Number puzzles: olympiad question 8

Worked solution

Have a go first, then read the solution step by step.

1. 1.b numbers from a = b lots of a + (0 + 1 + … + (b − 1)), so a ※ b = a × b + b × (b − 1) ÷ 2; likewise b ※ a = a × b + a × (a − 1) ÷ 2.
2. 2.Subtract and a × b cancels: 390 − 245 = [b × (b − 1) − a × (a − 1)] ÷ 2 = (b − a) × (a + b − 1) ÷ 2, so (b − a) × (a + b − 1) = 290.
3. 3.b − a and a + b differ by 2a, so b − a and a + b − 1 are one odd, one even, the first smaller. 290 splits that way as 1 × 290, 2 × 145, 5 × 58, 10 × 29.
4. 4.Try each: b − a = 1, a + b = 291 → a = 145, b = 146, a ※ b = 31755, no; b − a = 2, a + b = 146 → a = 72, b = 74, a ※ b = 8029, no; b − a = 5, a + b = 59 → a = 27, b = 32, a ※ b = 1360, no; b − a = 10, a + b = 30 → a = 10, b = 20, a ※ b = 390 ✓.
5. 5.So a = 10 and b = 20.
6. 6.Check: 10 ※ 20 = 10 × 20 + 20 × 19 ÷ 2 = 390 ✓; 20 ※ 10 = 10 × 20 + 10 × 9 ÷ 2 = 245 ✓
7. ⚠️Common mistake: assuming a ※ b and b ※ a are equal (※ can't be swapped); or missing a split of 290, stopping at the first, without checking a ※ b = 390.

Answer: 20