Competitions
Compete head-to-head, climb the castle, and train on the full olympiad ladder.
Ascend the castle floor by floor — each level raises the difficulty.
Play nowHead-to-head, one problem at a time, against the clock.
Play nowRace the clock — solve as many as you can on one life. Starts hard.
Play nowCompetition problems
Random practice is untimed and endless, mixing problems from the papers with extra hand-checked practice problems (with diagrams); numbered practice contests are full-length and timed like the real thing. USAMO and Putnam practice tests follow the real sitting format, and every problem ends with a full worked solution.
Middle-school contest: arithmetic, algebra, geometry, counting and number theory. Four original papers, every answer machine-checked and every problem with a worked solution; each paper climbs from a gentle start to three very hard finishers.
Full high-school syllabus, trig, logs and complex numbers included. Sixteen original papers with worked solutions, ramping from a real #1 to AIME-level last problems.
The AMC 10 syllabus with the dial turned up: four original papers that start where a real paper gets hard (about #13) and finish at mid-AIME difficulty. Worked solutions throughout.
Four original papers that open at AMC 12 #15 level and end at hard-AIME difficulty. Every answer machine-checked, every problem with a worked solution.
Four original fifteen-problem papers that open around AIME #7 and finish beyond the hardest real #15. Integer answers, every one machine-checked, every problem with a worked solution.
Fast, tricky middle-school problems across every topic, the first rung of the MathCounts ladder. Eight original thirty-problem papers with worked solutions, pitched at Chapter level: approachable openers ramping to the hardest Chapter problems, a clear step below State.
The Target round, run the real way: four pairs of two problems, six minutes per pair, hand each pair in before the next opens. Four original papers with worked solutions; the second problem of each pair is the harder one and #8 is the hardest. Calculator allowed.
Three original eight-problem papers in four timed pairs, six minutes each: the second problem of each pair is the harder one, and #8 goes beyond the hardest National Target problems. Worked solutions throughout.
The summit: three original eight-problem papers in four timed pairs, six minutes each. The second problem of each pair is the harder one, and #8 is AIME-level. Worked solutions throughout.
MathCastle’s own invitational: two original fifteen-problem papers of fun, tricky problems, ramping from an approachable start to AIME-level finishers. Worked solutions throughout.
Pure number theory: two original ten-problem papers (primes, orders, totients, valuations, Diophantine equations, digits), from a real opener to olympiad-flavoured finishers. Worked solutions throughout.
National olympiad level in the real two-day format: three problems, hand them in, then three more, each day ramping to a very hard closer. Four original papers, each a compute-form olympiad set with full proofs as solutions and one generating-function problem.
The hardest undergraduate competition, in the real A/B format: a six-problem morning sitting, then a six-problem afternoon sitting, each climbing from A1/B1 to a very hard A6/B6. Four original papers with complete worked solutions.
AP Calculus practice exams
Original AP-style exams: limits, derivatives, integrals, related rates, and free-response questions with multiple parts — timed, with worked solutions citing every rule used.
Limits, the power/product/chain rules, motion, area, and a full free-response problem.
Implicit differentiation, average value, optimization, related rates, and the FTC.