Contest Strategies · Chapter 45

Additional Techniques and Strategies

Win points without new math: eliminate choices, spot patterns in small cases, dodge silly mistakes, plan your 75 minutes and know when a guess beats a blank.

What's this chapter about?

45.1 Meta-solving Techniques
45.2 Silly Mistakes
45.3 Other Strategies to Maximize Your Score

You'll need

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45.10%

Meta-solving Techniques

In this section you will be able to find answers without fully solving: cross out choices with quick checks, spot patterns in small cases, and use the freedom a problem gives you.

Five answer choices are five clues. If two of them are odd and the answer must be even, you just halved your guess, and you haven’t solved anything yet.

You don’t always need the exact answer. A quick check, like parity, a units digit, a remainder or a rough size, can cross out wrong choices in seconds. Every problem in the trainer is a real AMC 10/12 problem from this book.

Look:a quick check shrinks the list. Then you solve, or guess from what’s left. It never replaces solving when you can.

Exploreplay first, formulas later

Problem 4 of 17

AMC 10/12 · Problem 6.4.3

Suppose that 13 cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the 13 cards be picked up in exactly two passes?

\boxed{7}\ \boxed{11}\ \boxed{8}\ \boxed{6}\ \boxed{4}\ \boxed{5}\ \boxed{9}\ \boxed{12}\ \boxed{1}\ \boxed{13}\ \boxed{10}\ \boxed{2}\ \boxed{3}

Step 1. Pick a reason chip, then tap a choice to cross it out.

Step 2. Solve, then lock in a survivor.

Right answer survived 0 of 0Final answers right: 0 · Good reasons: 0 of 0

Try this: Eliminate with parity only. Pick the Parity chip, peek at the clue, and cross out what it kills. How many choices are left?
Try this: Go to Problem 8 (digit sums). Which single chip leaves just one choice?
Try this: Switch to Engineering induction. Compute n = 1, 2, 3, guess the big case before checking, then add one more small case.
Try this: Switch to Use freedom. Assume an equilateral triangle, then test another one. Why does Problem 3 break?

Name it

Definition 45.1.1

Meta-solving is finding the answer to a problem without actually solving it.

Remark 45.1.2

These techniques may not work for all problems. They are especially useful when the problem provides answer choices.

Remark 45.1.3 · Meta-Solving Warning

Don't get too carried away with these techniques to the point where you don't even try to solve the problem legitimately.

Concept 45.1.4 · Engineering Induction

Engineering Induction is the process of trying and finding the value for small cases and assuming it's true for larger ones.

Steps for Engineering Induction problems:

Try small cases.
Look for a pattern amongst those small cases (there may not always be one).
Assume the pattern can continue for larger cases and find the answer.

Concept 45.1.6 · Looking for unique properties of numbers

Rather than computing the exact answer, you can compute unique properties of your answer. Then eliminate all answer choices that don't have the property, and you'll be left with 1 answer choice (or possibly more, in which case you can just guess from the remaining ones).

Some unique properties you can look for in your answer choices and try to compute are:

Units digit
Last 2 digits
Parity (even, odd)
Perfect square or not
Prime/composite
Modulus (remainders when divided by 3, 4, 5, etc.)
Denominators of common fractions (or what they must divide)
Multiples/factors of numbers
etc.

Concept 45.1.8

Look for the answer choices that are the "odd one out", different from all the others.

Look for outliers (primes, large/small numbers, odd/even numbers, powers of 2, etc.).

Concept 45.1.9 · Trying all the Option choices

In some problems, you can:

Try all the answer choices in the conditions of the problem.
Look at the conditions in the problem and see which of the answer choices could work.
etc.

After doing so, you will either have a better guess or the exact answer.

Concept 45.1.10 · Elimination of Option Choices

Tying closely to the previous technique, you can also usually eliminate answer choices based on:

Unique properties of numbers (see above)
How large or small the number must be

Concept 45.1.12 · Estimation of Answer

Often in problems (especially geometry) you can easily find an approximate answer and see which answer choice most closely matches what you got.

In geometry, a common strategy is to mark out areas approximately equal to areas you know.

Concept 45.1.13 · Using Freedom in Problems

Assuming facts when you have freedom in the problem statement can be very useful.

Essentially, as long as the problem is not telling you "this fact is not true" (so whatever assumption you make still satisfies the problem's conditions), you can assume the fact is true to simplify your problem and make it really easy to solve.

For example, if you are asked to find some universal ratio in a triangle and you aren't specifically told that the triangle isn't equilateral, you can just assume the triangle is equilateral and solve the rest of the problem from there.

side a2

Drag the purple dot or use the stepper to resize. The ratios never change.

Area =

\sqrt{3}

≈ 1.73

h = \tfrac{\sqrt3}{2}a,\quad \text{Area} = \tfrac{\sqrt3}{4}a^2

Equilateral: $\text{Area} = \tfrac{\sqrt3}{4}a^2$
45-45-90: $\text{Area} = \tfrac12 a^2$
30-60-90: $\text{Area} = \tfrac{\sqrt3}{2}a^2$

Checkpoint for section 45.1

Get 2 of 3 right to master this section. One try each, no hints.

A counting problem's answer is a multiple of $6$ (the count has a factor $3!$ ). Which choice can it be?

Engineering induction: small cases give $f(1) = 1$ , $f(2) = 4$ , $f(3) = 12$ , $f(4) = 32$ . Assuming the pattern $f(n) = n \cdot 2^{n-1}$ keeps going, what is $f(5)$ ?

A problem asks for a ratio in triangle $ABC$ with $AB = 5$ and $AC = 9$ . Can you assume the triangle is equilateral to make it easier?

45.20%

Silly Mistakes

In this section you will be able to catch the six kinds of silly mistakes before they cost you points.

Most lost points are not hard problems. They are easy problems answered wrong: $87 \cdot 93$ worked out as 8,191, or a 0 counted when the problem said “positive”.

Silly mistakes come in six families: arithmetic, mathematical, reading, missing the final step, false assumptions, and casework. Each family has its own fix.

Look:the trap answer is often one of the choices. Misreading “inclusive” gives exactly the number the test-writers expected you to get wrong.

Exploreplay first, formulas later

Try this: Switch to "Which mistake?" and sort the wrong answers into families. Which one surprised you?
Try this: Reading trainer: find the statement with "non-negative". What does it add that "positive" doesn't?
Try this: Open each family in "Before you submit" and tick the checks you'll really do. Which family bites you most?

Name it

Concept 45.2.1 · Avoiding Arithmetic Errors

A good way to avoid arithmetic errors is to:

Do your computation 2 ways (e.g. if you have to do $87 \cdot 93$ , multiply them once with 87 on top and once with 93 on top).
Be more organized, and write more steps.

Concept 45.2.2 · Avoiding Mathematical Errors

An easy way to avoid mathematical errors is to:

Do your work neatly!
Make a box on scratch paper for each problem.
Don't skip steps.
Check your work (following the tip above makes it easier).
Do your steps methodically.
Try to substitute your answer back into the problem (if you can).
Try an alternate solution to confirm your answer.
Estimate what the answer has to be, and see if your answer is close to your estimate.

Concept 45.2.3 · Avoid Reading Errors

Reading the problem wrong is one of the most common mistakes. Oftentimes you might forget about important key words like:

inclusive, except
even, odd
prime, composite
integer, natural, real, complex

Some strategies to avoid them are:

After solving the problem, reread the question part of the problem to make sure you are answering what it asks for!
Underline key words while reading. (If you take the test online, note the important words on your scratch paper instead.)

Concept 45.2.4 · Avoid Missing a Final Step Errors

Sometimes you might be so caught up in moving forward in the test that you forget an important step at the end.

For example, you might think "I'll multiply whatever answer I get by 5", and then you find that answer but forget to multiply by 5. A way to avoid this is:

Write "Remember …" big and bold on your scratch paper or the question paper.

Concept 45.2.6 · Avoid Making False Assumptions

Oftentimes you might just think something is true and assume it for the rest of the problem, when really it was false. Proving all your assumptions can be too time-consuming. However, we recommend at least seeing some sort of reasoning for why your assumption should be correct (unless, of course, you are using one of the meta-solving techniques).

Concept 45.2.7 · Avoid Casework Errors

Many times, in casework problems you might:

Undercount possibilities
Overcount possibilities
Miss edge/extreme cases

Some strategies to avoid them are:

Be methodical in your casework.
Make sure all your cases work:
- An easy way to do this is to try a few examples in each case to see if they actually work.
- Especially, make sure to check if extremes work.
- Make sure your cases are disjoint, so you're not overcounting anything common to two cases.
- Make sure your cases cover all the possibilities the problem asks for.
- Solve the problem in multiple ways (for example, by both casework and complementary counting).

Checkpoint for section 45.2

Get 2 of 3 right to master this section. One try each, no hints.

45.30%

Other Strategies to Maximize Your Score

In this section you will be able to plan your 75 minutes, know when a guess beats a blank, and walk in relaxed.

75 minutes, 25 problems. A right answer is worth 6, a blank 1.5, a wrong answer 0. Guessing is a math problem too.

That’s 3 minutes per question on average, but they aren’t equal. Easy ones should go faster, so the hard ones get more. When one question eats your time, star it and move on.

Look:a blind guess among 5 choices is worth

\frac{6}{5} = 1.2

points on average, less than the

1.5

for a blank. Cross out 2 choices first and a guess is worth

\frac{6}{3} = 2

Exploreplay first, formulas later

Try this: Drag the handles so Questions 1–10 go fast and you keep a few minutes to review. What does the advice say?
Try this: Take the Mock test. When Question 7 looks long and bashy, try ★ Star it and move on. Then compare the robots in Replay.
Try this: In the score planner, find two different splits that reach 111.
Try this: Open the guess calculator and cross out 2 choices. Is a guess now worth more than 1.5?

Name it

Concept 45.3.1 · Plan Your Time

Make sure to plan your time.

Questions 1–10 are generally easy, 11–20 are medium, 21–25 are hard.
Sometimes one of the early questions can be hard or bashy, or a later question can be easy.
Don't get stuck on a question. Move on and come back to it later.
Star any question you are unsure about but feel you can solve, or any question you solved but aren't confident about.
Budget your time properly.
Leave enough time for the last 10 problems.
Leave some time to review starred problems and check your work.

Concept 45.3.2 · Guessing Strategies

If you want to guess, keep these things in mind:

You get 1.5 points for leaving a question blank, so if you don't know how to solve the problem, just leave it blank.
Try using the meta-solving strategies above.
Only if you can narrow down the choices to 2–3 options should you make an educated guess.

Right17

Wrong0

Blank8

Score: 114 / 150Reaches the target, with 9 points to spare.

AIME target (example)USAJMO/USAMO target (example)

The defaults are the book's numbers (about 105, aim for 110+; 120 for USAJMO/USAMO). Cutoffs change every year, so check this year's and type them in.

Fewest right answers that reach 105

Right	Wrong	Blank	Score
15	0	10	105
16	1	8	108
16	2	7	106.5
16	3	6	105
17	4	4	108
17	5	3	106.5
17	6	2	105

Concept 45.3.3 · Test Day Strategies

Here are some test day strategies:

Don't try to cram or study on the day before the test.
- Just review a few formulas or strategies from this book.
Be relaxed.
- Get a good night's sleep.
- Eat a healthy meal.
- Meditate.
- Eat dark chocolate.
- Listen to music.
- Or whatever else makes you feel relaxed.

My AMC 10/12 test-day card

The day before

Be relaxed

Guessing

When you're stuck

AMC 10/12 A and B

Last, but most important: don't stress out too much! It's just a math contest, and you'll have many more chances.

Concept 45.3.4 · Problem Solving Strategies

Try to remember these problem solving strategies:

When you are stuck, try to use the information in the problem you haven't used yet.
When solving a problem, think of what technique would likely be at play.
Don't get too stuck with one approach to a problem. Move on, and come back to the problem with a fresh perspective.
In problems that seem complex, try small cases to look for a pattern that can show you how to approach the problem and what patterns may exist (this is similar to engineering induction).

Concept 45.3.5 · AIME Qualification

To qualify for the AIME, based on recent cutoff scores (when the book was written), you will need a score of at least 105. However, you should aim for 110+ to be safe. For this, you'll need a 17-0-8 split (right-wrong-skipped), an 18-5-2 split, a 19-6-0 split, or something better.

These are the book's numbers. Cutoffs change every year, so check this year's cutoff and type it into the score planner.

Right18

Wrong5

Blank2

Score: 111 / 150Reaches the target, with 6 points to spare.

6 \times 18 + 1.5 \times 2 + 0 \times 5 = 111

AIME target (example)USAJMO/USAMO target (example)

The defaults are the book's numbers (about 105, aim for 110+; 120 for USAJMO/USAMO). Cutoffs change every year, so check this year's and type them in.

Fewest right answers that reach 105

Right	Wrong	Blank	Score
15	0	10	105
16	1	8	108
16	2	7	106.5
16	3	6	105
17	4	4	108
17	5	3	106.5
17	6	2	105

Concept 45.3.6 · USAJMO/USAMO Qualification

To qualify for the USAJMO/USAMO, you'll probably need a score of at least 120 to have a good chance (of course, this also depends heavily on your AIME score).

The book's number: check this year's cutoffs.

Right19

Wrong1

Blank5

Score: 121.5 / 150Reaches the target, with 1.5 points to spare.

6 \times 19 + 1.5 \times 5 + 0 \times 1 = 121.5

AIME target (example)USAJMO/USAMO target (example)

The defaults are the book's numbers (about 105, aim for 110+; 120 for USAJMO/USAMO). Cutoffs change every year, so check this year's and type them in.

Fewest right answers that reach 120

Right	Wrong	Blank	Score
19	0	6	123
19	1	5	121.5
19	2	4	120
20	3	2	123
20	4	1	121.5
20	5	0	120

Remark 45.3.7

Another strategy for the AMC 10/12 B: if you are expecting an AIME-qualifying score on the AMC 10/12 A, try to solve more problems and take a "higher risk, higher reward" approach on the B. If you didn't do well on the A, take the safe path.

Remark 45.3.8

Last, but most important: don't stress out too much about how you will do! It's just a math contest, and you'll probably have many more opportunities in the future.

Checkpoint for section 45.3

Get 2 of 3 right to master this section. One try each, no hints.