Difference between revisions of "2009 AMC 12B Problems/Problem 19"
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x^2-20x+20=1 | x^2-20x+20=1 | ||
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x^2-20x+19=0 | x^2-20x+19=0 | ||
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(x-1)(x-19)=0 | (x-1)(x-19)=0 | ||
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x=1, x=19 | x=1, x=19 |
Revision as of 22:49, 7 February 2011
Contents
Problem
For each positive integer , let
. What is the sum of all values of
that are prime numbers?
Solution
Solution 1
To find the answer it was enough to play around with . One can easily find that
is a prime, then
becomes negative for
between
and
, and then
is again a prime number. And as
is already the largest option, the answer must be
.
Solution 2
We will now show a complete solution, with a proof that no other values are prime.
Consider the function , then obviously
.
The roots of are:
We can then write , and thus
.
We would now like to factor the right hand side further, using the formula . To do this, we need to express both constants as squares of some other constants. Luckily, we have a pretty good idea how they look like.
We are looking for rational and
such that
. Expanding the left hand side and comparing coefficients, we get
and
. We can easily guess (or compute) the solution
,
.
Hence , and we can easily verify that also
.
We now know the complete factorization of :
As the final step, we can now combine the factors in a different way, in order to get rid of the square roots.
We have ,
and
.
Hence we obtain the factorization .
For both terms are positive and larger than one, hence
is not prime. For
the second factor is positive and the first one is negative, hence
is not a prime. The remaining cases are
and
. In both cases,
is indeed a prime, and their sum is
.
Solution 3
Instead of doing the hard work, we can try to guess the factorization. One good approach:
We can make the observation that looks similar to
with the exception of the
term. In fact, we have
. But then we notice that it differs from the desired expression by a square:
.
Now we can use the formula to obtain the same factorization as in the previous solution, without all the work.
Solution 4
After arriving at the factorization , a more mathematical approach would be to realize that the second factor is always positive when x is a positive integer. Therefore, in order for
to be prime, the first factor has to be
.
We can set it equal to 1 and solve for :
Substituting these values into the second factor and adding would give the answer.
See Also
2009 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 |