Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 9"
(Created page with "== Problem == == Solution == == See also == {{UNCO Math Contest box|year=2018|n=II|num-b=8|num-a=10}} [[Category:]]") |
(→Problem) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | + | Call a set of integers Grassilian if each of its elements is at least as large as the number of | |
| − | + | elements in the set. For example, the three-element set <math>\{2, 48, 100\}</math> is not Grassilian, but the | |
| + | six-element set <math>\{6, 10, 11, 20, 33, 39\}</math> is Grassilian. Let <math>G(n)</math> be the number of Grassilian subsets of <math>\{1, 2, 3, ..., n\}</math>. (By definition, the empty set is a subset of every set and is Grassilian.) | ||
| + | (a) Find <math>G(3)</math>, <math>G(4)</math>, and <math>G(5)</math>. | ||
| + | (b) Find a recursion formula for <math>G(n + 1)</math>. That is, find a formula that expresses <math>G(n + 1)</math> in | ||
| + | terms of <math>G(n), G(n 1),\;dots</math> | ||
| + | (c) Give an explanation that shows that the formula you give is correct. | ||
== Solution == | == Solution == | ||
Revision as of 01:31, 14 January 2019
Problem
Call a set of integers Grassilian if each of its elements is at least as large as the number of
elements in the set. For example, the three-element set 
is not Grassilian, but the
six-element set 
is Grassilian. Let 
be the number of Grassilian subsets of 
. (By definition, the empty set is a subset of every set and is Grassilian.)
(a) Find 
, 
, and 
.
(b) Find a recursion formula for 
. That is, find a formula that expresses in
terms of 
(c) Give an explanation that shows that the formula you give is correct.
Solution
See also
| 2018 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
[[Category:]]
