Difference between revisions of "2020 AMC 10A Problems/Problem 14"
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− | ==Problem== | + | == Problem == |
− | Real numbers <math>x</math> and <math>y</math> satisfy <math>x + y = 4</math> and <math>x \cdot y = -2</math>. What is the value of<cmath>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?</cmath> | + | Real numbers <math>x</math> and <math>y</math> satisfy <math>x + y = 4</math> and <math>x \cdot y = -2</math>. What is the value of |
+ | |||
+ | <cmath>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?</cmath> | ||
+ | |||
<math>\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480</math> | <math>\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480</math> | ||
− | == Solution == | + | == Solutions == |
+ | |||
+ | === Solution 1 === | ||
<cmath>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y=x+\frac{x^3}{y^2}+y+\frac{y^3}{x^2}=\frac{x^3}{x^2}+\frac{y^3}{x^2}+\frac{y^3}{y^2}+\frac{x^3}{y^2}</cmath> | <cmath>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y=x+\frac{x^3}{y^2}+y+\frac{y^3}{x^2}=\frac{x^3}{x^2}+\frac{y^3}{x^2}+\frac{y^3}{y^2}+\frac{x^3}{y^2}</cmath> | ||
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From the givens, it can be concluded that <math>x^2y^2=4</math>. Also, <cmath>(x+y)^2=x^2+2xy+y^2=16</cmath> This means that <math>x^2+y^2=20</math>. Substituting this information into <math>\frac{(x^2+y^2)(x+y)(x^2-xy+y^2)}{x^2y^2}</math>, we have <math>\frac{(20)(4)(22)}{4}=20\cdot 22=\boxed{\textbf{(D)}\ 440}</math>. ~PCChess | From the givens, it can be concluded that <math>x^2y^2=4</math>. Also, <cmath>(x+y)^2=x^2+2xy+y^2=16</cmath> This means that <math>x^2+y^2=20</math>. Substituting this information into <math>\frac{(x^2+y^2)(x+y)(x^2-xy+y^2)}{x^2y^2}</math>, we have <math>\frac{(20)(4)(22)}{4}=20\cdot 22=\boxed{\textbf{(D)}\ 440}</math>. ~PCChess | ||
− | == Solution 2 == | + | === Solution 2 === |
− | As above, we need to calculate <math>\frac{(x^2+y^2)(x^3+y^3)}{x^2y^2}</math>. Note that <math>x,y,</math> are the roots of <math>x^2-4x-2</math> and so <math>x^3=4x^2+2x</math> and <math>y^3=4y^2+2y</math>. Thus <math>x^3+y^3=4(x^2+y^2)+2(x+y)=4(20)+2(4)=88</math> where <math>x^2+y^2=20</math> and <math>x^2y^2=4</math> as in the previous solution. Thus the answer is <math>\frac{(20)(88)}{4}=\boxed{\textbf{(D)}\ 440}</math>. | + | As above, we need to calculate <math>\frac{(x^2+y^2)(x^3+y^3)}{x^2y^2}</math>. Note that <math>x,y,</math> are the roots of <math>x^2-4x-2</math> and so <math>x^3=4x^2+2x</math> and <math>y^3=4y^2+2y</math>. Thus <math>x^3+y^3=4(x^2+y^2)+2(x+y)=4(20)+2(4)=88</math> where <math>x^2+y^2=20</math> and <math>x^2y^2=4</math> as in the previous solution. Thus the answer is <math>\frac{(20)(88)}{4}=\boxed{\textbf{(D)}\ 440}</math>. Note(<math>x^2+y^2=(x+y)^2-2xy=20</math>, and <math>x^2y^2 = (xy)^2 = 4</math>) |
− | == Solution 3 == | + | === Solution 3 === |
Note that <math>( x^3 + y^3 ) ( \frac{1}{y^2} + \frac{1}{x^2} ) = x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y.</math> Now, we only need to find the values of <math>x^3 + y^3</math> and <math>\frac{1}{y^2} + \frac{1}{x^2}.</math> <br> | Note that <math>( x^3 + y^3 ) ( \frac{1}{y^2} + \frac{1}{x^2} ) = x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y.</math> Now, we only need to find the values of <math>x^3 + y^3</math> and <math>\frac{1}{y^2} + \frac{1}{x^2}.</math> <br> | ||
<br> | <br> | ||
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Solving the original equation, we get <math>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y = (88)(5) = \boxed{\textbf{(D)}\ 440}.</math> ~[[User:emerald_block|emerald_block]] | Solving the original equation, we get <math>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y = (88)(5) = \boxed{\textbf{(D)}\ 440}.</math> ~[[User:emerald_block|emerald_block]] | ||
− | + | === Solution 4 (Bashing) === | |
− | ==Solution 4 (Bashing)== | ||
This is basically bashing using Vieta's formulas to find <math> x </math> and <math> y </math> (which I highly do not recommend, I only wrote this solution for fun). | This is basically bashing using Vieta's formulas to find <math> x </math> and <math> y </math> (which I highly do not recommend, I only wrote this solution for fun). | ||
− | + | We use Vieta's to find a quadratic relating <math> x </math> and <math> y </math>. We set <math> x </math> and <math> y </math> to be the roots of the quadratic <math> Q ( n ) = n^2 - 4n - 2 </math> (because <math> x + y = 4 </math>, and <math> xy = -2 </math>). We can solve the quadratic to get the roots <math> 2 + \sqrt{6} </math> and <math> 2 - \sqrt{6} </math>. <math> x </math> and <math> y </math> are "interchangeable" (WLOG, without loss of generality, we could have <math>x</math> equal one and <math>y</math> equal the other), meaning that it doesn't matter which solution <math> x </math> or <math> y </math> is, because it'll return the same result when plugged in. So we plug in <math> 2 + \sqrt{6} </math> for <math> x </math> and <math> 2 - \sqrt{6} </math> and get <math> \boxed{\textbf{(D)}\ 440} </math> as our answer. | |
− | We use Vieta's to find a quadratic relating <math> x </math> and <math> y </math>. We set <math> x </math> and <math> y </math> to be the roots of the quadratic <math> Q ( n ) = n^2 - 4n - 2 </math> (because <math> x + y = 4 </math>, and <math> xy = -2 </math>). We can solve the quadratic to get the roots <math> 2 + \sqrt{6} </math> and <math> 2 - \sqrt{6} </math>. <math> x </math> and <math> y </math> are "interchangeable", meaning that it doesn't matter which solution <math> x </math> or <math> y </math> is, because it'll return the same result when plugged in. So we plug in <math> 2 + \sqrt{6} </math> for <math> x </math> and <math> 2 - \sqrt{6} </math> and get <math> \boxed{\textbf{(D)}\ 440} </math> as our answer. | ||
− | |||
~Baolan | ~Baolan | ||
+ | ~<B+ | ||
− | ==Solution 5 (Bashing Part 2)== | + | === Solution 5 (Bashing Part 2) === |
This usually wouldn't work for most problems like this, but we're lucky that we can quickly expand and factor this expression in this question. | This usually wouldn't work for most problems like this, but we're lucky that we can quickly expand and factor this expression in this question. | ||
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= \boxed{\textbf{(D)} 440} </math>. | = \boxed{\textbf{(D)} 440} </math>. | ||
− | ==Solution 6 (Complete Binomial Theorem)== | + | === Solution 6 (Complete Binomial Theorem) === |
We first simplify the expression to <cmath>x + y + \frac{x^5 + y^5}{x^2y^2}.</cmath> | We first simplify the expression to <cmath>x + y + \frac{x^5 + y^5}{x^2y^2}.</cmath> | ||
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<cmath>4 + \frac{2(872)}{4}</cmath> | <cmath>4 + \frac{2(872)}{4}</cmath> | ||
which equals <math>\boxed{\textbf{(D)} 440}</math>. | which equals <math>\boxed{\textbf{(D)} 440}</math>. | ||
− | |||
~ fidgetboss_4000 | ~ fidgetboss_4000 | ||
− | ==Solution 7== | + | === Solution 7 === |
− | |||
As before, simplify the expression to <cmath>x + y + \frac{x^5 + y^5}{x^2y^2}.</cmath> | As before, simplify the expression to <cmath>x + y + \frac{x^5 + y^5}{x^2y^2}.</cmath> | ||
Since <math>x + y = 4</math> and <math>x^2y^2 = 4</math>, we substitute that in to obtain <cmath> 4 + \frac{x^5 + y^5}{4}.</cmath> | Since <math>x + y = 4</math> and <math>x^2y^2 = 4</math>, we substitute that in to obtain <cmath> 4 + \frac{x^5 + y^5}{4}.</cmath> | ||
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~Binderclips1 | ~Binderclips1 | ||
− | ==Video Solution== | + | |
+ | === Solution 8 === | ||
+ | We can use Newton Sums to solve this problem. | ||
+ | We start by noticing that we can rewrite the equation as <math>\frac{x^3}{y^2} + \frac{y^3}{x^2} + x + y.</math> | ||
+ | Then, we know that <math>x + y = 4,</math> so we have <math>\frac{x^3}{y^2} + \frac{y^3}{x^2} + 4.</math> | ||
+ | We can use the equation <math>x \cdot y = -2</math> to write <math>x = \frac{-2}{y}</math> and <math>y = \frac{-2}{x}.</math> | ||
+ | Next, we can plug in these values of <math>x</math> and <math>y</math> to get <math>\frac{x^3}{y^2} + \frac{y^3}{x^2} = \frac{x^5}{4} + \frac{y^5}{4},</math> which is the same as <cmath>\frac{x^3}{y^2} + \frac{y^3}{x^2} = \frac{x^5 + y^5}{4}.</cmath> | ||
+ | Then, we use Newton sums where <math>S_n</math> is the elementary symmetric sum of the sequence and <math>P_n</math> is the power sum (<math>x^n + y^n</math>). Using this, we can make the following Newton sums: | ||
+ | <cmath>P_1 = S_1</cmath> | ||
+ | <cmath>P_2 = P_1 S_1 - 2S_2</cmath> | ||
+ | <cmath>P_3 = P_2 S_1 - P_1 S_2</cmath> | ||
+ | <cmath>P_4 = P_3 S_1 - P_2 S_2</cmath> | ||
+ | <cmath>P_5 = P_4 S_1 - P_3 S_2.</cmath> | ||
+ | We also know that <math>S_1</math> is 4 because <math>x + y</math> is four, and we know that <math>S_2</math> is <math>-2</math> because <math>x \cdot y</math> is <math>-2</math> as well. | ||
+ | Then, we can plug in values! We have | ||
+ | <cmath>P_1 = S_1 = 4</cmath> | ||
+ | <cmath>P_2 = P_1 S_1 - 2S_2 = 16 - (-4) = 20</cmath> | ||
+ | <cmath>P_3 = P_2 S_1 - P_1 S_2 = 80 - (-8) = 88</cmath> | ||
+ | <cmath>P_4 = P_3 S_1 - P_2 S_2 = 88 \cdot 4 - (-40) = 392</cmath> | ||
+ | <cmath>P_5 = P_4 S_1 - P_3 S_2 = 392 \cdot 4 - (-2) \cdot 88 = 1744.</cmath> | ||
+ | We earlier noted that <math>\frac{x^3}{y^2} + \frac{y^3}{x^2} = \frac{x^5 + y^5}{4},</math> so we have that this equals <math>\frac{1744}{4},</math> or <math>436.</math> Then, plugging this back into the original equation, this is <math>436 + 4</math> or <math>440,</math> so our answer is <math>\boxed{\textbf{(D)}\ 440}.</math> | ||
+ | |||
+ | ~Coolpeep | ||
+ | |||
+ | === Solution 9 === | ||
+ | As in the first solution, we get the expression to be <math>\frac{x^3+y^3}{x^2} + \frac{x^3+y^3}{y^2}.</math> | ||
+ | |||
+ | Then, since the numerators are the same, we can put the two fractions as a common denominator and multiply the numerator by <math>x^2y^2.</math> This gets us <math>\frac{(x^2 + y^2)(x^3 + y^3)}{x^2y^2}.</math> | ||
+ | |||
+ | Now, since we know <math>x+y=4</math> and <math>xy=-2,</math> instead of solving for <math>x</math> and <math>y,</math> we will try to manipulate the above expression them into a manner that we can substitute the sum and product that we know. Also, another form of <math>x^3+y^3</math> is <math>(x+y)(x^2-xy+y^2).</math> | ||
+ | |||
+ | Thus, we can convert the current expression to <math>\frac{(x^2 + y^2)(x+y)(x^2-xy+y^2)}{x^2y^2}.</math> | ||
+ | |||
+ | Doing some algebraic multiplications, we get <math>\frac{((x+y)^2 - 2xy)(x+y)((x+y)^2 - 3xy)}{(xy)^2}.</math> | ||
+ | |||
+ | Since we know <math>x+y=4</math> and <math>xy=-2,</math> we have <math>\frac{(16-(-4))(4)(16-(-6))}{4} = \frac{20 \cdot 4 \cdot 22}{4} = 20 \cdot 22 = 440.</math> | ||
+ | |||
+ | Therefore the answer is <math>\boxed{\textbf{(D)} 440}.</math> | ||
+ | |||
+ | ~mathboy282 | ||
+ | |||
+ | === Solution 10 (Algebra Bash) === | ||
+ | |||
+ | We give all of the terms in this expression a common denominator. <math>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y = \frac{x^3y^2 + x^5 + y^5 + x^2y^3}{x^2y^2}</math>. We can find <math>x^2y^2 = (xy)^2 = (-2)^2 = 4</math> and we can find <math>x^3y^2 + x^2y^3 = x^2y^2(x + y) = 16</math>. Our expression <math>\frac{xy^2 + x^5 + y^5 + x^2y}{x^2y^2}</math> is now <math>\frac{16 + x^5 + y^5}{4}</math>. Now all we need to find is <math>x^5 + y^5</math>. Using the binomial theorem, <math>(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 = 4^5 = 1024</math>. The extra terms that we don't need is <math>5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 = 5xy(x^3 + y^3) + 10x^2y^2(x + y) = -10(x^3 + y^3) + 160</math>. What's <math>x^3 + y^3</math>? We use the same method. Using the binomial theorem, <math>(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3 + y^3 + 3xy(x + y) = x^3 + y^3 - 24 = 4^3 = 64</math>. Now we know that <math>x^3 + y^3 = 64 + 24 = 88</math>, and plugging that into <math>-10(x^3 + y^3) + 160</math> gives <math>-10(88) + 160 = -880 + 160 = -720</math>. Now we see that those extra terms have a sum of <math>-720</math>. Thus <math>(x + y)^5 = x^5 + y^5 - 720 = 1024</math> so <math>x^5 + y^5 = 1024 + 720 = 1744</math>. Remember our goal: we want to find <math>\frac{-8 + x^5 + y^5}{4}</math>. Using <math>x^5 + y^5 = 1744</math>, <math>\frac{16 + x^5 + y^5}{4} = \frac{16 + 1744}{4} = \frac{1760}{4} = \boxed{\textbf{(D)}\ 440}</math>. | ||
+ | |||
+ | ~ [https://artofproblemsolving.com/wiki/index.php/User:Yrock Yrock] | ||
+ | |||
+ | === Solution 11=== | ||
+ | |||
+ | Since we know <math>x+y=4</math>, we can simplify to <math>4 + \frac{x^3}{y^2} + \frac{y^3}{x^2}</math>. Then, when you add the fractions, you get <math>\frac{x^5 + y^5}{x^2y^2} = \frac{x^5 + y^5}{4}</math>. To find <math>x^5 + y^5</math>, we expand <math>(x+y)^5 = 4^5</math> and then simplify. <math>x^5+y^5+5x^4y+5xy^4+10x^3y^2+10x^2y^3 = 1024</math>. We can use the fact that <math>xy = -2</math> to simplify to <math>x^5 + y^5-10x^3 - 10y^3 + 40x + 40y = 1024</math>. We can factor to get <math>x^5 + y^5 -10(x^3 + y^3) +40(x+y)</math>. To find <math>x^3 + y^3</math>, we simplify <math>(x+y)^3 = 4^3</math>. <math>x^3+y^3+3x^2y+3y^2x = 64</math>. We can use the same process as we did before and get <math>x^3 + y^3 -6(x+y) = 64</math>(using the fact that <math>xy = -2</math>). Therefore, <math>x^3 + y^3 = 88</math>. Now, plugging it back in to our other expansion, we get <math>x^5 + y^5 + 160 - 880 = 1024</math>. <math>x^5 +y^5 = 1744</math>. Now to get the final answer, all we need to do is to plug it back into the original equation and get <math>\frac{1744}{4} + 4 = \boxed{\textbf{(D)}\ 440}</math>. | ||
+ | |||
+ | ~idk12345678 | ||
+ | |||
+ | == Video Solutions == | ||
+ | |||
+ | === Video Solution 1 === | ||
+ | |||
+ | https://www.youtube.com/watch?v=x4cF3o3Fzj8&t=376s | ||
+ | |||
+ | ~Education, The Study of Everything | ||
+ | |||
+ | === Video Solution 2 === | ||
+ | |||
+ | https://youtu.be/PNkRlUKWCzg | ||
+ | |||
+ | === Video Solution 3 === | ||
+ | |||
+ | https://www.youtube.com/watch?v=jlRmDrL_jmk | ||
+ | ~Mathematical Dexterity (Don't Worry, Be Hoppy!) | ||
+ | |||
+ | === Video Solution 4 === | ||
+ | |||
https://youtu.be/ZGwAasE32Y4 | https://youtu.be/ZGwAasE32Y4 | ||
~IceMatrix | ~IceMatrix | ||
− | == | + | === Video Solution 5 === |
+ | |||
+ | https://youtu.be/XEtzvxfFEJk | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | === Video Solution 6 === | ||
+ | |||
+ | https://youtu.be/ba6w1OhXqOQ?t=3551 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | == See Also == | ||
{{AMC10 box|year=2020|ab=A|num-b=13|num-a=15}} | {{AMC10 box|year=2020|ab=A|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 15:18, 15 August 2024
Contents
Problem
Real numbers and
satisfy
and
. What is the value of
Solutions
Solution 1
Continuing to combine
From the givens, it can be concluded that
. Also,
This means that
. Substituting this information into
, we have
. ~PCChess
Solution 2
As above, we need to calculate . Note that
are the roots of
and so
and
. Thus
where
and
as in the previous solution. Thus the answer is
. Note(
, and
)
Solution 3
Note that Now, we only need to find the values of
and
Recall that and that
We are able to solve the second equation, and doing so gets us
Plugging this into the first equation, we get
In order to find the value of we find a common denominator so that we can add them together. This gets us
Recalling that
and solving this equation, we get
Plugging this into the first equation, we get
Solving the original equation, we get ~emerald_block
Solution 4 (Bashing)
This is basically bashing using Vieta's formulas to find and
(which I highly do not recommend, I only wrote this solution for fun).
We use Vieta's to find a quadratic relating and
. We set
and
to be the roots of the quadratic
(because
, and
). We can solve the quadratic to get the roots
and
.
and
are "interchangeable" (WLOG, without loss of generality, we could have
equal one and
equal the other), meaning that it doesn't matter which solution
or
is, because it'll return the same result when plugged in. So we plug in
for
and
and get
as our answer.
~Baolan
~<B+
Solution 5 (Bashing Part 2)
This usually wouldn't work for most problems like this, but we're lucky that we can quickly expand and factor this expression in this question.
We first change the original expression to , because
. This is equal to
. We can factor and reduce
to
. Now our expression is just
. We factor
to get
. So the answer would be
.
Solution 6 (Complete Binomial Theorem)
We first simplify the expression to
Then, we can solve for
and
given the system of equations in the problem.
Since
we can substitute
for
.
Thus, this becomes the equation
Multiplying both sides by
, we obtain
or
By the quadratic formula we obtain
.
We also easily find that given
,
equals the conjugate of
.
Thus, plugging our values in for
and
, our expression equals
By the binomial theorem, we observe that every second terms of the expansions
and
will cancel out (since a positive plus a negative of the same absolute value equals zero). We also observe that the other terms not canceling out are doubled when summing the expansions of
.
Thus, our expression equals
which equals
which equals
.
~ fidgetboss_4000
Solution 7
As before, simplify the expression to
Since
and
, we substitute that in to obtain
Now, we must solve for
. Start by squaring
, to obtain
Simplifying,
. Squaring once more, we obtain
Once again simplifying,
. Now, to obtain the fifth powers of
and
, we multiply both sides by
.
We now have
, or
We now solve for
.
, so
.
Plugging this back into
, we find that
, so we have
. This equals 440, so our answer is
.
~Binderclips1
Solution 8
We can use Newton Sums to solve this problem.
We start by noticing that we can rewrite the equation as
Then, we know that
so we have
We can use the equation
to write
and
Next, we can plug in these values of
and
to get
which is the same as
Then, we use Newton sums where
is the elementary symmetric sum of the sequence and
is the power sum (
). Using this, we can make the following Newton sums:
We also know that
is 4 because
is four, and we know that
is
because
is
as well.
Then, we can plug in values! We have
We earlier noted that
so we have that this equals
or
Then, plugging this back into the original equation, this is
or
so our answer is
~Coolpeep
Solution 9
As in the first solution, we get the expression to be
Then, since the numerators are the same, we can put the two fractions as a common denominator and multiply the numerator by This gets us
Now, since we know and
instead of solving for
and
we will try to manipulate the above expression them into a manner that we can substitute the sum and product that we know. Also, another form of
is
Thus, we can convert the current expression to
Doing some algebraic multiplications, we get
Since we know and
we have
Therefore the answer is
~mathboy282
Solution 10 (Algebra Bash)
We give all of the terms in this expression a common denominator. . We can find
and we can find
. Our expression
is now
. Now all we need to find is
. Using the binomial theorem,
. The extra terms that we don't need is
. What's
? We use the same method. Using the binomial theorem,
. Now we know that
, and plugging that into
gives
. Now we see that those extra terms have a sum of
. Thus
so
. Remember our goal: we want to find
. Using
,
.
~ Yrock
Solution 11
Since we know , we can simplify to
. Then, when you add the fractions, you get
. To find
, we expand
and then simplify.
. We can use the fact that
to simplify to
. We can factor to get
. To find
, we simplify
.
. We can use the same process as we did before and get
(using the fact that
). Therefore,
. Now, plugging it back in to our other expansion, we get
.
. Now to get the final answer, all we need to do is to plug it back into the original equation and get
.
~idk12345678
Video Solutions
Video Solution 1
https://www.youtube.com/watch?v=x4cF3o3Fzj8&t=376s
~Education, The Study of Everything
Video Solution 2
Video Solution 3
https://www.youtube.com/watch?v=jlRmDrL_jmk ~Mathematical Dexterity (Don't Worry, Be Hoppy!)
Video Solution 4
~IceMatrix
Video Solution 5
~savannahsolver
Video Solution 6
https://youtu.be/ba6w1OhXqOQ?t=3551
~ pi_is_3.14
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |
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