Vietnam's math problem enters International Math Olympiad after nearly 40 years

After nearly 40 years, Vietnam has another math problem included in the International Math Olympiad, by teacher Tran Quang Hung, High School for the Gifted in Natural Sciences.

The information was shared by Mr. Hung with VnExpress on July 19. His math problem was question 2 in the IMO exam on day 1. The content is as follows:

"Let Ω and Γ be circles with centers M and N, respectively, such that the radius of Ω is less than the radius of Γ. Suppose Ω and Γ intersect at two distinct points A and B. Line MN intersects Ω at C and Γ at D, so that C, M, N, D lie on MN in that order. Let P be the circumcentre of triangle ACD. Line AP meets Ω again at E≠A and meets Γ again at F≠A. Let H be the orthocenter of triangle PMN.

Prove that the line through H parallel to AP is tangent to the circumcircle of triangle BEF.

(The orthocenter of a triangle is the point of intersection of its altitudes)".

Translation:

"Given circles Ω and Γ with centers M and N respectively so that the radius of Ω is smaller than the radius of Γ. Suppose circles Ω and Γ intersect at distinct points A and B. Line MN intersects Ω at point C and intersects Γ at point D, so that the order of points on that line is C, M, N and D respectively. Let P be the center of the circle circumscribing triangle ACD. Line AP intersects Ω again at point E ≠ A. Line AP intersects Γ again at point F ≠ A. Let H be the orthocenter of triangle PMN.

Prove that the line passing through H and parallel to AP is tangent to the circle circumscribing triangle BEF.

(The orthocenter of a triangle is the intersection of its altitudes.)".

This is the fourth time Vietnam has had a problem selected for the official IMO exam, according to the Ministry of Education and Training . The first problem in the IMO exam was in 1977, by author Phan Duc Chinh. The second problem was in 1982, by teacher Van Nhu Cuong. The most recent time was in 1987, the problem used was by author Nguyen Minh Duc.

In addition to the official Math exam in this year's exam, Mr. Hung also had two Geometry problems shortlisted for IMO 2022 and IMO 2019.

Teacher Tran Quang Hung (left) with teacher Nguyen Chu Gia Vuong (head of delegation) and students attending IMO 2025. Photo: Provided by the character

Mr. Tran Quang Hung is currently a teacher at the High School for Gifted Students in Natural Sciences (under the University of Natural Sciences, Vietnam National University, Hanoi). He has many years of experience teaching elementary geometry to specialized Math classes and teaching Olympic geometry to national and international teams for gifted students.

Associate Professor Dr. Nguyen Vu Luong, Chairman of the Science and Training Council, High School for the Gifted in Natural Sciences, assessed that the math problem of teacher Tran Quang Hung being chosen "is worthy".

After many years of working together, Mr. Luong commented that Mr. Hung has a special talent for geometry and is diligent in researching this field. Therefore, Mr. Hung's geometry exams are often different, creative, and have a high content of knowledge.

"That doesn't mean Hung's questions will require students to draw dozens of circles, which are complicated and cumbersome. The questions are difficult in the sense that sometimes the drawings are simple, but require students to have a deep understanding and apply many geometric results to solve them. That's why students are very afraid of Mr. Hung's questions but still like to study with him," said Mr. Luong.

Regarding the process, about four months before the exam, the head of each country's delegation will collect proposed problems, the author does not necessarily have to be a member of the delegation but only needs to be from their own country, and then send them to the question selection committee of the host country.

The host country will select about 30 entries, and put them on the IMO short list. A few days before the exam, the delegation leaders vote to select the 6 official entries.

Vietnam in top 10 IMO 2025

The International Mathematical Olympiad has been held annually since 1959. Vietnam first participated in 1974. IMO 2025 took place in Australia from July 10 to 20, attracting more than 630 contestants from 110 countries and territories.

Each day, candidates must solve three problems in 4.5 hours. The maximum score for each problem is 7. Candidates can receive the questions in their mother tongue, but must register in advance and be approved by the organizing committee.

This year's Vietnamese delegation had 6 students participating, won two gold medals, three silver and one bronze, ranked 9th overall.

Vo Trong Khai, grade 12, Phan Boi Chau High School for the Gifted, Nghe An province: Gold Medal (hometown: old Nghi Xuan district, Ha Tinh province).

Student Tran Minh Hoang, grade 12, Ha Tinh Specialized High School, Ha Tinh province: Gold Medal (hometown: old Nghi Xuan district, Ha Tinh province).

Nguyen Dang Dung, grade 12, High School for Gifted Students in Natural Sciences, University of Natural Sciences, Vietnam National University, Hanoi: Silver Medal.

Student Nguyen Dinh Tung, grade 11, High School for Gifted Students in Natural Sciences, University of Natural Sciences, Vietnam National University, Hanoi: Silver Medal.

Student Le Phan Duc Man, grade 12, Le Hong Phong High School for the Gifted, Ho Chi Minh City: Silver Medal

Student Truong Thanh Xuan, grade 11, Bac Ninh High School for the Gifted, Bac Ninh province: Bronze Medal.

Source: https://baohatinh.vn/bai-toan-cua-viet-nam-vao-de-thi-olympic-toan-quoc-te-sau-gan-40-nam-post292009.html