Difference between revisions of "2017 AMC 12A Problems/Problem 21"
(→Solution 1) |
(→Solution 1) |
||
Line 35: | Line 35: | ||
with each <math>a_i</math> in <math>S</math>. | with each <math>a_i</math> in <math>S</math>. | ||
− | Since <math>x</math> and the parenthesized term are both integers, | + | Since <math>x</math> and the parenthesized term are both integers, <math>x</math> must be a factor of <math>a_0</math>. However, <math>a_0</math> is in <math>S</math> and every number in <math>S</math> already has all of its factors in <math>S</math>. Therefore, <math>x</math> must be in <math>S</math> and <math>S</math> cannot be expanded. <math>\{-10,-5,-2,-1,0,1,2,5,10\}</math> has <math>9</math> elements <math>\to \boxed{\textbf{(D)}}</math> |
==See Also== | ==See Also== | ||
{{AMC12 box|year=2017|ab=A|num-b=20|num-a=22}} | {{AMC12 box|year=2017|ab=A|num-b=20|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 23:38, 8 February 2017
Problem
A set is constructed as follows. To begin,
. Repeatedly, as long as possible, if
is an integer root of some polynomial
for some
, all of whose coefficients
are elements of
, then
is put into
. When no more elements can be added to
, how many elements does
have?
Solution 1
At first, .
has root
, so now
.
has root
, so now
.
has root
, so now
.
has root
, so now
.
has root
, so now
.
has root
, so now
.
has root
, so now
.
At this point, no more elements can be added to . To see this, let
=
with each in
.
Since and the parenthesized term are both integers,
must be a factor of
. However,
is in
and every number in
already has all of its factors in
. Therefore,
must be in
and
cannot be expanded.
has
elements
See Also
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.