Tournament of Towns problems and solutions

The International Mathematics Tournament of Towns is a mathematics problem solving competition in which towns throughout the world can participate on an equal basis.

Students participate in their own towns which involves minimal transport and administrative costs.

The Tournament is conducted each year in two stages - Autumn and Spring (northern hemisphere time). The southern hemisphere academic year coincides with this structure.

Each stage has two papers, an "O" level and an "A" level, which are spaced roughly one week apart. The A level paper is more difficult, but offers more points. Students and their towns may participate in either stages or levels, or in all levels and stages.

The Tournament is open to all high school students, with the highest age of students being about 17 years old.

Students are awarded points for their best three questions in each paper, and their annual score is based on their best score in any of the four papers for the year.

There are two versions of each paper, known as the Senior and Junior papers. Students in Years 10 and 11 (the final two years of high school in the Russian nomenclature) are classified as Senior participants and therefore attempt the Senior paper. So that Year 10 students are not disadvantaged their scores are multiplied by 5/4. Younger students, in Years 9 and below, attempt the Junior paper. To ensure that the scoring is fair to all levels of students, Year 8 students have their scores multiplied by 4/3, Year 7 students have their scores multiplied by 3/2 and Year 6 students and below have their scores multiplied by 2.

A town's score is based on the average of its best N students, where the population of the town is N hundred thousand (see below for definition of town). There is a minimum of N=5. If a population is less than 500,000 then the score is multiplied by an appropriate compensatory factor.

Students who exceed a certain minimum score are awarded a Diploma by the Russian Academy of Sciences. Local organising committees also present their own awards. The Tournament is managed by a central committee in Moscow, which is a subcommittee of the Russian Academy of Sciences.

You can find problems and solutions here.