[obm-l] Problemas da IMO

[gugu@xxxxxxx]

Prova da IMO retirada do Site http://www.mathlinks.go.ro/

O Problema 1 é nois que mandou...


First Day - 44th IMO 2003 Japan 

1. Let A be a 101-element subset of the set S={1,2,3,...,1000000}. Prove that 
there exist numbers t_1, t_2, ..., t_{100} in S such that the sets 

Aj = { x + tj | x is in A } for each j = 1, 2, ..., 100 

are pairwise disjoint. 


2. Find all pairs of positive integers (a,b) such that the number 

a^2 / ( 2ab^2-b^3+1) is also a positive integer. 

3. Given is a convex hexagon with the property that the segment connecting the 
middle points of each pair of opposite sides in the hexagon is  sqrt(3) / 2 
times the sum of those sides' sum. 

Prove that the hexagon has all its angles equal to 120. 


Second Day - 44th IMO 2003 Japan 

4. Given is a cyclic quadrilateral ABCD and let P, Q, R be feet of the 
altitudes from D to AB, BC and CA respectively. Prove that if PR = RQ then the 
interior angle bisectors of the angles < ABC and < ADC are concurrent on AC. 

5. Let x1 <= x2 <= ... <= xn be real numbers, n>2. 

a) Prove the following inequality: 

(sum  ni,j=1 | xi - xj | ) 2 <= 2/3 ( n^2 - 1 )sum ni,j=1 ( xi - xj)^2 

b) Prove that the equality in the inequality above is obtained if and only if 
the sequence (xk) is an arithemetical progression. 

6. Prove that for each given prime p there exists a prime q such that n^p - p 
is not divisible by q for each positive integer n. 



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