Difference between revisions of "2015 UMO Problems/Problem 5"

(Created page with "==Problem == Find, with proof, all positive integers <math>n</math> with <math>2\le n\le 20</math> such that the greatest common divisor of the coefficients of <math>(x+y)^n-...")
 
 
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==Problem ==
 
==Problem ==
  
Find, with proof, all positive integers <math>n</math> with <math>2\le n\le 20</math> such that the greatest common divisor of the coefficients of <math>(x+y)^n-x^n-y^n</math>
+
A <math>3 \times 3</math> grid is filled with integers (positive or negative) such that the product of the integers
is equal to exactly <math>3</math>.
+
in any row or column is equal to <math>20</math>. For example, one possible grid is:
  
 +
<math>\begin{bmatrix}
 +
  1 & -5& -4 \
 +
    10  & -2 &-1 \
 +
  2& 2& 5
 +
\end{bmatrix}</math>
  
 +
In how many ways can this be done?
  
 
== Solution ==
 
== Solution ==
  
 
== See Also ==
 
== See Also ==
{{UMO box|year=2015|num-b=2|num-a=4}}
+
{{UMO box|year=2015|num-b=4|num-a=6}}
  
[[Category:]]
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[[Category:Intermediate Combinatorics Problems]]

Latest revision as of 02:00, 6 November 2015

Problem

A $3 \times 3$ grid is filled with integers (positive or negative) such that the product of the integers in any row or column is equal to $20$. For example, one possible grid is:

$\begin{bmatrix}    1 & -5& -4 \\     10  & -2 &-1 \\    2& 2& 5 \end{bmatrix}$

In how many ways can this be done?

Solution

See Also

2015 UMO (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6
All UMO Problems and Solutions