ISL 2003

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$ .

Proposed by A. Di Pisquale & D. Matthews, Australia

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Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions:

• $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$ ;

• $f(x)<f(y)$ for all $1\le x<y$ .

Proposed by Hojoo Lee, Korea

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Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$ , $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$ .

Let $M=\max\{z_2,\ldots,z_{2n}\}$ . Prove that $$\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right).$$

Proposed by Reid Barton, USA

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Let $ABC$ be an isosceles triangle with $AC=BC$ , whose incentre is $I$ . Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$ . The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$ , respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$ , respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$ .

Proposed by Hojoo Lee

Let $m$ be a fixed integer greater than $1$ . The sequence $x_0$ , $x_1$ , $x_2$ , $\ldots$ is defined as follows: $$x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}$$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .

Proposed by Marcin Kuczma, Poland

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The sequence $a_0$ , $a_1$ , $a_2,$ $\ldots$ is defined as follows: $$a_0=2, a_{k+1}=2a_k^2-1 \text{for }k \geq 0.$$ Prove that if an odd prime $p$ divides $a_n$ , then $2^{n+3}$ divides $p^2-1$ .

Here is a nice problem:

Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that :

1. $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u

Edited by Orl.

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Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:

(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;

(ii) for any nonempty subset of $A$ , the product of its elements is not a perfect $p$ -th power.

What is the largest possible number of elements in $A$ ?

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