ISL 2001

Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$ .

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Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ , satisfying $$f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)$$ for all $x,y$ .

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Find all positive integers $a_1, a_2, \ldots, a_n$ such that

$$\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n},$$ where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$ .

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A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$ . Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

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Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$ , $0 \leq j \leq n$ , $x_j$ equals the number of times $j$ appears in the sequence.

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For a positive integer $n$ define a sequence of zeros and ones to be balanced if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are neighbors if you can move one of the $2n$ symbols of $a$ to another position to form $b$ . For instance, when $n = 4$ , the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$ .

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A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each $n$ , show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$ .

IMO ShortList 2001, combinatorics problem 7, alternative

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Let $ABC$ be a triangle with centroid $G$ . Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$ .

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Let $M$ be a point in the interior of triangle $ABC$ . Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$ . Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define

$$p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.$$

Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

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Let $O$ be an interior point of acute triangle $ABC$ . Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$ . Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$ , and $CA_1B_1$ .

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Consider the system \begin{align*}x + y &= z + u,2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$ .

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Let $a_1 = 11^{11}, a_2 = 12^{12}, a_3 = 13^{13}$ , and $a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.$ Determine $a_{14^{14}}$ .

Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$ .

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