐⭐ minimum. Leave the rare ones (T15-A5 well-ordering, T17-A1 Descartes) until last β€” they appear maybe 1 in 4 papers.
"How do I tell if I'm really ⭐⭐⭐ vs being optimistic?"
3-star test: pick any random past-paper problem on that skill. Solve it from scratch in under 90 seconds with no notes, no calculator, no false starts. If you needed to think for 30+ sec, you're ⭐⭐. If you needed to look anything up, you're ⭐. Be brutal β€” optimistic ratings cost you marks on competition day.
"What if a skill appears in two weeks (e.g., Circle theorems in W9 and W17)?"
Rate them separately. W9 (T9-A3, T9-A4, T9-A5) tests basic recognition β€” "is this cyclic?", "what's the inscribed angle?". W17 (T17-A1, T17-A2, T17-A3) tests advanced synthesis β€” Descartes, inversion. You can be ⭐⭐⭐ on W9 and ⭐ on W17 β€” that's fine, very different difficulty tiers.
"Should I prioritise proof skills (W11, W15) or computational ones?"
Look at AIMO mark distribution: Q1-Q8 are ~70% of marks and are computational. Q9-Q10 are ~30% of marks and are proof. If you're targeting Distinction (top 25%), focus on the computational skills first. If you're targeting High Distinction (top 5%) or Prize, you need proof skills near ⭐⭐⭐ or you can't crack Q9-Q10. Most students should spend 70% of sprint time on computational.
"How long should one drill session be?"
45 min focused, then break. Pattern: 5 min warm-up (re-read the part, 1-2 worked examples) β†’ 35 min solo solving (5-8 problems on one skill) β†’ 5 min review (look at solutions, mark errors). Beyond 60 min you start grinding without learning. Two short sessions per day > one long session.
"Should I keep practising during exam week?"
Day -7 to -3: yes, light practice, 30 min/day max. Day -2 to -1: STOP NEW LEARNING. Only review your own notes, the formula sheet (W21 Part 4), and re-read your worst-area Re-study links. Day 0 (exam day): a 10-min warm-up problem to wake the brain, then go in fresh. Cramming the night before destroys recall.
"My weak count keeps growing as I rate. Should I stop?"
No β€” keep rating. It feels worse before it feels better. Knowing you're ⭐ on 20 skills is way more useful than thinking you're ⭐⭐ on all 80. After rating, you'll find the top 5 are usually a small cluster (often counting + proof + 3D), which is fixable in 7 days. Honest diagnosis is the whole point of this dashboard.
"Are W16 (Mock Marathon) skills tested here?"
No β€” W16 was pure mock-paper practice with no new atomic skills introduced. Your W16 performance, however, is your best signal for what's really weak. Look back at your W16 mock scores: any topic where you scored <50% should be re-rated ⭐ here regardless of what you "feel" you know.