*Below is an analysis and commentary on the “killer” problem of the IMO 2017 exam by Dr. Tran Nam Dung (Deputy Secretary of Pi Mathematics Magazine) sent to Dan Tri newspaper:*

When the results of IMO 2017 examination are updated after the first day of marking, one thing is What people pay attention to is the score results of exercises 3 and 6, the two “golden deciding” exercises of the exam. According to Mr. Nguyen Khac Minh (specialist of the Department of Examinations – Ministry Education and Training in charge of Olympic Mathematics and also a member of the IMO 2017 examination committee), when choosing topic 3, it was considered “easier” than with lesson 6.

However, up to now, while lesson 6 has had many 7’s (of which the Korean team alone had 3 7’s), in lesson 3 there is only one 7’s. Australia and a few 1’s, the rest are all 0’s. So what kind of “killer” problem is problem 3?

Problem 3 proposed by Austria is a combinatorial problem of the type Just reading the problem and understanding the requirements of the problem is not simple.

The “killer” problem in the IMO 2017 exam.

The difficult point of the problem is predicting the results and how to move forward. We must understand that the rabbit’s path, the hunter does not know, the positioning device is also randomly placed, only knows that it is no more than 1 meter away from the rabbit. Meanwhile, the rabbit knows where the hunter is.</p >

If you predict a positive answer, the problem will be very difficult because there are two uncertainties. If we predict that the answer is negative, the problem will be easier because according to the logic of negation, we only need to show **a case** where the rabbit can run away from the hunter, which means we You can proactively place the positioning device wherever you want (just need to satisfy the distance conditions). This is the main idea in the solution below based on oneplusone’s brief solution on the artofproblemsolving forum. (According to the information we know, oneplusone is the nickname on the forum of Lim Jeck, a Singaporean student who has participated in the International Mathematical Olympiad 5 times with 3 gold medals, 1 silver medal, 1 bronze medal, of which he won in 2012. Absolute score 42/42.)

**For solution see here.**

**Comments:**

This is a difficult problem. Just reading to understand the problem’s requirements is a problem. Compiling reviews on the artofproblemsolving forum, we see the words “the problem is too difficult and cannot be solved in the exam room!” “terrible problem!” “It’s hard to solve in the exam room!”. But the difficulty in this problem is the orientation, not the technique. If we determine the right direction (the answer is “no”), we can continue to think about building a rabbit’s movement strategy to “run away and fly away”. The most troublesome point here is the locator.

Note that in case the answer is negative, we can proactively place the locator wherever we want (which we would not be obtained if “intended” to prove an affirmative answer). That’s why in its solution oneplusone gave the initiative to the rabbit. It can be said that this is the key point to solve the problem.

Once we have an orientation to build a distance running strategy, we can start from small distances for example 2 3 4 to build from Then find a general strategy for rabbits. That is also a common solution strategy for catch-up problems.

Finally, we can add that in this problem there is hardly a chance to earn component points.

**Dr. Tran Nam Dung**

TS. Tran Nam Dung used to be a math student at Phan Chau Trinh High School in Da Nang; Won the silver medal at the International Mathematical Olympiad (IMO) in 1983 in Paris, the capital of France.

After that, Tran Nam Dung went to Russia to be a student and graduate student at General University. Moscow’s name is Lomonosov and is currently a lecturer in the Department of Mathematics and Informatics at Ho Chi Minh City University of Natural Sciences. Member of the Executive Committee of the Vietnam Mathematical Association – Deputy Secretary of Pi Magazine – Mathematics magazine for students of the Vietnam Mathematical Association.

Dr. Tran Nam Dung contributed to compiling a set of 10th grade math textbooks including 4 volumes (Vietnam Education Publishing House 2009).

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