Difference between revisions of "2002 IMO Problems/Problem 1"
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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. | <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. | ||
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+ | ==Solution== | ||
+ | {{solution}} | ||
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+ | ==See Also== | ||
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+ | {{IMO box|year=2002|before=First Question|num-a=2}} |
Revision as of 00:26, 19 November 2023
Problem
is the set of all
with
non-negative integers such that
. Each element of
is colored red or blue, so that if
is red and
, then
is also red. A type
subset of
has
blue elements with different first member and a type
subset of
has
blue elements with different second member. Show that there are the same number of type
and type
subsets.
Solution
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See Also
2002 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |