In many Olympic-level math competitions and tests, students are often not allowed to use calculators. This rule encourages participants to develop strong mental math skills and a deeper understanding of numerical properties.

When faced with large numbers or complex expressions, students learn to simplify problems by breaking them down into smaller, more manageable parts. For example, they might factor numbers into primes, use properties like distributivity, or recognize patterns such as powers and multiples.

This approach not only makes calculations easier but also helps reveal underlying relationships within the problem, enabling more efficient and accurate problem-solving without relying on electronic tools.

Let’s solve 72 + 73 + 66 + 75 + 67 + 69 by expressing each number as 70 plus or minus a small number:

• Write each number as 70 plus or minus a difference:

• 72 = 70 + 2
• 73 = 70 + 3
• 66 = 70 – 4
• 75 = 70 + 5
• 67 = 70 – 3
• 69 = 70 – 1

• Add all the 70s together:
  70 + 70 + 70 + 70 + 70 + 70 = 6 × 70 = 420
• Add all the differences:
  (+2) + (+3) + (-4) + (+5) + (-3) + (-1) = 2 + 3 – 4 + 5 – 3 – 1
• Calculate the sum of differences step by step:
  2 + 3 = 5
  5 – 4 = 1
  1 + 5 = 6
  6 – 3 = 3
  3 – 1 = 2
• Add the total difference to the sum of 70s:
  420 + 2 = 422

Therefore, 72 + 73 + 66 + 75 + 67 + 69 = 422.

ANOTHER OLIMPIC MATH PROBLEM

Calculating this expression without a calculator:

(55555 * 77777) / 12345432

becomes easier when you recognize certain patterns.

First, note that 55555 can be written as 5 * 11111,

and similarly, 77777 as 7 * 11111.

Also, know that 123454321 is the product of 11111 * 11111.

Using these facts, rewrite the expression as

(5 * 11111) * (7 * 11111) / 11111*11111.

This simplification reduces the problem to straightforward multiplication and division, making the calculation much simpler.

Now, simply cancel the identical numbers on both sides of the fraction line, leaving you with 7 * 5, which equals 35.