Cím:	Hungarian National Competition in Mathematics, 2001/2002
Füzet:	2003/októberi melléklet, 22 - 23. oldal	PDF  |  MathML
Témakör(ök):	OKTV

A szöveg csak Firefox böngészőben jelenik meg helyesen. Használja a fenti PDF file-ra mutató link-et a letöltésre. Second round   1. The hyperbolas with equations $xy=1$ and $xy=-1$ are drawn on the same set of coordinate axes. Prove that if the intersection of a circle of radius $R$ centred at the origin with the hyperbolas form the vertex set of a regular polygon then the area of the polygon is equal to $R^4$.   2. Find the real numbers $x$, $y$, $z$, $t$ satisfying the following simultaneous equation and inequality: $\begin{array}{c}{\displaystyle {\displaystyle \begin{array}{c}\hfill {\displaystyle x+y+z=\frac{3}{2}\mathrm{,}}\hfill \\ \hfill {\displaystyle \sqrt[]{4x-1}+\sqrt[]{4y-1}+\sqrt[]{4z-1}\ge 2+3^{\sqrt[]{t-2}}\mathrm{.}}\hfill \end{array}}}\end{array}$   3. Let $f\left(x\right)$ denote a real valued function defined on the set of real numbers different from 0 and 1. Determine the function $f\left(x\right)$, such that the equation $\begin{array}{c}{\displaystyle f\left(x\right)+k{x}^{2}f\left(\frac{1}{x}\right)=\frac{x}{x+1}}\end{array}$ is satisfied by every $x$ in its domain, where $k$ is a constant with $0<k^2\ne 1$. For what elements $x$ in the domain is $f\left(x\right)=0$?   4. Let $O_A$ denote the point where the bisector of the interior angle at vertex $A$ of a triangle $ABC$ first intersects the inscribed circle. Obtain the points $O_B$ and $O_C$ similarly, on the interior angle bisectors from $B$ and $C$. Let $k_A$ be the circle about $O_A$, tangent to $AB$ and $CA$. Similarly, let $k_B$ denote the circle about $O_B$ touching $BC$ and $AB$ and let $k_C$ denote the circle about $O_C$ touching $CA$ and $BC$. Prove that the three lines different from the sides of the triangle that touch two of the circles $k_A$, $k_B$, $k_C$ externally are concurrent.   Third (final) round   1. The excircle touching the side $AB$ of a triangle $ABC$ touches $AB$ at $P$ and the extension of $AC$ at $Q$. The excircle drawn to side $BC$ touches the extension of $AC$ at $U$ and the extension of $AB$ at $X$. Prove that the intersection of the lines $PQ$ and $UX$ is equidistant from the lines $AB$ and $BC$.   2. Is there an $n$-sided polygon in which the number of acute angles is $\begin{array}{c}{\displaystyle n^2-30n+236?}\end{array}$   3. Let $n$ be a fixed integer greater than 1. Find real numbers $x_1\mathrm{,}{x}_2\mathrm{,}\text{...}\mathrm{,}{x}_n$ such that $\begin{array}{c}{\displaystyle {\displaystyle \begin{array}{c}\hfill {\displaystyle x_1+x_2+\text{...}+x_n=2\left(n-1\right)\mathrm{,}}\hfill \\ \hfill {\displaystyle {\left(x_1-1\right)}^2+{\left(x_2-1\right)}^2+\text{...}+{\left(x_n-1\right)}^2=n\mathrm{,}}\hfill \end{array}}}\end{array}$ and $x_n$ is as large as possible.   Schools with advanced mathematics programme   First round   1. Let $a=1+\sqrt[]{5}$. Evaluate $\left(4-a\right)\cdot \sqrt[]{2+a}\cdot \sqrt[3]{a}\cdot \sqrt[6]{3a+4}$.   2. $ABCD$ is a trapezium with parallel sides $AB$ and $CD$. Let $E$ and $F$ be interior points on the sides $AD$ and $BC$, respectively. Show that if the lines $AF$ and $EC$ are parallel then so are the lines $EB$ and $DF$.   3. Given that the representation of a certain power of two consists of identical digits in a particular number system, prove that the representation has at most two digits.   4. Is there a non-constant polynomial with integer coefficients that assigns a value of the form $k!$ to every positive integer (where $k$ is a positive integer)?   5. Determine the expected value of the second largest number drawn in lottery in which 5 numbers are drawn at random from 1 to 90 inclusive. [The expected value: Consider the second largest number in every possible set of numbers drawn, and take their arithmetic mean. (A certain number is counted as many times as it occurs as second largest number in the different draws.)]   Second (final) round   1. Each $x_i$ out of the numbers $x_1\mathrm{,}{x}_2\mathrm{,}\text{...}\mathrm{,}{x}_n$ ($n\ge 1$) equals the sum of the squares of the remaining numbers $x_j$. Determine all possible such sequences of $n$ numbers.   2. Select an interior point on each side of a parallelogram. Prove that the perimeter of the quadrilateral determined by the four points is at least twice as long as the shorter diagonal of the parallelogram.   3. Find a subset $H$ of the set of positive integers that has the following two properties: (i) every sufficiently large positive integer can be expressed as a sum of at most 100 (not necessarily different) elements of $H$; (ii) 2002 is the smallest number $k$, such that every sufficiently large positive integer can be expressed as a sum of exactly $k$ (not necessarily different) elements of $H$.