# Worldwide collection of final examinations in mathematics

I am interested in comparing mathematics (and also physics) final examinations from all around the world for the highest secondary school qualification in the particular country. Especially I am interested in countries which ranked high in TIMSS/PISA.

For the German Abitur (I am a mathematics and physics teacher in Germany), there are some web sites which collect problems and solutions of the Abitur exams of the last years for example:

http://ne.lo-net2.de/selbstlernmaterial/m/abi/abiindex.html (for Mathematics for different federal states)
http://ne.lo-net2.de/selbstlernmaterial/p/abi/abiindex.html (for Physics)

So I am looking for similar links for the original problems (and possibly solutions) of the final examinations of other countries (or other examinations roughly equivalent to the German Abitur). Ideally it would be great to have a version which was translated to English (or German) but having the exam in the original language would be sufficient.

I am also interested in comparing older (or even historical) exams of different countries if available (though I didn't find a good link for this even for German exams).

Please add if Calculators, Graphical Calculators or Computer Algebra Systems are allowed. Same for formula sheets/booklets, etc.

Clearly one might find such resources using a search engine, but because of the different languages and school systems this is not so easy. Because of the international user base, this site seems to be a good place to ask this question.

Edit: It would be useful if you could add some background about the exam system of the particular country or state.

In the case of Germany:

The German exams linked above are part of the German Abitur, which allows you to study every subject in university if you pass it ("Allgemeine Hochschulreife"). The Abitur is slightly different in each federal state, but you can roughly say this:

• Main subjects of the math exams are calculus, vector geometry (and very little linear algebra) and elementary probability theory (depending on the year and the federal state one of the latter might be left out)

• Over the last 10 years most states changed to a system, where students graduate after the 12th grade and everybody has to take a mathematics course in grades 11 and 12 with 4x 45 min lessons a week. Based on this course (and implicitly content of the lower grades) they have to write the exams at the end of the 12th grade.

• Before that most states had a system where students graduated after the 13th grade and were allowed to choose between a typcically 3x45 min per week math course in 12 and 13 ("Grundkurs") or a 5x45 min per week math course ("Leistungskurs") with different exams at the end of the 13th year. In the Leistungskurs, typically some very basic real analysis was part of the curriculum (convergence of sequences, continuity (using sequences or $\epsilon$-$\delta$ test), some proof of theorems of calculus, proof by induction etc)

• In the last 10 years many states introduced graphical calculators ("GTR") or even computer algebra systems ("CAS") which were allowed in the exams. Often there was a part without any calculator and a part with a GTR or CAS. Before that students had typically only a usual scientific calculator. Usually a table of formulas is available for the students in the exams. One might add that in the case of Baden-Württemberg the GTR and CAS are abandoned for the coming Abitur 2019 (on "Allgemeinbildenen Gymnasien" and 2017 on "Beruflichen Gymnasien").

• You may want to look at the entrance exams used for Moscow State University. See the results of this google search: mathematics "entrance exam" "Moscow State University – Dave L Renfro Dec 23 '14 at 15:12
• The International Baccalaureate Diploma Programme has final exams in each of its four levels of mathematics. The exams are called "papers," and a search like "ib math hl 2 past papers" should give some examples. Note: These exams are not usually used nation-wide or state-wide, but since they are used by participating schools in 100+ countries, you might consider them in your comparison. – Xi Yu Dec 29 '14 at 5:48

The Sixth Term Examination Papers are national and public examinations taken by 18 year old school leavers (pre-university), intending to study mathematics at one of the more selective/over-subscribed UK universities.

Although there is a close association with Cambridge University - and are considered to be entrance papers to read mathematics there - several UK universities now ask for grades (although not particularly high) in mathematics STEP.

Because STEP are public examinations open to any candidate, they are based on what is considered to be "core" syllabus concepts. So in some sense, the range of questions is quite narrow, but in practice the questions are usually much more demanding than standard A level, with each one drawing upon a number of different syllabus areas and often requiring the application of standard techniques in novel/interesting ways.

Much information is available here here and in a study booklet, solutions and past papers are available.

• Taken by 18 year old school leavers for university entrance (Abitur equivalent)
• No calculator, books or other material is permitted, a formula sheet is provided.
• Examinations are three hours.
• Candidates attempt 6 questions, out of a possible 14 (or so). Questions can be taken from pure, statistics and mechanics.

For standard A Level mathematics, the UK consists of 7 (yes, 7) examination boards - but information is available here. Both STEP and A Level are directly comparable to the German Abitur.

Around 1997-1998 I became aware of some neat math problems on the internet. This was back in the day when you could pretty much find everything of mathematical interest on the internet with search engines and by knowing some places to visit where sites of interest were collected, although in this particular case I might have been led to the problems by a Russian mathematician (student of E. P. Dolzhenko) that I was in correspondence with. At the time I was teaching high school (a specialized math-science boarding high school for gifted students), and in one of the classes (multivariable calculus) I had only 5 or 6 students, all very strong (possibly all now have a graduate degree in a science field, and one has a Ph.D. in mathematics from Rice University), so we did these "time-outs" two or three times a week for the last 15 or 20 minutes of class to work on the problems that I had found on the internet. (Just now I had to go back to some old LaTeX class handouts to get the URLs, then use the Internet Archive to locate valid URLs for them.) The problems are in Russian, but many can be understood without being able to read Russian. It also helped in my case that one of my students was fluent in Russian.

1995 problems

1996 problems

1997 problems

1998 problems

Regarding mathematics problems on the Moscow entrance exams, much has been written about them lately. The following seem to be especially useful discussions of these problems:

Alexander Shen, Entrance Examinations to the Mekh-mat, The Mathematical Intelligencer 16 #4 (1994), 6-10.

Tanya Khovanova and Alexey Radul, Jewish Problems, arXiv:1110.1556v2, 15 October 2011, 21 pages.

Jay Egenhoff, Math as a Tool of Anti-Semitism, The Mathematics Enthusiast 11 #3, 649-664.

Tanya Khovanova's Math Blog, A Math Exam’s Hidden Agenda, 11 May 2011.

• (Side-note: The Khovanova & Radul arXiv piece was published in AMM, as mentioned in MESE 4114. The citation is: Khovanova, T., & Radul, A. (2012). Killer problems. American Mathematical Monthly, 119(10), 815-823.) – Benjamin Dickman Dec 24 '14 at 4:32

For the United States (NYC, specifically): You can find the archive of Regents Examinations here.

(See also the Regent Examinations wikipage if you are unfamiliar with these assessments.)

The mathematics examinations are available from 1866 through today, though the archive is incomplete.

One prominent researcher of the math regents is Eileen Donoghue, cf. MESE 1820 and MESE 369.

Here is an example of the start of the earliest test in the archive (1866):

And here is an example of the start of a recent test in the archive (2014):

• Very interesting. Thanks! I am not familiar with this system and though I have read the wiki page, I have some questions about this: So this are exams to earn a high school degree which are not written at the end of the 12th grade but distributed over the 9th to the 12th grade? However a high school diploma doesn't allow you to study mathematics at university (like a german abitur), you have to take a SAT test or something like that (is that correct?). Reading the exams I get the impression that calculus, vector geometry and probability is not part of the curriculum? – Julia Dec 24 '14 at 9:32
• @Julia The US education system varies greatly throughout the country. The Regents Examinations are specific to New York; more information is available here (NY HS diploma) and here (math ed in NY). Finally: To read NYC's Dept of Ed summary of HS academic policy (e.g., as related to the Regents) see here. – Benjamin Dickman Dec 24 '14 at 10:29
• @Julia: The "SAT test or something like that" is simply a test that many colleges and universities require for admissions (because high school grades can vary greatly from school to school for "equivalent performance"). In the US one doesn't apply to a university to study a particular subject, that comes later. In many cases you don't choose your major area until the end of your 2nd year, although of course if you're planning to major in mathematics you'd want to begin taking mathematics right away and if you're planning to major in music you'd want to begin taking music right away. – Dave L Renfro Dec 27 '14 at 11:56

Sorry, I do not have the ability to translate these; however, for the high school mathematics exam of China, this website works, search for 会考数学. That means huikao (high school final exam) and shuxue (mathematics).

For the physics use this and from there you can choose which region of China and subject you would like.

For the college entrance exam: 高考

• Your first link is specific to Zhejiang province; your last gaokao link (the last one) is specific to Shanghai (which is traditionally the easiest college entrance examination in China; FWIW I believe Henan's gaokao is often the most difficult). – Benjamin Dickman Dec 27 '14 at 12:20
• @BenjaminDickman My Chinese language skill is poor, but I know this site has several different provinces exams, if you can make this answer better please do. – El Santi Dec 27 '14 at 12:29
• Searching here for mathematics (数学) will show you other examples from the former link; a similar approach here will let you explore different gaokaos in the latter. (To some extent, I think your response should just be a comment till a fluent speaker posts a full answer; but, admittedly, I do not foresee myself doing as much anytime soon...) – Benjamin Dickman Dec 27 '14 at 12:55
• Yes, it would be better if a person who is fluent posted an answer, I was just giving a resource that I know of, I didn't think that my answer would be very good, but it was too long for a comment. – El Santi Dec 28 '14 at 13:31