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Difference between revisions of "1989 AIME Problems/Problem 8"

(→‎Solution 3 (Finite Differences by Arithmetic))
(→‎Solution 7 (Similar to Solutions 3 and 4))
 
(34 intermediate revisions by 6 users not shown)
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== Solution 1 (Quadratic Function) ==
 
== Solution 1 (Quadratic Function) ==
−
Note that each equation is of the form <cmath>f(k)=k^2x_1+(k+1)^2x_2+(k+2)^2x_3+(k+3)^2x_4+(k+4)^2x_5+(k+5)^2x_6+(k+6)^2x_7,</cmath> for some <math>k\in\{1,2,3\}.</math>
+
Note that each given equation is of the form <cmath>f(k)=k^2x_1+(k+1)^2x_2+(k+2)^2x_3+(k+3)^2x_4+(k+4)^2x_5+(k+5)^2x_6+(k+6)^2x_7</cmath> for some <math>k\in\{1,2,3\}.</math>
  
 
When we expand <math>f(k)</math> and combine like terms, we obtain a quadratic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x_4,x_5,x_6,</math> and <math>x_7.</math>  
 
When we expand <math>f(k)</math> and combine like terms, we obtain a quadratic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x_4,x_5,x_6,</math> and <math>x_7.</math>  
Line 30: Line 30:
 
Finally, the answer is <cmath>f(4)=16a+4b+c=\boxed{334}.</cmath>
 
Finally, the answer is <cmath>f(4)=16a+4b+c=\boxed{334}.</cmath>
  
−
~Azjps (Fundamental Logic)
+
~Azjps ~MRENTHUSIASM
−
 
 
−
~MRENTHUSIASM (Reconstruction)
 
  
 
== Solution 2 (Linear Combination) ==
 
== Solution 2 (Linear Combination) ==
Line 49: Line 47:
 
Finally, applying the Transitive Property to <math>(8)</math> and <math>(9)</math> gives <math>S-234=100,</math> from which <math>S=\boxed{334}.</math>
 
Finally, applying the Transitive Property to <math>(8)</math> and <math>(9)</math> gives <math>S-234=100,</math> from which <math>S=\boxed{334}.</math>
  
−
~Duohead (Fundamental Logic)
+
~Duohead ~MRENTHUSIASM
−
 
 
−
~MRENTHUSIASM (Reconstruction)
 
  
 
== Solution 3 (Finite Differences by Arithmetic) ==
 
== Solution 3 (Finite Differences by Arithmetic) ==
−
Note that the second differences of all quadratic sequences must be constant (but nonzero).
+
Note that the second differences of all quadratic sequences must be constant (but nonzero). One example is the following sequence of perfect squares:
−
 
 
−
One example, the perfect square sequence, is shown below:
 
  
 
<asy>
 
<asy>
Line 63: Line 57:
  
 
for (real i=1; i<=10; ++i) {
 
for (real i=1; i<=10; ++i) {
−
   label("$"+string(i^2)+"$",(i-1,0));
+
   label("\boldmath{$"+string(i^2)+"$}",(i-1,0));
 
}
 
}
  
Line 75: Line 69:
  
 
for (real i=1; i<=9; ++i) {
 
for (real i=1; i<=9; ++i) {
−
   draw((0.05+(i-1),-0.1)--(28/65+(i-1),-0.65),red);
+
   draw((0.1+(i-1),-0.15)--(0.4+(i-1),-0.6),red);
 
}
 
}
  
 
for (real i=1; i<=8; ++i) {
 
for (real i=1; i<=8; ++i) {
−
   draw((37/65+(i-1),-0.85)--(0.95+(i-1),-1.4),red);
+
   draw((0.6+(i-1),-0.9)--(0.9+(i-1),-1.35),red);
 
}
 
}
  
 
for (real i=1; i<=9; ++i) {
 
for (real i=1; i<=9; ++i) {
−
   draw((37/65+(i-1),-0.65)--(0.95+(i-1),-0.1),red);
+
   draw((0.6+(i-1),-0.6)--(0.9+(i-1),-0.15),red);
 
}
 
}
  
 
for (real i=1; i<=8; ++i) {
 
for (real i=1; i<=8; ++i) {
−
   draw((0.05+i,-1.4)--(28/65+i,-0.85),red);
+
   draw((0.1+i,-1.35)--(0.4+i,-0.9),red);
 
}
 
}
  
−
label("First Differences",(-0.75,-0.75),align=W);
+
label("\textbf{First Differences}",(-0.75,-0.75),align=W);
−
label("Second Differences",(-0.75,-1.5),align=W);
+
label("\textbf{Second Differences}",(-0.75,-1.5),align=W);
 
</asy>
 
</asy>
  
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It follows that the second differences of <math>(1,12,123,S)</math> must be constant, as shown below:
 
It follows that the second differences of <math>(1,12,123,S)</math> must be constant, as shown below:
  
−
<b>NO EDIT PLEASE. DIAGRAM WILL BE READY SOON</b>
+
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(10cm);
 +
 
 +
label("\boldmath{$1$}",(0,0));
 +
label("\boldmath{$12$}",(1,0));
 +
label("\boldmath{$123$}",(2,0));
 +
label("\boldmath{$S$}",(3,0));
 +
 
 +
label("$11$",(0.5,-0.75));
 +
label("$111$",(1.5,-0.75));
 +
label("$d_1$",(2.5,-0.75));
 +
 
 +
label("$100$",(1,-1.5));
 +
label("$d_2$",(2,-1.5));
 +
 
 +
for (real i=1; i<=3; ++i) {
 +
  draw((0.1+(i-1),-0.15)--(0.4+(i-1),-0.6),red);
 +
}
  
−
Finally, the answer is <math>S=123+211=\boxed{334}.</math>
+
for (real i=1; i<=2; ++i) {
 +
  draw((0.6+(i-1),-0.9)--(0.9+(i-1),-1.35),red);
 +
}
  
 +
for (real i=1; i<=3; ++i) {
 +
  draw((0.6+(i-1),-0.6)--(0.9+(i-1),-0.15),red);
 +
}
 +
 +
for (real i=1; i<=2; ++i) {
 +
  draw((0.1+i,-1.35)--(0.4+i,-0.9),red);
 +
}
 +
 +
label("\textbf{First Differences}",(-0.75,-0.75),align=W);
 +
label("\textbf{Second Differences}",(-0.75,-1.5),align=W);
 +
</asy>
 +
 +
Finally, we have <math>d_2=100,</math> from which
 +
<cmath>\begin{align*}
 +
S&=123+d_1 \\
 +
&=123+(111+d_2) \\
 +
&=\boxed{334}.
 +
\end{align*}</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
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Alternatively, applying finite differences, one obtains <cmath>c_4 = {3 \choose 2}f(2) - {3 \choose 1}f(1) + {3 \choose 0}f(0) =334.</cmath>
 
Alternatively, applying finite differences, one obtains <cmath>c_4 = {3 \choose 2}f(2) - {3 \choose 1}f(1) + {3 \choose 0}f(0) =334.</cmath>
  
−
==Solution 5 (Very Cheap: Not Recommended)==
+
== Solution 5 (Assumption) ==
 +
The idea is to multiply the first, second and third equations by <math>a,b,</math> and <math>c,</math> respectively.
 +
 
 +
We can only consider the coefficients of <math>x_1,x_2,</math> and <math>x_3:</math>
 +
<cmath>\begin{align}
 +
a+4b+9c&=16, \\
 +
4a+9b+16c&=25, \\
 +
9a+16b+25c&=36.
 +
\end{align}</cmath>
 +
Subtracting <math>(1)</math> from <math>(2),</math> we get <cmath>3a+5b+7c=9. \hspace{15mm}(4)</cmath>
 +
Subtracting <math>3\cdot(4)</math> from <math>(3),</math> we get <cmath>b+4c=9. \hspace{25.5mm}(5)</cmath>
 +
Subtracting <math>(1)</math> from <math>4\cdot(5),</math> we get <cmath>7c-a=20. \hspace{23mm}(6)</cmath>
 +
From <math>(5)</math> and <math>(6),</math> we have <math>(a,b,c)=(7c-20,9-4c,c).</math> Substituting this into <math>(2)</math> gives <math>(a,b,c)=(1,-3,3).</math>
 +
 
 +
Therefore, the answer is <math>1\cdot1+12\cdot(-3) + 123\cdot3 = \boxed{334}.</math>
 +
 
 +
==Solution 6 (Assumption)==
 
We let <math>(x_4,x_5,x_6,x_7)=(0,0,0,0)</math>. Thus, we have  
 
We let <math>(x_4,x_5,x_6,x_7)=(0,0,0,0)</math>. Thus, we have  
−
<cmath>\begin{cases}  
+
<cmath>\begin{align*}  
−
x_1+4x_2+9x_3&=1\\
+
x_1+4x_2+9x_3&=1,\\
−
4x_1+9x_2+16x_3&=12\\
+
4x_1+9x_2+16x_3&=12,\\
−
9x_1+16x_2+25x_3&=123\\  
+
9x_1+16x_2+25x_3&=123.\\  
−
\end{cases}</cmath>
+
\end{align*}</cmath>
 
Grinding this out, we have <math>(x_1,x_2,x_3)=\left(\frac{797}{4},-229,\frac{319}{4}\right)</math> which gives <math>\boxed{334}</math> as our final answer.  
 
Grinding this out, we have <math>(x_1,x_2,x_3)=\left(\frac{797}{4},-229,\frac{319}{4}\right)</math> which gives <math>\boxed{334}</math> as our final answer.  
  
−
-Pleaseletmewin
+
~Pleaseletmewin
 +
 
 +
==Solution 7 (Similar to Solutions 3 and 4)==
 +
 
 +
Let <math>s_n = n^2</math> be the sequence of perfect squares.
 +
By either expanding or via finite differences, one can prove the miraculous recursion
 +
<cmath>s_n = 3s_{n-1} - 3s_{n-2} + s_{n-3}.</cmath>
 +
Hence, the answer is simply
 +
<cmath>3 \cdot 123 - 3 \cdot 12 + 1 = \boxed{334}.</cmath>
 +
 
 +
I first saw this in a Mathologer video: https://www.youtube.com/watch?v=4AuV93LOPcE
 +
 
 +
~Ritwin
  
 
==Video Solution==
 
==Video Solution==

Latest revision as of 03:15, 20 November 2023

Problem

Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that \begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.

Solution 1 (Quadratic Function)

Note that each given equation is of the form \[f(k)=k^2x_1+(k+1)^2x_2+(k+2)^2x_3+(k+3)^2x_4+(k+4)^2x_5+(k+5)^2x_6+(k+6)^2x_7\] for some $k\in\{1,2,3\}.$

When we expand $f(k)$ and combine like terms, we obtain a quadratic function of $k:$ \[f(k)=ak^2+bk+c,\] where $a,b,$ and $c$ are linear combinations of $x_1,x_2,x_3,x_4,x_5,x_6,$ and $x_7.$

We are given that \begin{alignat*}{10} f(1)&=\phantom{42}a+b+c&&=1, \\ f(2)&=4a+2b+c&&=12, \\ f(3)&=9a+3b+c&&=123, \end{alignat*} and we wish to find $f(4).$

We eliminate $c$ by subtracting the first equation from the second, then subtracting the second equation from the third: \begin{align*} 3a+b&=11, \\ 5a+b&=111. \end{align*} By either substitution or elimination, we get $a=50$ and $b=-139.$ Substituting these back produces $c=90.$

Finally, the answer is \[f(4)=16a+4b+c=\boxed{334}.\]

~Azjps ~MRENTHUSIASM

Solution 2 (Linear Combination)

For simplicity purposes, we number the given equations $(1),(2),$ and $(3),$ in that order. Let \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7=S. \hspace{29.5mm}(4)\] Subtracting $(1)$ from $(2),$ subtracting $(2)$ from $(3),$ and subtracting $(3)$ from $(4),$ we obtain the following equations, respectively: \begin{align*} 3x_1 + 5x_2 + 7x_3 + 9x_4 + 11x_5 + 13x_6 + 15x_7 &=11, \hspace{20mm}&(5) \\ 5x_1 + 7x_2 + 9x_3 + 11x_4 + 13x_5 + 15x_6 + 17x_7 &=111, &(6) \\ 7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7 &=S-123. &(7) \\ \end{align*} Subtracting $(5)$ from $(6)$ and subtracting $(6)$ from $(7),$ we obtain the following equations, respectively: \begin{align*} 2x_1+2x_2+2x_3+2x_4+2x_5+2x_6+2x_7&=100, &(8) \\ 2x_1+2x_2+2x_3+2x_4+2x_5+2x_6+2x_7&=S-234. \hspace{20mm}&(9) \end{align*} Finally, applying the Transitive Property to $(8)$ and $(9)$ gives $S-234=100,$ from which $S=\boxed{334}.$

~Duohead ~MRENTHUSIASM

Solution 3 (Finite Differences by Arithmetic)

Note that the second differences of all quadratic sequences must be constant (but nonzero). One example is the following sequence of perfect squares:

[asy] /* Made by MRENTHUSIASM */ size(20cm); for (real i=1; i<=10; ++i) { label("\boldmath{$"+string(i^2)+"$}",(i-1,0)); } for (real i=1; i<=9; ++i) { label("$"+string(1+2*i)+"$",(i-0.5,-0.75)); } for (real i=1; i<=8; ++i) { label("$2$",(i,-1.5)); } for (real i=1; i<=9; ++i) { draw((0.1+(i-1),-0.15)--(0.4+(i-1),-0.6),red); } for (real i=1; i<=8; ++i) { draw((0.6+(i-1),-0.9)--(0.9+(i-1),-1.35),red); } for (real i=1; i<=9; ++i) { draw((0.6+(i-1),-0.6)--(0.9+(i-1),-0.15),red); } for (real i=1; i<=8; ++i) { draw((0.1+i,-1.35)--(0.4+i,-0.9),red); } label("\textbf{First Differences}",(-0.75,-0.75),align=W); label("\textbf{Second Differences}",(-0.75,-1.5),align=W); [/asy]

Label equations $(1),(2),(3),$ and $(4)$ as Solution 2 does. Since the coefficients of $x_1,x_2,x_3,x_4,x_5,x_6,x_7,$ or $(1,4,9,16),(4,9,16,25),(9,16,25,36),(16,25,36,49),(25,36,49,64),(36,49,64,81),(49,64,81,100),$ respectively, all form quadratic sequences with second differences $2,$ we conclude that the second differences of equations $(1),(2),(3),(4)$ must be constant.

It follows that the second differences of $(1,12,123,S)$ must be constant, as shown below:

[asy] /* Made by MRENTHUSIASM */ size(10cm); label("\boldmath{$1$}",(0,0)); label("\boldmath{$12$}",(1,0)); label("\boldmath{$123$}",(2,0)); label("\boldmath{$S$}",(3,0)); label("$11$",(0.5,-0.75)); label("$111$",(1.5,-0.75)); label("$d_1$",(2.5,-0.75)); label("$100$",(1,-1.5)); label("$d_2$",(2,-1.5)); for (real i=1; i<=3; ++i) { draw((0.1+(i-1),-0.15)--(0.4+(i-1),-0.6),red); } for (real i=1; i<=2; ++i) { draw((0.6+(i-1),-0.9)--(0.9+(i-1),-1.35),red); } for (real i=1; i<=3; ++i) { draw((0.6+(i-1),-0.6)--(0.9+(i-1),-0.15),red); } for (real i=1; i<=2; ++i) { draw((0.1+i,-1.35)--(0.4+i,-0.9),red); } label("\textbf{First Differences}",(-0.75,-0.75),align=W); label("\textbf{Second Differences}",(-0.75,-1.5),align=W); [/asy]

Finally, we have $d_2=100,$ from which \begin{align*} S&=123+d_1 \\ &=123+(111+d_2) \\ &=\boxed{334}. \end{align*} ~MRENTHUSIASM

Solution 4 (Finite Differences by Algebra)

Notice that we may rewrite the equations in the more compact form as: \begin{align*} \sum_{i=1}^{7}i^2x_i&=c_1, \\ \sum_{i=1}^{7}(i+1)^2x_i&=c_2, \\ \sum_{i=1}^{7}(i+2)^2x_i&=c_3, \\ \sum_{i=1}^{7}(i+3)^2x_i&=c_4, \end{align*} where $c_1=1, c_2=12, c_3=123,$ and $c_4$ is what we are trying to find.

Now consider the polynomial given by $f(z) = \sum_{i=1}^7 (i+z)^2x_i$ (we are only treating the $x_i$ as coefficients).

Notice that $f$ is in fact a quadratic. We are given $f(0)=c_1, f(1)=c_2, f(2)=c_3$ and are asked to find $f(3)=c_4$. Using the concept of finite differences (a prototype of differentiation) we find that the second differences of consecutive values is constant, so that by arithmetic operations we find $c_4=\boxed{334}$.

Alternatively, applying finite differences, one obtains \[c_4 = {3 \choose 2}f(2) - {3 \choose 1}f(1) + {3 \choose 0}f(0) =334.\]

Solution 5 (Assumption)

The idea is to multiply the first, second and third equations by $a,b,$ and $c,$ respectively.

We can only consider the coefficients of $x_1,x_2,$ and $x_3:$ \begin{align} a+4b+9c&=16, \\ 4a+9b+16c&=25, \\ 9a+16b+25c&=36. \end{align} Subtracting $(1)$ from $(2),$ we get \[3a+5b+7c=9. \hspace{15mm}(4)\] Subtracting $3\cdot(4)$ from $(3),$ we get \[b+4c=9. \hspace{25.5mm}(5)\] Subtracting $(1)$ from $4\cdot(5),$ we get \[7c-a=20. \hspace{23mm}(6)\] From $(5)$ and $(6),$ we have $(a,b,c)=(7c-20,9-4c,c).$ Substituting this into $(2)$ gives $(a,b,c)=(1,-3,3).$

Therefore, the answer is $1\cdot1+12\cdot(-3) + 123\cdot3 = \boxed{334}.$

Solution 6 (Assumption)

We let $(x_4,x_5,x_6,x_7)=(0,0,0,0)$. Thus, we have \begin{align*} x_1+4x_2+9x_3&=1,\\ 4x_1+9x_2+16x_3&=12,\\ 9x_1+16x_2+25x_3&=123.\\ \end{align*} Grinding this out, we have $(x_1,x_2,x_3)=\left(\frac{797}{4},-229,\frac{319}{4}\right)$ which gives $\boxed{334}$ as our final answer.

~Pleaseletmewin

Solution 7 (Similar to Solutions 3 and 4)

Let $s_n = n^2$ be the sequence of perfect squares. By either expanding or via finite differences, one can prove the miraculous recursion \[s_n = 3s_{n-1} - 3s_{n-2} + s_{n-3}.\] Hence, the answer is simply \[3 \cdot 123 - 3 \cdot 12 + 1 = \boxed{334}.\]

I first saw this in a Mathologer video: https://www.youtube.com/watch?v=4AuV93LOPcE

~Ritwin

Video Solution

https://www.youtube.com/watch?v=4mOROTEkvWI ~ MathEx

See also

1989 AIME (Problems • Answer Key • Resources)
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Problem 7 		Followed by
Problem 9
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