Additional Techniques and Strategies

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

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:

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

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.

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.).

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.

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

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.

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.

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.

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.

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.)

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.

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).

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).

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.

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.

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.

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).

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.

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.