SIMO Contributions

The following is extracted from my correspondence with several ex-IMO participants on my proposals for the 1996 SIMO Camp Oral Quiz. The problems which were subsequently selected for the quiz are marked with asteriks, and post-Quiz remarks were added for some of the problems. Click here for more information about SIMO.

Part I

*1. Let M be the maximum possible value of x₁x₂ + x₂x₃ + ... + x₅x₁ where x₁, x₂, ..., x₅ is a permutation of 1,2,3,4,5 and let N be the number of permutations for which this maximum is attained. Evaluate M + N.

Comment(s): Quite original I suppose. The style is typically AIME.

*2. Calculate the number of ways of choosing 4 numbers from the set {1,2,...,11} such that at least 2 of the numbers are consecutive.

Comment(s): Not original. A simple counting problem, yet not too easy.

3. What is the difference between the number of bronze medals and the number of silver medals Sinagapore has won from the 34th IMO till the 36th IMO?

Comment(s): The trainees must be more familiar with the history of Singapore's participation in the IMO. If they are really dedicated to the training, they should be equipped with such knowledge. The motive behind selecting these 3 IMOs is that they are recent and that it yield an answer of 0. No, I'm not trying to put myself in the limelight.

4. How many Singaporeans have represented Singapore in the IMOs so far?

Comment(s): A question with much room for changes depending on our objective. The problem is somewhat tricky in that the trainees might not have included the team leaders and deputy team leaders. However, there is also the question of whether the observers (!!!) ought to be included in the count. Anyway, this question arises from last year's IMO, as Mr Tai and I attempted to count the number of Singaporeans who have represented Singapore as contestants. I choose to use the phrase 'represented Singapore' instead of 'taken part' or 'participated' because of ZZZ's ambiguous citizenship status. As we all known, he took part in the 1993 IMO as a Briton. Anyway, it's a very tough question, unless we were to provide some of these statistics during some night session. Of course, the scope could be limited to the recent IMOs, e.g. from 1990 onwards.

Part II

*1. The primitive Pythagorean triangle with sides 2547 and 4004 and hypotenuse 40085 has area 50945094, which is an 8-digit number of the form abcdabcd. Find a Pythagorean triangle whose area is of the form abcabc.

Comment(s): The original problem - from Crux - asked for another primitive Pythagorean triangle whose area is of the form abcdabcd. However, this would demand tedious calculations. So far, I have found only 1 triangle satisfying the property stated in the adapted problem, namely the one with sides 663, 606 and hypotenuse 905.

After the Quiz: A simpler solution was found: namely the triangle with sides 2002, 336, and hypotenuse 2030.

*2. Determine the remainder when (x⁴-1)(x²-1) is divided by 1+x+x².

Comment(s): A special case of a generalisation of a problem proposed by Murray Klamkin.

After the Quiz: Some of the trainees obtained an answer by substituting specific values of x, as expected.

3. The captain of a ship arrived at a circular treasure island knowing that a treasure was buried at the mid-pt of the orthocenters of the triangles ABC and DEF, where A, B, C, D, E, and F are the locations of 6 trees on the edge of the island. The captain was able to locate these 6 trees, but he did not know which tree corresponded to which letter. What is the maximum possible number of holes which the captain dug before he retrieved the treasure (assuming that the captain is intelligent and familiar with Eucliean geometry)?

Comment(s): No, this is not a trick question. Nonetheless, I like this problem a lot. It is a problem from the 1995 International Tournament of the Towns [like we all know, I'm a very resourceful person :)], translated from Chinese.

After the Quiz: This problem was used as a training problem. The quickest and most elegant solution - via the use of vectors - eluded many of the ex-IMO participants and the trainees.

In response to the need for more original problems for SIMO training and the efforts to set up a pool of problems for our consideration, I have decided to submit the following possibly perplexing problems, though they are not very original:

1.
Prove that the following inequality holds for all positive real numbers x, y, z:
z sqrt(x^2 + y^2 - xy) + x sqrt(y^2 + z^2 - yz) >= y sqrt(x^2 + z^2 + xz)
When does equality hold?

2. (TeX format, hopefully)
Prove that the following inequality holds for all positive real numbers $x,y,z$:
$ ({\frac{x+y}{2}})^2 ({\frac{z+y}{2}})^2 ({\frac{x+z}{2}})^2 \ge ({\frac{xy+yz+xz}{3})}^3 $