Three mathematical problems authored by Vietnamese mathematicians have been selected for the International Mathematical Olympiad (IMO) exams over the years.
Among them, a problem by the late Associate Professor Van Nhu Cuong is recognized as one of the most challenging and intriguing in IMO history.
1. Problem by Phan Duc Chinh - IMO 1977
The first Vietnamese-authored problem selected for the IMO was by Phan Duc Chinh. It was included as problem number 2 in the 1977 exam:
“In a finite sequence of real numbers, the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.”
Phan Duc Chinh was an influential figure in Vietnamese mathematics education. He was one of the first teachers at the A0 specialized mathematics class of the University of Hanoi (now the specialized mathematics class at the High School for Gifted Students, under the University of Science, Vietnam National University, Hanoi). He played a significant role in training many students who later won international mathematics medals.
2. Problem by Van Nhu Cuong - IMO 1982
Another Vietnamese problem selected for the IMO was authored by Van Nhu Cuong. It was problem number 6 in the 1982 exam:
“Let S be a square with side length 100. Let L be a path within S composed of line segments \(A_0A_1\), \(A_1A_2\), ..., \(A_{n-1}A_n\) with \(A_0 ≠ A_n\). Suppose that for every point P on the boundary of S, there is a point of L at a distance from P no greater than 1/2. Prove that there are two points X and Y of L such that the distance between X and Y is not greater than 1 and the length of the part of L which lies between X and Y is not smaller than 198.”
The problem by the late Associate Professor Van Nhu Cuong was praised for its difficulty and originality. According to Professor Tran Van Nhung, former Deputy Minister of Education and Training, many countries wanted to exclude this problem from the IMO, but the chairman insisted on keeping it, describing it as "very interesting."
Van Nhu Cuong was a prominent educator, textbook author, and founder of Vietnam's first private high school, Luong The Vinh High School in Hanoi. His problem, originally presented in a poetic context involving a village and a river, was later adapted to a more mathematical language for the IMO.
Original Problem by late Associate Professor Van Nhu Cuong:
"Once upon a time, there was a square village with each side 100 km long. A river meandered through the village. Any point in the village was no more than 0.5 km from the river. Prove that there are two points on the river such that the straight-line distance between them is no more than 1 km, but the distance along the river is no less than 198 km." (Assume the river's width is negligible.)
3. Problem by Nguyen Minh Duc - IMO 1987
The third problem selected was authored by Nguyen Minh Duc and was problem number 4 in the 1987 IMO:
“Prove that there is no function f from the set of non-negative integers into itself such that \(f(f(n)) = n + 1987\) for every n.”
Nguyen Minh Duc, a former student of the High School for Gifted Students in Natural Sciences, won a Silver Medal at the 1975 IMO. Before retiring, he was a research officer at the Institute of Information Technology under the Vietnam Academy of Science and Technology.
The International Mathematical Olympiad (IMO) has been held annually since 1959, with Vietnam participating since 1974. Before each competition, the team leaders of participating countries submit problems to the host country, which then selects a shortlist of about 30 problems. Shortly before the competition, team leaders vote on the final six problems to be included in that year’s exam.
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