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Difference between revisions of "1986 IMO Problems"

(Created page with "== Day 1 == === Problem 1 === Let <math>d</math> be any positive integer not equal to <math>2, 5</math> or <math>13</math>. Show that one can find distinct <math>a,b</math> ...")
 
 
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Let <math>d</math> be any positive integer not equal to <math>2, 5</math> or <math>13</math>. Show that one can find distinct <math>a,b</math> in the set <math>\{2,5,13,d\}</math> such that <math>ab-1</math> is not a perfect square.
 
Let <math>d</math> be any positive integer not equal to <math>2, 5</math> or <math>13</math>. Show that one can find distinct <math>a,b</math> in the set <math>\{2,5,13,d\}</math> such that <math>ab-1</math> is not a perfect square.
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[[1986 IMO Problems/Problem 1|Solution]]
  
 
=== Problem 2 ===
 
=== Problem 2 ===
  
 
Given a point <math>P_0</math> in the plane of the triangle <math>A_1A_2A_3</math>. Define  <math>A_s=A_{s-3}</math> for all <math>s\ge4</math>. Construct a set of points <math>P_1,P_2,P_3,\ldots</math> such that <math>P_{k+1}</math> is the image of <math>P_k</math> under a rotation center <math>A_{k+1}</math> through an angle <math>120^o</math> clockwise for <math>k=0,1,2,\ldots</math>. Prove that if <math>P_{1986}=P_0</math>, then the triangle <math>A_1A_2A_3</math> is equilateral.
 
Given a point <math>P_0</math> in the plane of the triangle <math>A_1A_2A_3</math>. Define  <math>A_s=A_{s-3}</math> for all <math>s\ge4</math>. Construct a set of points <math>P_1,P_2,P_3,\ldots</math> such that <math>P_{k+1}</math> is the image of <math>P_k</math> under a rotation center <math>A_{k+1}</math> through an angle <math>120^o</math> clockwise for <math>k=0,1,2,\ldots</math>. Prove that if <math>P_{1986}=P_0</math>, then the triangle <math>A_1A_2A_3</math> is equilateral.
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[[1986 IMO Problems/Problem 2|Solution]]
  
 
=== Problem 3 ===
 
=== Problem 3 ===
  
 
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers <math>x,y,z</math> respectively, and <math>y<0</math>, then the following operation is allowed: <math>x,y,z</math> are replaced by <math>x+y,-y,z+y</math> respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
 
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers <math>x,y,z</math> respectively, and <math>y<0</math>, then the following operation is allowed: <math>x,y,z</math> are replaced by <math>x+y,-y,z+y</math> respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
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[[1986 IMO Problems/Problem 3|Solution]]
  
 
== Day 2 ==
 
== Day 2 ==
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Let <math>A,B</math> be adjacent vertices of a regular <math>n</math>-gon (<math>n\ge5</math>) with center <math>O</math>. A triangle <math>XYZ</math>, which is congruent to and initially coincides with <math>OAB</math>, moves in the plane in such a way that <math>Y</math> and <math>Z</math> each trace out the whole boundary of the polygon, with <math>X</math> remaining inside the polygon. Find the locus of <math>X</math>.
 
Let <math>A,B</math> be adjacent vertices of a regular <math>n</math>-gon (<math>n\ge5</math>) with center <math>O</math>. A triangle <math>XYZ</math>, which is congruent to and initially coincides with <math>OAB</math>, moves in the plane in such a way that <math>Y</math> and <math>Z</math> each trace out the whole boundary of the polygon, with <math>X</math> remaining inside the polygon. Find the locus of <math>X</math>.
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[[1986 IMO Problems/Problem 4|Solution]]
  
 
=== Problem 5 ===
 
=== Problem 5 ===
  
 
Find all functions <math>f</math> defined on the non-negative reals and taking non-negative real values such that: <math>f(2)=0,f(x)\ne0</math> for <math>0\le x<2</math>, and <math>f(xf(y))f(y)=f(x+y)</math> for all <math>x,y</math>.
 
Find all functions <math>f</math> defined on the non-negative reals and taking non-negative real values such that: <math>f(2)=0,f(x)\ne0</math> for <math>0\le x<2</math>, and <math>f(xf(y))f(y)=f(x+y)</math> for all <math>x,y</math>.
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[[1986 IMO Problems/Problem 5|Solution]]
  
 
=== Problem 6 ===
 
=== Problem 6 ===
  
 
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line <math>L</math> parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on <math>L</math> is not greater than <math>1</math>?
 
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line <math>L</math> parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on <math>L</math> is not greater than <math>1</math>?
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[[1986 IMO Problems/Problem 6|Solution]]
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* [[1986 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1986 IMO 1986 Problems on the Resources page]
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]]
  
 
{{IMO box|year=1986|before=[[1985 IMO]]|after=[[1987 IMO]]}}
 
{{IMO box|year=1986|before=[[1985 IMO]]|after=[[1987 IMO]]}}

Latest revision as of 23:04, 29 January 2021

Day 1

Problem 1

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Solution

Problem 2

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

Solution

Problem 3

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Solution

Day 2

Problem 4

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Solution

Problem 5

Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.

Solution

Problem 6

Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

Solution

1986 IMO (Problems) • Resources
Preceded by
1985 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1987 IMO
All IMO Problems and Solutions
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