Difference between revisions of "1982 IMO Problems"

Latest revision as of 22:10, 29 January 2021

Day 1

Problem 1

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $f(m+n)-f(m)-f(n)=0 \text{ or } 1.$ Determine $f(1982)$ .

Solution

Problem 2

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$ , $a_{2}$ , $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$ . Let $M_{i}$ be the midpoint of the side $a_{i}$ , and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$ . Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$ . Prove that the lines $M_{1}S_{1}$ , $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent. Solution

Problem 3

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$ .

a) Prove that for every such sequence there is an $n\ge1$ such that: ${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999.$

b) Find such a sequence such that for all $n$ : ${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4.$ Solution

Day 2

Problem 4

Prove that if $n$ is a positive integer such that the equation $x^3-3xy^2+y^3=n$ has a solution in integers $x,y$ , then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$ . Solution

Problem 5

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that ${AM\over AC}={CN\over CE}=r.$ Determine $r$ if $B,M$ and $N$ are collinear. Solution

Problem 6

Let $S$ be a square with sides length $100$ . Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$ . Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$ . Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$ . Solution

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 === Problem 1 ===
 The function <math>f(n)</math> is defined on the positive integers and takes non-negative integer values. <math>f(2)=0,f(3)>0,f(9999)=3333</math> and for all <math>m,n:</math> <cmath> f(m+n)-f(m)-f(n)=0 \text{ or } 1. </cmath> Determine <math>f(1982)</math>.
 [[1982 IMO Problems/Problem 1 | Solution]]

1982 IMO (Problems) • Resources
Preceded by 1981 IMO Problems	1 • 2 • 3 • 4 • 5 • 6	Followed by 1983 IMO Problems
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