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Difference between revisions of "2002 IMO Problems/Problem 1"

(Created page with "<math>S</math> is the set of all <math>(h,k)</math> with <math>h,k</math> non-negative integers such that <math>h + k < n</math>. Each element of <math>S</math> is colored red...")
 
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<math>S</math> is the set of all <math>(h,k)</math> with <math>h,k</math> non-negative integers such that <math>h + k < n</math>. Each element of <math>S</math> is colored red or blue, so that if <math>(h,k)</math> is red and <math>h′ ≤ h,k′ ≤ k</math>, then <math>(h′,k′)</math> is also red. A type <math>1</math> subset of <math>S</math> has <math>n</math> blue elements with different first member and a type <math>2</math> subset of <math>S</math> has <math>n</math> blue elements with different second member. Show that there are the same number of type <math>1</math> and type <math>2</math> subsets.
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<math>S</math> is the set of all <math>(h,k)</math> with <math>h,k</math> non-negative integers such that <math>h + k < n</math>. Each element of <math>S</math> is colored red or blue, so that if <math>(h,k)</math> is red and <math>h' \le h,k' \le k</math>, then <math>(h',k')</math> is also red. A type <math>1</math> subset of <math>S</math> has <math>n</math> blue elements with different first member and a type <math>2</math> subset of <math>S</math> has <math>n</math> blue elements with different second member. Show that there are the same number of type <math>1</math> and type <math>2</math> subsets.

Revision as of 00:25, 19 November 2023

$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h + k < n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h' \le h,k' \le k$, then $(h',k')$ is also red. A type $1$ subset of $S$ has $n$ blue elements with different first member and a type $2$ subset of $S$ has $n$ blue elements with different second member. Show that there are the same number of type $1$ and type $2$ subsets.

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