TMO 15

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Hi,

During the past several days, the 15^{th} Thailand Mathematics Olympiad was held in Nakhon Ratchasima. In this post, we are sharing our solutions, along with some comments, to the problems in the competition. But since it’s always good to try solving problems yourself before reading the solution, we’ve attached the solution file at the end of this post and give the non-spoiler version of comments to each problem first.

Note that we use the difficulty rates \beta as in our POTWs, that is, POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

P1 Let the incircle of triangle ABC tangent to BC, CA, AB at D, E, F respectively. Let P and Q be the midpoint of DF and DE respectively. Let PC intersect DE at R and BQ intersect DF at S. Prove that

  • a) Points B, C, P, Q lie on a circle.
  • b) Points P, Q, R, S lie on a circle.

Comment This is normal easy geometry, playing with details of the picture for enough time will solve the problem. We rate this \beta = 1.1.

P2 Show that there are no function f:\mathbb{R} \rightarrow \mathbb{R} that

f(x+f(y))=f(x)+y^2

for all real number x and y.

Comment This problem require some amount of insights in solving functional equation, i.e., you should know what to look for when trying to substitutes x and y in the equation. We rate this \beta =1.9.

P3 Karakade has three flash drives of each of the six capacities 1,2,4,8,16,32 GB. She gives each of her 6 servants three flash drives of different capacities. Prove that there are two capacities of which each servant got at most one of each type, or any two of servants have different sum of the capacity.

Comment Once you fully understand the structure of what the problem given and asked for, there remains only a few steps of deduction to reach the result we want. We rate this \beta= 1.4.

P4 Let a, b, c be nonzero real number such that a+b+c = 0. Find the maximum value of

\displaystyle \frac{a^2b^2c^2}{(a^2+ab+b^2)(b^2+bc+c^2)(c^2+ca+a^2)}.

Comment The key point is to guess the answer to the problem, which is not obvious at all. The experienced student also may notice some identity that makes the problem look easier to bound. We rate this \beta = 2.1.

P5 Let a, b be positive integer such that 5 \nmid a, b and 5^5 \mid a^5+b^5. Find the minimum value of a+b.

Comment Many students may immediately see that the problem is one-step-done by applying a famous lemma. We rate this $\beta = 1.5$ since lots of students will know the lemma.

P6 Let A be the set of all triples (x,y,z) of positive integers satisfying 2x^2+3y^3=4z^4.

  • a) Show that if (x,y,z)\in A then 6 divide all of x,y,z.
  • b) Show that A is an infinite set.

Comment The first part is just normal divisibility problem, and the second one is also not hard, the method used to show is very classical. We rate this \beta =1.5.

P7 Every number in set S=\{ 1,2,...,61\} is colored by one of 25 colors. It is permitted to use only some colors. Let m be the number of subset of S such that every member in the subset has the same colour. What is the minimum value of m?

Comment The method we presented in our solution is the common technique used when the constraints given is discrete. We rate this \beta = 1.6.

P8 There are 2n+1 tickets such that there is a number written on each ticket and all numbers are different. The sum of all number on tickets is more than 2330 but every sum of any n tickets have sum not exceed 1165. What is the maximum value of $n$?

Comment The key point is to efficiently use the conditions given to bound the quantity required in an ingenious way. We rate this \beta = 2.2.

P9 Let the incircle of \triangle{ABC} tangent to AB at D. Let P be a point on BC different from B and C. Let K and L be incenters of \triangle{ABP} and \triangle{ACP}, respectively. Suppose the circumcircle of triangle \triangle{KPL} cuts AP again at Q.

Comment Maybe there is beautiful synthetic solution, but our solution used heavy side-bashing with some help from trigonometry. We rate this, from the viewpoint of bashing, \beta = 2.5.

P10 Let a,b,c be nonzero real numbers. Suppose that functions f,g : \mathbb{R} \rightarrow \mathbb{R} satisfy

a f(x+y)+ b f(x-y) = c f(x) + g(y)

for all real numbers x and y such that y > 2018. Show that there exists a function h: \mathbb{R} \rightarrow \mathbb{R} such that

f(x+y)+f(x-y) = 2f(x) + h(y)

for all real numbers x and y.

Comment The solution we present is relatively short compared to other solutions we found; it seems there are many ways to attack the problem. However, this is easily the hardest problem on the test. We rate this \beta = 3.2.

In case you wish to see the solutions, they are available in our tests archive. Here is a direct link to the file.

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