APMO 2015: Problems, Solutions, Results and Preparation Guide

The Asian Pacific Mathematics Olympiad (APMO) is one of the most respected mathematics competitions for talented secondary-school students across the Asia-Pacific region. Every year, students face challenging proof-based problems designed to test mathematical creativity, logical reasoning, and problem-solving ability.

The 27th Asian Pacific Mathematics Olympiad (APMO 2015) was held in March 2015 and brought together students from 33 participating countries. A total of 299 students took part in the competition. The official APMO report records a mean score of 15.33, with a standard deviation of 8.78. The gold, silver, and bronze cutoffs were 25, 19, and 13 respectively.

For students preparing for mathematical olympiads today, APMO 2015 remains a useful collection of problems. The problems are particularly valuable because they require students to develop ideas rather than simply apply standard formulas.

What Is the APMO?

The Asian Pacific Mathematics Olympiad is an international mathematics competition involving countries from the Asia-Pacific region and beyond.

Unlike many school-level mathematics competitions, APMO focuses heavily on mathematical reasoning and proof. Students are expected to explain their ideas clearly and construct logically complete solutions.

The official APMO website provides problem statements and solutions from previous competitions, including the complete 2015 problem set and solution set.

This makes previous APMO papers excellent resources for students who want to move from routine problem solving toward olympiad-level mathematics.

APMO 2015 at a Glance

Here are some important facts about APMO 2015:

Detail	APMO 2015
Competition	27th Asian Pacific Mathematics Olympiad
Year	2015
Participating countries	33
Participating students	299
Mean score	15.33
Standard deviation	8.78
Gold cutoff	25
Silver cutoff	19
Bronze cutoff	13
Country rank 1	United States
Country rank 2	Korea
Country rank 3	Russia
Country rank 4	Singapore
Country rank 5	Japan

These figures come from the official APMO 2015 report.

APMO 2015 Problems

The APMO 2015 paper contains five olympiad problems. As with other APMO papers, the problems are designed to reward insight, persistence, and rigorous mathematical reasoning.

The official APMO problem archive provides both the 2015 problem statement and the corresponding official solution.

Students should resist the temptation to immediately look at the solution. A much better approach is to spend substantial time working on each problem independently.

For example, when attempting an APMO problem, a student can ask:

What information is given?
What exactly must be proved?
Can I test small cases?
Is there a useful pattern?
Can I introduce a helpful variable or construction?
Is the problem related to geometry, number theory, combinatorics, or inequalities?
Can I transform the problem into something simpler?
What would a complete proof need to establish?

These questions help develop the habits required for international olympiad mathematics.

APMO 2015 Results

The United States finished first in the country ranking in APMO 2015 with a total score of 298. Korea finished second with 279, Russia third with 266, Singapore fourth with 259, and Japan fifth with 256.

The top ten countries were:

United States of America
Republic of Korea
Russia
Singapore
Japan
Canada
Thailand
Taiwan
Australia
Brazil

The official report records 11 gold awards, 42 silver awards, 102 bronze awards, and 81 honorable mentions.

India participated with a team of 10 students in APMO 2015.

India finished 17th in the country ranking, with a total team score of 127. The Indian team received one silver medal, five bronze medals, and three honorable mentions.

The Japanese Mathematical Olympiad Foundation’s record also lists the individual Japanese results and confirms that Japan finished fifth among the participating countries.

India’s APMO 2015 result is particularly useful when looking at the development of olympiad mathematics in the country. Students preparing for competitions such as IOQM, RMO-level mathematics, national olympiads, and international competitions can use older APMO papers as advanced practice material.

Why Are APMO Problems Difficult?

One of the biggest differences between an APMO problem and a typical school mathematics question is that the solution is rarely obvious from a familiar formula.

A student may understand the mathematical topic involved and still struggle to find the key idea.

That is normal.

Olympiad mathematics tests several skills simultaneously:

1. Mathematical creativity

You may need to discover an unexpected construction, substitution, observation, or relationship.

2. Logical reasoning

A correct answer is not enough. The reasoning must establish why the result is true.

3. Persistence

Some olympiad problems take considerable time before the central idea becomes visible.

4. Proof writing

Students must learn to communicate mathematical arguments clearly and rigorously.

5. Pattern recognition

Small examples and special cases can sometimes reveal the structure of a problem.

This is why solving older APMO papers can be much more valuable than simply memorizing olympiad formulas.

How to Use the APMO 2015 Paper for Practice

If you are preparing for an olympiad, don’t simply download the APMO 2015 paper and read the solutions.

Use it as a genuine mock competition.

Step 1: Work without solutions

Print or open the five problems and attempt them without looking at the answers.

Give yourself enough uninterrupted time to think.

Step 2: Record your ideas

Write down unsuccessful approaches as well as successful ones.

An approach that does not work can still teach you something about the structure of the problem.

Step 3: Write a complete solution

Once you believe you have solved a problem, write the solution as if another mathematician will check it.

Avoid statements such as:

“Clearly this is true.”

Instead, explain why it is true.

Step 4: Compare with the official solution

After finishing your attempt, compare your argument with the official solution.

You do not necessarily need to reproduce the official approach. In olympiad mathematics, different correct solutions can often exist.

Step 5: Study the key idea

Ask yourself what made the official solution work.

Was it:

a clever substitution?
an invariant?
a geometric construction?
an inequality?
modular arithmetic?
an extremal argument?
a counting argument?
a clever case division?

Understanding the idea is more important than memorizing the solution.

APMO 2015 for Students Preparing for IMO

APMO problems can be excellent preparation for the International Mathematical Olympiad (IMO) because both competitions emphasize proof-based mathematical problem solving.

However, students should not treat APMO as simply another collection of questions to memorize.

The real benefit comes from learning how to attack unfamiliar problems.

A useful progression is:

School Mathematics → Olympiad Foundations → National Olympiad Problems → APMO → IMO-Level Problems

Students who are new to olympiad mathematics may find APMO problems challenging. That does not mean they should avoid them.

Instead, they can begin with selected problems and gradually increase the difficulty.

What Topics Appear in APMO Problems?

APMO problems can involve several major areas of olympiad mathematics, including:

Number theory
Geometry
Combinatorics
Algebra
Inequalities
Functional equations
Mathematical constructions
Counting and invariants

The important point is that a problem does not always announce its topic.

A question may look like an algebra problem but require a number-theoretic observation. A geometry problem may depend on a transformation or inequality.

This is one reason olympiad students need broad mathematical training.

Tips for Solving APMO Problems

If you are struggling with an APMO problem, don’t immediately search for the answer.

Try these techniques first.

Try small cases

If the problem contains integers, sequences, configurations, or combinatorial objects, examine small examples.

Small cases may reveal a pattern or suggest a conjecture.

Work backwards

If the problem asks you to prove a particular result, ask what would be sufficient to establish that result.

Sometimes working backwards reveals the missing intermediate statement.

Look for invariants

In many olympiad problems, something remains unchanged while other quantities change.

Identifying that invariant can dramatically simplify the problem.

Draw a diagram

For geometry problems, a carefully constructed diagram can reveal relationships that are difficult to see from the written statement alone.

Try a simpler version

If the original problem seems overwhelming, simplify one condition and investigate what happens.

This can reveal the underlying mechanism of the problem.

Write everything down

Olympiad problem solving is not purely mental work.

Writing equations, diagrams, cases, and observations often makes the next step easier to see.

APMO 2015 Problems and Solutions

Students looking for the original APMO 2015 paper should use the official APMO archive, which provides the problem statements and official solutions by year.

For your preparation, it is useful to keep the problem paper and solution paper separate.

Try the problems first.

Then use the solutions as a learning resource.

You can also maintain a notebook containing:

Problem → First Idea → Failed Attempts → Key Observation → Final Proof → What I Learned

This simple method can significantly improve olympiad problem-solving skills over time.

Who Should Solve APMO 2015?

APMO 2015 is most suitable for students who already have some experience with olympiad mathematics.

It can be useful for:

Students preparing for national mathematics olympiads
IOQM and advanced olympiad students
Students preparing for international competitions
Students transitioning toward IMO-level problems
Mathematics students who enjoy proof-based problems
Teachers looking for challenging enrichment problems

You do not need to solve all five problems immediately.

Start with one problem, spend serious time on it, and learn from the process.

Frequently Asked Questions About APMO 2015

When was APMO 2015 held?

APMO 2015 was the 27th edition of the Asian Pacific Mathematics Olympiad. The Japanese Mathematical Olympiad Foundation records the Japanese competition date as March 10, 2015.

How many countries participated in APMO 2015?

There were 33 participating countries and 299 participating students.

How many problems were in APMO 2015?

The APMO 2015 paper contained five problems. The official APMO archive provides the problem statements and solutions.

What was the APMO 2015 gold cutoff?

The gold cutoff was 25 points. The silver cutoff was 19 and the bronze cutoff was 13.

Which country ranked first in APMO 2015?

The United States of America ranked first with a total team score of 298.

What was India’s rank in APMO 2015?

India ranked 17th, with a team total of 127 points. India received one silver medal, five bronze medals, and three honorable mentions.

Where can I find APMO 2015 problems and solutions?

The official APMO website provides an archive of problem statements and official solutions, including the 2015 competition.

APMO 2015 is more than an old mathematics competition paper. It is a valuable training resource for students who want to develop the ability to solve unfamiliar mathematical problems.

The best way to use the paper is not to rush through all five questions. Instead, choose a problem, spend time exploring it, write down your ideas, and only then study the official solution.

Over time, this process builds something that ordinary textbook exercises often cannot: the ability to think mathematically when the method is not obvious.

For students preparing for higher-level mathematics competitions, working through historical APMO papers—including APMO 2015—can become an important part of a long-term olympiad training program.

Ready for the challenge? Start with the APMO 2015 problems, attempt them without looking at the solutions, and see how far your ideas can take you.