Back-to-back USAMO and IMO problems in one day!
by shiningsunnyday, Jun 23, 2016, 2:47 PM
1997 IMO P4 wrote:
An 
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that silver matrices exist for infinitely many values of 
.
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that silver matrices exist for infinitely many values of 
.
it's trivial, and for 
does the trick. After some experimenting, 
works with 
(made by Interm CP)
Similarly, the bottom-left 
matrix is the same except with the 
being replaced by a 
with the assumption that we can find a 
silver matrix. We construct a 
by first inserting the 
being added to each entry of the original 
term replaced by a 
and column 
(by the inductive hypothesis), the upper right and bottom left matrices will consist of 
This however, excludes 
which can be easily fixed by replacing each 
in the bottom left matrix with 
For example, this algorithm renders us the following 
silver matrix: 
Applying this algorithm, the inductive step is complete.
P33 of 101 wrote:
Let 
be distinct positive integers. Find the maximum value of 
where 
is a real number within 
be distinct positive integers. Find the maximum value of 
where 
is a real number within 
which we can clean up a bit by substituting 
![$$\Rightarrow x^n(1-x^{m-n})=[y^n(1-y)^{m-n}]^{\frac{1}{m-n}}.$$](http://latex.artofproblemsolving.com/a/5/9/a59276b1156e186e81a054ecb99dfdbd704562c3.png)
We wish to apply 
,
which, unfortunately, doesn't get rid of the 
term due to the lack of symmetry. Instead, consider what will happen if we apply ![$$[(m-n)y]^n [n(1-y)]^{m-n} \le \left(\frac{n(m-n)y+n(m-n)(1-y)}{m} \right)^m=\left(\frac{n(m-n))}{m} \right)^m.$$](http://latex.artofproblemsolving.com/b/d/9/bd9ed060dad044996256af8dadb3aa0f1f13c047.png)
Now we're home, since the previous term is the original expression multiplied by 
Therefore, the answer is 
This post has been edited 1 time. Last edited by shiningsunnyday, Jun 23, 2016, 2:48 PM
May 2017
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