Five reasons I used Tetsuryokukai's University of Tokyo mathematics past papers

A recommendation based on my own preparation

When I was preparing for the University of Tokyo entrance examination, my main mathematics past-paper collection was Tetsuryokukai's University of Tokyo Mathematics Problem Book. I attended a different preparatory school, but found this collection especially helpful.

This review reflects the editions discussed in the original 2018 article. It compares them with the Akahon books published by Kyogakusha and the Aohon books associated with Sundai. Those names cover multiple books and editions: the comments below are not a claim that every current Akahon or Aohon has the same organization or explanations.

Several medical-school classmates told me they had also used Tetsuryokukai's collection. That was an informal conversation within a selected group, not a representative survey of successful applicants. It does not establish that using the book causes admission, or that most applicants need it.

For available editions, see Tetsuryokukai's official publications page. Check the covered years and contents before choosing a volume.

1. Many alternative solutions

The feature I valued most was the number of approaches presented. In the editions I used, some problems had six or seven solutions. Instead of marking an answer and moving on, I could compare coordinate geometry, vectors, calculus, and more economical observations on the same problem.

A limited supply of past papers can then provide much more practice. Ask what changes between two approaches: the variable chosen, the quantity held fixed, the symmetry used, or the amount of calculation required.

The alternative solutions also had star ratings indicating how difficult they were to discover or carry out. That helped me distinguish a realistic examination approach from an elegant method better read afterward. You do not need to memorize every advanced alternative to benefit from the collection.

2. Explanations of how to find a strategy

A finished solution can be correct yet leave an important question unanswered: How was I supposed to think of that?

I liked the explanations leading from the problem statement to a useful plan. The editions I compared sometimes gave only a short hint or an occasional supplementary discussion, whereas this collection devoted more attention to the route toward the solution.

When reviewing, pause before the algebra and reconstruct that route yourself. Which condition suggests a recurrence? Which symmetry makes a coordinate choice useful? Which expression should be treated as a constant? Learning those decisions is more transferable than remembering a page of calculations.

3. Concise, complete model answers

I also found the model answers good examples of mathematical writing: enough explanation to justify the conclusion, without unnecessarily long detours.

After understanding a solution, close the book and write your own version. Then compare whether you stated the domain, explained any case split, checked equality conditions, and proved that all requested possibilities were covered. The point is to learn clear reasoning, not to reproduce the author's wording.

The original review described these as answers sufficient for full marks. That is the author's assessment of their completeness, not a guarantee of an examiner's scoring decision.

4. Problems grouped by examination year

For full-paper practice, organization by year has two advantages.

First, it lets you practice choosing an order. Read the whole paper, identify accessible problems, and decide where to invest time. Solving isolated questions cannot fully reproduce that decision.

Second, a topic heading can give away the intended method. A probability question placed under “sequences,” for example, practically advertises a probability recurrence. A complete yearly paper asks you to recognize that structure yourself.

Topic-based collections still have a useful role when repairing a weakness. My preference for year-by-year organization concerns realistic paper practice, rather than a claim that topic practice has no value. You can combine both, while keeping some unseen yearly papers for timed attempts.

5. Difficulty ratings for different skills

The editions I used assessed creativity, calculation, logical reasoning, and overall difficulty separately, with A/B/C ratings. That breakdown was more informative than a single label such as “hard.”

Suppose a volume problem is rated demanding in calculation but modest in creativity. If you could not even set up its integral, the useful response is to revisit cross-sections and the connection between area and volume, rather than merely saying that the problem required a brilliant idea. If you formed the integral correctly but could not evaluate it, the weakness is different.

Use the ratings to guide a diagnosis, and verify it against your actual work. A rating is an editorial judgment; it does not replace examining the step at which you became stuck.

How I would use the collection

Attempt a yearly paper without topic hints, review the main solution, compare a few relevant alternatives, and rewrite the argument in a concise form. Record the decision you missed, then revisit the problem after an interval.

The original article regarded the higher purchase cost as a drawback. Prices and editions change, so compare the actual books available to you and choose one whose explanations you can use consistently. My recommendation rests on the five learning features above, not on a claim that an expensive book is necessary for admission.