Challenging algebra/geometry reference source for secondary school students

If you want to bridge that gap between standard curriculum and competitive math, you should look into older, classic problem books rather than modern textbooks. Standard textbooks are designed to build confidence through repetition, which is why they often feel too easy for students aiming for the UKMT. To build the "mathematical muscle" needed for the IMC, you need problems that require non-obvious manipulations rather than just applying a formula. I'd suggest hunting down older editions of books by authors like Hall and Knight. While they might seem dated, their algebraic proofs and problem sets are incredibly rigorous. They don't hold your hand; they force you to understand the underlying logic of how variables interact. This is exactly the kind of "fundamental" training that helps when a student sees a complex polynomial in a competition and needs to know how to break it down instantly. Another great route is looking into "Problem Solving" collections specifically designed for math circles. These aren't organized by topic like a textbook, but by the type of thinking required. This forces students to stop looking for a specific chapter to solve a problem and instead start looking for patterns. It’s a much harder way to learn, but it’s the most effective way to prepare for the jump from GCSE to A-Level. If you're looking for geometry specifically, focus on books that emphasize Euclidean proofs rather than just area calculations. If they can't prove why a property exists, they'll struggle when a competition question twists that property into something unrecognizable. Try to find resources that focus on "construction" as well, as that builds a much stronger spatial intuition for algebra-heavy geometry problems.