Blog archive
Sorted newest first — every guide ends with practice problems and full solutions.
If n+1 pigeons sleep in n holes, one hole is crowded — and that single sentence wins a shocking number of olympiad problems. Here is how to spot the pigeons hidden inside a statement.
Last digits, divisibility, and equations that look impossible — almost all of them collapse once you stop tracking numbers and start tracking remainders.
Most olympiad geometry starts with a fact you already know from school. The skill is knowing which fact to apply next — and how to keep the chain short.
When a problem asks whether a configuration can ever be reached, stop simulating and start looking for a quantity that never changes — or always changes the same way.
You do not need to recall every named inequality. You need to recognise the shape "a sum is minimised when the parts are equal" and know when the equality case can be attained.
Combinatorics is not enumeration. It is finding a structure that turns a messy count into a clean product, a symmetry, or two counts of the same thing.
f(x + y) = f(x) + f(y) looks forbidding until you try x = y = 0. Most functional equations give up their secrets to a small set of standard substitutions.
Knowing why 3, 9, 11 and 7 work turns a memorised trick into a tool you can rebuild in ten seconds — including for bases and moduli nobody taught you.
