1, you can view the award-winning certificates and medals obtained by the coach, which is a direct proof of guiding students to achieve excellent results in the competition.

2. We can consider whether the coach has published any papers related to the math contest, which can further prove his professional knowledge and guiding ability in this field.

3. We can consider whether the coach has coached students to win the gold medal in the International Mathematical Olympiad, or independently coached students to win multiple international mathematical olympiads, or participated in guiding students to win multiple international mathematical olympiads.

4. The coach has a large number of students who won the first prize in the National Mathematics League, helping them to enter the world famous schools, which is also a strong proof of the gold medal level of the coach.