USAMO 2003
by utkarshgupta, Aug 7, 2015, 2:32 PM
So here are the problems I recently tried.
These proofs are informal (incomplete to be more precise)
Problem 1
Prove that for every positive integer
there exists an -digit number divisible by 
all of whose digits are odd.
Problem 2 (USAMO 2003) :
A convex polygon
in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
Problem 5 :
Let
, 
, 
be positive real numbers. Prove that
![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.
\]](//latex.artofproblemsolving.com/0/5/e/05e90cc0c9d60f8943608b690f75d20bcd4f6220.png)
These proofs are informal (incomplete to be more precise)
Problem 1
Prove that for every positive integer
there exists an -digit number divisible by 
all of whose digits are odd.
is trivial.
be such an 
mod 
for some 
where 
mod 
where 
appropriately (depending on 
Problem 2 (USAMO 2003) :
A convex polygon
in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
be some diagonal of the polygon.
with 
is a rational polygon.
is a rational number.
is a rational number.
is a rational number.Problem 5 :
Let
, 
, 
be positive real numbers. Prove that![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.
\]](http://latex.artofproblemsolving.com/0/5/e/05e90cc0c9d60f8943608b690f75d20bcd4f6220.png)
This post has been edited 3 times. Last edited by utkarshgupta, Aug 7, 2015, 2:49 PM
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