# Day $$1$$

Tuesday, July 18, 2017

Problem 1. For each integer  $$a_0\gt1$$, define the sequence $$a_0$$, $$a_1$$, $$a_2$$, $$\dotsc$$ by:

$a_{n+1} = \begin{cases}\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\a_n + 3 & \text{otherwise.}\end{cases}\quad \text{for each}\; n\geqslant 0.$

Determine all values of $$a_0$$ so that there exists a number $$A$$ such that $$a_n = A$$ for infinitely many values of $$n$$.

Problem 2. Let $$\Bbb R$$ be the set of real numbers. Determine all functions $$f\colon \Bbb R \rightarrow \Bbb R$$ such that, for all real numbers $$x$$ and $$y$$,

$f\big(f(x)f(y)\big) + f(x+y) = f(xy).$

Problem 3. A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point, $$A_0$$ , and the hunter’s starting point, $$B_0$$ are the same. After $$n-1$$ rounds of the game, the rabbit is at point $$A_{n-1}$$ and the hunter is at point $$B_{n-1}$$ . in the $$n^{\text{th}}$$ round of the game, three things occur in order:

i) The rabbit moves invisibly to a point $$A_n$$ such that the distance between $$A_{n-1}$$ and $$A_n$$ is exactly $$1$$ .

ii) A tracking device reports a point $$P_n$$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $$P_n$$ and $$A_n$$ is at most $$1$$ .

iii) The hunter moves visibly to a point $$B_n$$ such that the distance between $$B_{n-1}$$ and $$B_n$$ is exactly $$1$$ .

Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $$10^9$$ rounds, she can ensure that the distance between her and the rabbit is at most $$100$$ ?

# Day $$2$$

Wednesday, July 19, 2017

Problem 4. Let $$R$$ and $$S$$ be different points on a circle $$\Omega$$ such that $$RS$$ is not a diameter. Let $$\ell$$ be the tangent line to at $$R$$. Point $$T$$ is such that $$S$$ is the midpoint of the line segment $$RT$$. Point $$J$$ is chosen on the shorter arc $$RS$$ of $$\Omega$$ so that the circumcircle  $$\Gamma$$ of triangle $$JST$$ intersects $$\ell$$ at two distinct points. Let $$A$$ be the common point of $$\Gamma$$ and $$\ell$$ that is closer to $$R$$. Line $$AJ$$ meets $$\Omega$$ again at $$K$$. Prove that the line $$KT$$ is tangent to $$\Gamma$$.

Problem 5. An integer $$N\geqslant2$$ is given. A collection of $$N(N + 1)$$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $$N(N-1)$$ players from this row leaving a new row of $$2N$$ players in which the following $$N$$ conditions hold:

(1) no one stands between the two tallest players,

(2) no one stands between the third and fourth tallest players,

$$\vdots$$

(N) no one stands between the two shortest players.

Show that this is always possible.

Problem 6. An ordered pair $$(x, y)$$ of integers is a primitive point if the greatest common divisor of $$x$$ and $$y$$ is $$1$$. Given a finite set $$S$$ of primitive points, prove that there exist a positive integer $$n$$ and integers $$a_0$$, $$a_1$$ , $$\dotsc$$, $$a_n$$ such that, for each $$(x, y)$$ in $$S$$, we have:

$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \dotsb + a_{n-1}xy^{n-1} + a_ny^n = 1.$

Tagged with:

This site uses Akismet to reduce spam. Learn how your comment data is processed.