Difference between revisions of "2017 AMC 12A Problems/Problem 21"
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At first, <math>S=\{0,10\}</math>. | At first, <math>S=\{0,10\}</math>. | ||
− | + | <cmath>\begin{align*} | |
− | < | + | 10x+10 & \quad\text{has root}\quad x=-1, & \quad\text{so now}\quad & S=\{-1,0,10\}. \\ |
− | + | -x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+10 & \quad\text{has root}\quad x=1, & \quad\text{so now}\quad & S=\{-1,0,1,10\}. \\ | |
− | + | x+10 & \quad\text{has root}\quad x=-10, & \quad\text{so now}\quad & S=\{-10,-1,0,1,10\}. \\ | |
− | + | x^4-x^2-x+10 & \quad\text{has root}\quad x=2, & \quad\text{so now}\quad & S=\{-10,-1,0,1,2,10\}. \\ | |
− | + | x^4-x^2+x+10 & \quad\text{has root}\quad x=-2, & \quad\text{so now}\quad & S=\{-10,-2,-1,0,1,2,10\}. \\ | |
− | + | 2x-10 & \quad\text{has root}\quad x=5, & \quad\text{so now}\quad & S=\{-10,-2,-1,0,1,2,5,10\}. \\ | |
− | + | 2x+10 & \quad\text{has root}\quad x=-5, & \quad\text{so now}\quad & S=\{-10,-5,-2,-1,0,1,2,5,10\}. \\ | |
− | + | \end{align*}</cmath> | |
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At this point, no more elements can be added to <math>S</math>. To see this, let | At this point, no more elements can be added to <math>S</math>. To see this, let |
Revision as of 08:58, 9 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, .
At this point, no more elements can be added to . To see this, let
with each in
.
is a factor of
, and
is in
, so
has to be a factor of some element in
. There are no such integers left, so there can be no more additional elements.
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.