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Difference between revisions of "1981 IMO Problems"

 
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=== Problem 3 ===
 
=== Problem 3 ===
  
−
Determine the maximum value of <math> \displaystyle m^3 + n^3 </math>, where <math> \displaystyle m </math> and <math> \displaystyle n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math> \displaystyle ( n^2 - mn - m^2 )^2 = 1 </math>.
+
Determine the maximum value of <math> \displaystyle m^2 + n^2 </math>, where <math> \displaystyle m </math> and <math> \displaystyle n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math> \displaystyle ( n^2 - mn - m^2 )^2 = 1 </math>.
  
 
[[1981 IMO Problems/Problem 3 | Solution]]
 
[[1981 IMO Problems/Problem 3 | Solution]]
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=== Problem 4 ===
 
=== Problem 4 ===
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 +
(a) For which values of <math> \displaystyle n>2</math> is there a set of <math>\displaystyle n</math> consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining <math>\displaystyle n-1</math> numbers?
 +
 +
(b) For which values of <math>\displaystyle n>2</math> is there exactly one set having the stated property?
  
 
[[1981 IMO Problems/Problem 4 | Solution]]
 
[[1981 IMO Problems/Problem 4 | Solution]]
  
 
=== Problem 5 ===
 
=== Problem 5 ===
 +
 +
Three congruent circles have a common point <math> \displaystyle O </math> and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point <math>\displaystyle O </math> are collinear.
  
 
[[1981 IMO Problems/Problem 5 | Solution]]
 
[[1981 IMO Problems/Problem 5 | Solution]]
  
 
=== Problem 6 ===
 
=== Problem 6 ===
 +
 +
The function <math>\displaystyle f(x,y)</math> satisfies
 +
 +
(1) <math> \displaystyle f(0,y)=y+1, </math>
 +
 +
(2) <math> \displaystyle f(x+1,0)=f(x,1), </math>
 +
 +
(3) <math> \displaystyle f(x+1,y+1)=f(x,f(x+1,y)), </math>
 +
 +
for all non-negative integers <math> \displaystyle x,y </math>. Determine <math> \displaystyle f(4,1981) </math>.
  
 
[[1981 IMO Problems/Problem 6 | Solution]]
 
[[1981 IMO Problems/Problem 6 | Solution]]

Latest revision as of 14:09, 29 October 2006

Problems of the 22nd IMO 1981 U.S.A.

Day I

Problem 1

$\displaystyle P$ is a point inside a given triangle $\displaystyle ABC$. $\displaystyle D, E, F$ are the feet of the perpendiculars from $\displaystyle P$ to the lines $\displaystyle BC, CA, AB$, respectively. Find all $\displaystyle P$ for which

$\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$

is least.

Solution

Problem 2

Let $\displaystyle 1 \le r \le n$ and consider all subsets of $\displaystyle r$ elements of the set $\{ 1, 2, \ldots , n \}$. Each of these subsets has a smallest member. Let $\displaystyle F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that

$F(n,r) = \frac{n+1}{r+1}.$

Solution

Problem 3

Determine the maximum value of $\displaystyle m^2 + n^2$, where $\displaystyle m$ and $\displaystyle n$ are integers satisfying $m, n \in \{ 1,2, \ldots , 1981 \}$ and $\displaystyle ( n^2 - mn - m^2 )^2 = 1$.

Solution

Day II

Problem 4

(a) For which values of $\displaystyle n>2$ is there a set of $\displaystyle n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $\displaystyle n-1$ numbers?

(b) For which values of $\displaystyle n>2$ is there exactly one set having the stated property?

Solution

Problem 5

Three congruent circles have a common point $\displaystyle O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $\displaystyle O$ are collinear.

Solution

Problem 6

The function $\displaystyle f(x,y)$ satisfies

(1) $\displaystyle f(0,y)=y+1,$

(2) $\displaystyle f(x+1,0)=f(x,1),$

(3) $\displaystyle f(x+1,y+1)=f(x,f(x+1,y)),$

for all non-negative integers $\displaystyle x,y$. Determine $\displaystyle f(4,1981)$.

Solution

Resources

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