For German Maths Olympiad papers, first check the competition name. Mathematik-Olympiade papers are organised by grade and round. Bundeswettbewerb Mathematik (BWM) has a separate set of proof problems. IMO-Auswahlwettbewerb papers belong to the invitation-based international-team selection process. They are not interchangeable.
Use the Germany entry and IMO-selection guide for participation. Below are the official paper sources, a verified historical paper-and-solution pair, and original exercises to help you write a proof.
Choose the right German paper collection
| Your aim | Choose | Check before starting |
|---|---|---|
| Practise your school Olympiad stage | Mathematik-Olympiade | Olympiad number, grade (Klasse) and round (Stufe or Runde) |
| Work on extended written proofs | Bundeswettbewerb Mathematik | Year and first or second round |
| Prepare for advanced IMO selection | IMO-Auswahlwettbewerb | Year and selection paper; entry is a separate invitation process |
Open the official tasks and solutions directory. The page has separate sections for all three programmes. Most of these resources are in German.
Mathematik-Olympiade: grade first, then round
Choose your grade before judging the difficulty of a paper. A national-final paper for an older grade is not a sensible first diagnostic for a younger pupil. Within your grade, work through a released school-round paper before moving to regional or state problems.
- Schulrunde: school round.
- Regionalrunde: regional round.
- Landesrunde: state round.
- Bundesrunde: national round.
The official public-release page lists when questions and solutions become available. As checked on 26 September 2026, it covers the 66th Olympiad, 2026–27. For grades 5–12, school-round questions are scheduled to become public on 1 October 2026 and solutions on 30 October 2026. These are online release dates, not your school’s hand-in deadlines.
Current competition material is distributed through schools and coordinators. Some teacher archives require access credentials. Ask your teacher for authorised resources; a restricted archive is not a public download collection. Do not use solutions or outside assistance on an active competition submission where the rules prohibit them.
A verified historical BWM paper with official solutions
The official directory provides BWM material from 2000 onwards. Here is one matched pair you can use without confusing the round or version:
- BWM 2025, round 1 — question sheet (German PDF, 2 pages).
- BWM 2025, round 1 — final official solutions (German PDF, 15 pages).
This is a past competition. The March 2025 submission date printed in the question sheet is historical. Open the question sheet first; keep the solutions closed until you have made a serious attempt. The official solutions offer useful examples of how a complete argument is written.
IMO selection papers
Use the “Internationale Mathematik-Olympiade” section of the official directory for released IMO-selection papers and available solutions. Check the year and paper separately. Downloading a paper is open study; participating in selection requires qualification and an invitation. The Germany guide explains the route.
A short German paper glossary
- Aufgaben
- Problems or question paper.
- Lösungen
- Solutions.
- Klasse / Klassenstufe
- School grade.
- Runde / Stufe
- Round or stage.
- Vorläufige Version
- Provisional version.
- Endgültige Version
- Final version.
Two original exercises in writing a proof
These are original teaching exercises, not copied German competition problems. They bridge familiar arithmetic and the habit of explaining why a statement always holds.
1. Divisibility needs an argument for every integer
Prove that n³ − n is divisible by 6 for every integer n.
Hint
Factorise the expression. What must be true of any three consecutive integers?
Worked proof
Factorise: n³ − n = n(n² − 1) = (n − 1)n(n + 1). These are three consecutive integers.
At least one is even, so their product is divisible by 2. Their remainders on division by 3 are 0, 1 and 2 in some order, so one factor is divisible by 3.
Because 2 and 3 are coprime, a number divisible by both is divisible by 6. Hence 6 divides n³ − n. This also covers zero and negative integers: the factorisation and divisibility statements still hold.
What makes this a proof? Checking n = 1, 2 and 3 suggests a pattern. The consecutive-integer argument establishes it for every integer n.
2. Find the right boxes
Seven distinct integers are chosen from 1, 2, …, 12. Prove that two of the chosen integers have sum 13.
Hint
Pair up the twelve integers so that every pair sums to 13.
Worked proof
Partition the integers into six pairs:
{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 8}, {6, 7}.
Every integer belongs to exactly one pair. If we chose at most one integer from each pair, we could choose at most six integers. We have chosen seven, so at least one pair must contribute both of its integers. Those two integers sum to 13.
Why seven? Six do not guarantee the conclusion. Choosing 1, 2, 3, 4, 5 and 6 gives no pair with sum 13. Thus seven is the smallest number that forces it.
How to review a written solution
- State the target. Write what must be proved, including the permitted values of every variable.
- Try a few examples. Use them to find a possible idea, not as a substitute for proof.
- Name the key step. It might be factorisation, a congruence, a useful pairing or triangle congruence.
- Compare with the official solution. Identify the first step you could not justify. Close the solution and write a complete proof yourself.
Questions students ask
Does “Bundesrunde” mean Bundeswettbewerb Mathematik?
No. Bundesrunde is the national round of Mathematik-Olympiade. Bundeswettbewerb Mathematik is a separate competition. Match the full programme name on the document.
Why can I see a release date but no current question PDF?
The official site releases material on its published schedule. Your school may distribute a current-round paper earlier through the competition’s authorised route. A public-release date does not change your school deadline.
Are these official English translations?
No. This guide explains how to find the German originals. Our worked examples are original English teaching material and are not translations of the linked papers.
Official resources checked 26 September 2026. Independent guide; not affiliated with Mathematik-Olympiaden e.V., Bildung & Begabung or Germany’s IMO team. Copyright in official papers remains with its respective holders; files are linked at their original source.