Original problem of Mr. Van Nhu Cuong at the International Mathematical Olympiad

In 1982, the International Mathematical Olympiad (IMO) was held in Budapest (Hungary). The Vietnamese delegation contributed a Geometry problem by Mr. Van Nhu Cuong, along with comments from two other math teachers, Professor Hoang Xuan Sinh and Professor Doan Quynh.

This was the most difficult problem of the exam that year, and was almost eliminated.

The Vietnamese delegation has participated in IMO competitions 43 times and has mostly achieved high results.

In the history of IMO, there were 3 problems from the Vietnamese delegation that were selected to be included in the exam. They were IMO 1977 with the problem of Associate Professor Phan Duc Chinh, IMO 1982 by Associate Professor Van Nhu Cuong and IMO 1987 by Dr. Nguyen Minh Duc, Silver Medal IMO 1975).

Professor Tran Van Nhung, currently Secretary General of the State Council for Professor Titles, recounted the story of IMO 1982:

That year, the Vietnamese delegation was led by Professor Hoang Xuan Sinh and Professor Doan Quynh as deputy head. Vietnam's math problems were very difficult and unique. Many countries wanted to remove them from the 6 problems in the exam. But the President of IMO that year - Hungarian Academician Professor R. Afred, Director of the Institute of Mathematics, Hungarian Academy of Sciences - not only praised them but also decided to keep them.

Only 20 contestants of the competition were able to solve this test. Among them was Le Tu Quoc Thang from the Vietnamese delegation. He also won the Gold Medal with a score of 42/42, while the Vietnamese delegation ranked 5/30 participating countries.

He studied at university in Russia and then worked in Russia, Germany, Italy, and the United States. Until 2004, Le Tu Quoc Thang was a professor at the Georgia Institute of Technology (one of the five strongest schools in the United States in engineering). Currently, he is one of the world's leading experts on differential topology, 3-dimensional manifolds, knot theory, and quasicrystals.

Speaking with VietNamNet, Professor Thang said that in 1982, after the Vietnamese delegation returned with results, Vietnam Television interviewed relevant people.

Mr. Van Nhu Cuong presented his original problem. This problem was a little different from the IMO exam (it was made easier). After that, Mr. Thang also presented the solution on television.

Professor Thang shared with VietNamNet the original problem that Mr. Van Nhu Cuong presented as follows:

Once upon a time (in Nghe An) there was a square village with each side measuring 100km. There was a river running through the village. Any point in the village was no more than 0.5km from the river.

Prove that there are 2 points on the river whose distance as the crow flies is not more than 1 km, but the distance along the river is not less than 198 km.

(We assume the river has negligible width).

And below is question number 6 in the 1982 IMO math problem.

Let S be a square with side length 100, and L be a non-intersecting zigzag line formed by line segments A0A1, A1A2…,An-1An with A0#An. Suppose that for every point P on the boundary of S there is a point in L that is no more than ½ away from P. Prove that: There exist 2 points X and Y in L such that the distance between X and Y does not exceed 1, and the length of the zigzag line L between X and Y is not less than 198.

Question number 6 in the 1982 IMO exam. Photo: IMO website

According to VNN