ISL 1998

Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$ . Prove that

$$\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}.$$

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Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that

$$\frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}.$$

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Let $x,y$ and $z$ be positive real numbers such that $xyz=1$ . Prove that

$$\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.$$

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Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$ . Let $D$ be the point on the side $BC$ such that $CD=2BD$ . The segment $AD$ is extended to $E$ so that $AD=DE$ . Prove that $$\angle ECB+180^{\circ }=2\angle EBC.$$

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Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$ . The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$ . Let $E$ be the reflection of $A$ in the line $BC$ , let $X$ be the foot of the perpendicular from $A$ to $BE$ , and let $Y$ be the midpoint of the segment $AX$ . Let the line $BY$ intersect the circle $\omega$ again at $Z$ .

Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$ .

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Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor$ for all positive integers $n$ . (Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .)

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Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ .

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A sequence of integers $a_{1},a_{2},a_{3},\ldots$ is defined as follows: $a_{1} = 1$ and for $n\geq 1$ , $a_{n + 1}$ is the smallest integer greater than $a_{n}$ such that $a_{i} + a_{j}\neq 3a_{k}$ for any $i,j$ and $k$ in $\{1,2,3,\ldots ,n + 1\}$ , not necessarily distinct. Determine $a_{1998}$ .

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Prove that for each positive integer $n$ , there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

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Let $a_{0},a_{1},a_{2},\ldots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$ , where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$ .

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