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Difference between revisions of "2021 AMC 10B Problems/Problem 8"

m (→‎Solution 3 (Illustrations of Solutions 1 and 2))
 
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<math>\textbf{(A)} ~367 \qquad\textbf{(B)} ~368 \qquad\textbf{(C)} ~369 \qquad\textbf{(D)} ~379 \qquad\textbf{(E)} ~380</math>
 
<math>\textbf{(A)} ~367 \qquad\textbf{(B)} ~368 \qquad\textbf{(C)} ~369 \qquad\textbf{(D)} ~379 \qquad\textbf{(E)} ~380</math>
  
−
==Solution 1==
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==Solution 1 (Observations and Patterns: Considers Only 5 Squares)==
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In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are <math>D</math> and <math>E,</math> respectively. Note that the numbers in the yellow cells are consecutive odd perfect squares, as we can prove by induction.
−
By observing that the right-top corner of a square will always be a square, we know that the top right corner of the <math>15</math>x<math>15</math> grid is <math>225</math>. We can subtract <math>14</math> to get the value of the top-left corner; <math>211</math>. We can then find the value of the bottom left and right corners similarly. From there, we can find the value of the box on the far right in the 2nd row from the top by subtracting <math>13</math>, since the length of the side will be one box shorter. Similarly, we find the value for the box 2nd from the left and 2nd from the top, which is <math>157</math>. We know that the least number in the 2nd row will be <math>157</math>, and the greatest will be the number to its left, which is <math>1</math> less than <math>211</math>. We then sum <math>157</math> and <math>210</math> to get <math>\boxed{\mathbf{(A)}\ 367}</math>.
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<asy>  
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/* Made by MRENTHUSIASM */
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-Dynosol
 
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==Solution 2: Draw It Out==
 
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Drawing out the diagram, we get <math>\boxed{\mathbf{(A)}\ 367}</math>. Note that this should mainly be used just to check your answer.
 
−
 
 
−
~Taco12
 
−
 
 
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==Solution 3 (Illustrations of Solutions 1 and 2)==
 
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In both solutions below, note that the numbers along the yellow cells are consecutive odd perfect squares. We can show this by induction.  
 
−
 
 
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Two pictorial solutions follow from here:
 
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===Solution 3.1 (Illustration of Solution 1: Considers Only 5 Squares)===
 
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<asy> /* Made by MRENTHUSIASM */
 
 
size(11.5cm);
 
size(11.5cm);
  
Line 85: Line 70:
 
fill((i+0.5,i+0.5)--(i-0.5,i+0.5)--(i-0.5,i-0.5)--(i+0.5,i-0.5)--cycle,yellow);
 
fill((i+0.5,i+0.5)--(i-0.5,i+0.5)--(i-0.5,i-0.5)--(i+0.5,i-0.5)--cycle,yellow);
 
}
 
}
 +
 +
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
 +
fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
  
 
label("$A$",(14.5,14.5));
 
label("$A$",(14.5,14.5));
 
label("$B$",(13.5,13.5));
 
label("$B$",(13.5,13.5));
 
label("$C$",(0.5,14.5));
 
label("$C$",(0.5,14.5));
−
 
−
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
 
 
label("$E$",(1.5,13.5));
 
label("$E$",(1.5,13.5));
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fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
 
 
label("$D$",(0.5,13.5));
 
label("$D$",(0.5,13.5));
  
 
add(grid(15,15,linewidth(1.25)));
 
add(grid(15,15,linewidth(1.25)));
  
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draw((7.5,7.5)--(8.5,7.5)--(8.5,6.5)--(6.5,6.5)--(6.5,8.5)--(9.5,8.5)--(9.5,5.5)--(5.5,5.5)--(5.5,9.5)--(9.5,9.5),red+linewidth(1.125),EndArrow);
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draw((1.5,13.5)--(14.5,13.5)--(14.5,0.5)--(0.5,0.5)--(0.5,14.5)--(14.5,14.5),red+linewidth(1.125),EndArrow);
−
draw((12.5,12.5)--(13.5,12.5)--(13.5,1.5)--(1.5,1.5)--(1.5,13.5)--(14.5,13.5)--(14.5,0.5)--(0.5,0.5)--(0.5,14.5)--(14.5,14.5),red+linewidth(1.125),EndArrow);
 
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</asy>
 
</asy>
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By observations, we proceed as follows:
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In the diagram above, the red arrows indicate the progression of numbers. In the second row from the top, the greatest and the least numbers are <math>D</math> and <math>E,</math> respectively. By observations, we proceed as follows:
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<cmath>\begin{alignat*}{6}
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<cmath>\begin{array}{lllllll}
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A=15^2=225, \ B=13^2=169  
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A=15^2=225, \ B=13^2=169 &\Longrightarrow &C &= &A-14 &= &211 \\
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\quad &\implies \quad &C &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\  
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&\Longrightarrow &D &= &C-1 &= &210 \\
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\quad &\implies \quad &D &= &&C-1 &= 210& \\  
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&\Longrightarrow &E &= &B-12 &= &157
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\quad &\implies \quad &E &= &&B-12 &= 157&.
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\end{array}</cmath>
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\end{alignat*}</cmath>
 
Therefore, the answer is <math>D+E=\boxed{\textbf{(A)} ~367}.</math>
 
Therefore, the answer is <math>D+E=\boxed{\textbf{(A)} ~367}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
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===Solution 3.2 (Illustration of Solution 2: Draws All 225 Squares Out)===
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==Solution 2 (Observations and Patterns: Considers Only 7 Squares)==
−
<asy> /* Made by MRENTHUSIASM */
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In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are <math>C</math> and <math>G,</math> respectively.
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 
size(11.5cm);
 
size(11.5cm);
  
−
for (real i=7.5; i<=14.5; ++i)  
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fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
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{
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fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
−
fill((i+0.5,i+0.5)--(i-0.5,i+0.5)--(i-0.5,i-0.5)--(i+0.5,i-0.5)--cycle,yellow);
+
 
−
}
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label("$A$",(14.5,14.5));
 +
label("$B$",(0.5,14.5));
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label("$C$",(0.5,13.5));
 +
label("$D$",(0.5,0.5));
 +
label("$E$",(14.5,0.5));
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label("$F$",(14.5,13.5));
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label("$G$",(1.5,13.5));
 +
 
 +
add(grid(15,15,linewidth(1.25)));
 +
 
 +
draw((1.5,13.5)--(14.5,13.5)--(14.5,0.5)--(0.5,0.5)--(0.5,14.5)--(14.5,14.5),red+linewidth(1.125),EndArrow);
 +
</asy>
 +
By observations, we proceed as follows:
 +
<cmath>\begin{alignat*}{6}
 +
A=15^2=225
 +
\quad &\implies \quad &B &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\
 +
\quad &\implies \quad &C &= &&B-1  &= 210& \\
 +
\quad &\implies \quad &D &= &&C-13 &= 197& \\
 +
\quad &\implies \quad &E &= &&D-14 &= 183& \\
 +
\quad &\implies \quad &F &= &&E-13 &= 170& \\
 +
\quad &\implies \quad &G &= &&F-13 &= 157&.
 +
\end{alignat*}</cmath>
 +
Therefore, the answer is <math>C+G=\boxed{\textbf{(A)} ~367}.</math>
 +
 
 +
~MRENTHUSIASM ~Dynosol
 +
 
 +
==Solution 3 (Brute Force: Draws All 225 Squares Out)==
 +
From the full diagram below, the answer is <math>210+157=\boxed{\textbf{(A)} ~367}.</math>
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(11.5cm);
  
 
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
 
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
Line 154: Line 169:
 
}
 
}
 
</asy>
 
</asy>
 +
<b>This solution is extremely time-consuming and error-prone. Please do not try it in a real competition unless you have no other options.</b>
  
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From the full diagram above, the answer is <math>210+157=\boxed{\textbf{(A)} ~367}.</math>
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~MRENTHUSIASM ~Taco12
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−
~MRENTHUSIASM
 
  
 
== Video Solution by OmegaLearn (Using Pattern Finding) ==
 
== Video Solution by OmegaLearn (Using Pattern Finding) ==

Latest revision as of 22:06, 23 August 2024

Problem

Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top? [asy] /* Made by samrocksnature */ add(grid(7,7)); label("$\dots$", (0.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (2.5,0.5)); label("$\dots$", (3.5,0.5)); label("$\dots$", (4.5,0.5)); label("$\dots$", (5.5,0.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (0.5,1.5)); label("$\dots$", (0.5,2.5)); label("$\dots$", (0.5,3.5)); label("$\dots$", (0.5,4.5)); label("$\dots$", (0.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (6.5,1.5)); label("$\dots$", (6.5,2.5)); label("$\dots$", (6.5,3.5)); label("$\dots$", (6.5,4.5)); label("$\dots$", (6.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (1.5,6.5)); label("$\dots$", (2.5,6.5)); label("$\dots$", (3.5,6.5)); label("$\dots$", (4.5,6.5)); label("$\dots$", (5.5,6.5)); label("$\dots$", (6.5,6.5)); label("$17$", (1.5,1.5)); label("$18$", (1.5,2.5)); label("$19$", (1.5,3.5)); label("$20$", (1.5,4.5)); label("$21$", (1.5,5.5)); label("$16$", (2.5,1.5)); label("$5$", (2.5,2.5)); label("$6$", (2.5,3.5)); label("$7$", (2.5,4.5)); label("$22$", (2.5,5.5)); label("$15$", (3.5,1.5)); label("$4$", (3.5,2.5)); label("$1$", (3.5,3.5)); label("$8$", (3.5,4.5)); label("$23$", (3.5,5.5)); label("$14$", (4.5,1.5)); label("$3$", (4.5,2.5)); label("$2$", (4.5,3.5)); label("$9$", (4.5,4.5)); label("$24$", (4.5,5.5)); label("$13$", (5.5,1.5)); label("$12$", (5.5,2.5)); label("$11$", (5.5,3.5)); label("$10$", (5.5,4.5)); label("$25$", (5.5,5.5)); [/asy]

$\textbf{(A)} ~367 \qquad\textbf{(B)} ~368 \qquad\textbf{(C)} ~369 \qquad\textbf{(D)} ~379 \qquad\textbf{(E)} ~380$

Solution 1 (Observations and Patterns: Considers Only 5 Squares)

In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $D$ and $E,$ respectively. Note that the numbers in the yellow cells are consecutive odd perfect squares, as we can prove by induction. [asy] /* Made by MRENTHUSIASM */ size(11.5cm); for (real i=7.5; i<=14.5; ++i) { fill((i+0.5,i+0.5)--(i-0.5,i+0.5)--(i-0.5,i-0.5)--(i+0.5,i-0.5)--cycle,yellow); } fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green); fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green); label("$A$",(14.5,14.5)); label("$B$",(13.5,13.5)); label("$C$",(0.5,14.5)); label("$E$",(1.5,13.5)); label("$D$",(0.5,13.5)); add(grid(15,15,linewidth(1.25))); draw((1.5,13.5)--(14.5,13.5)--(14.5,0.5)--(0.5,0.5)--(0.5,14.5)--(14.5,14.5),red+linewidth(1.125),EndArrow); [/asy] By observations, we proceed as follows: \begin{alignat*}{6} A=15^2=225, \ B=13^2=169 \quad &\implies \quad &C &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\ \quad &\implies \quad &D &= &&C-1 &= 210& \\ \quad &\implies \quad &E &= &&B-12 &= 157&. \end{alignat*} Therefore, the answer is $D+E=\boxed{\textbf{(A)} ~367}.$

~MRENTHUSIASM

Solution 2 (Observations and Patterns: Considers Only 7 Squares)

In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $C$ and $G,$ respectively. [asy] /* Made by MRENTHUSIASM */ size(11.5cm); fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green); fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green); label("$A$",(14.5,14.5)); label("$B$",(0.5,14.5)); label("$C$",(0.5,13.5)); label("$D$",(0.5,0.5)); label("$E$",(14.5,0.5)); label("$F$",(14.5,13.5)); label("$G$",(1.5,13.5)); add(grid(15,15,linewidth(1.25))); draw((1.5,13.5)--(14.5,13.5)--(14.5,0.5)--(0.5,0.5)--(0.5,14.5)--(14.5,14.5),red+linewidth(1.125),EndArrow); [/asy] By observations, we proceed as follows: \begin{alignat*}{6} A=15^2=225 \quad &\implies \quad &B &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\ \quad &\implies \quad &C &= &&B-1 &= 210& \\ \quad &\implies \quad &D &= &&C-13 &= 197& \\ \quad &\implies \quad &E &= &&D-14 &= 183& \\ \quad &\implies \quad &F &= &&E-13 &= 170& \\ \quad &\implies \quad &G &= &&F-13 &= 157&. \end{alignat*} Therefore, the answer is $C+G=\boxed{\textbf{(A)} ~367}.$

~MRENTHUSIASM ~Dynosol

Solution 3 (Brute Force: Draws All 225 Squares Out)

From the full diagram below, the answer is $210+157=\boxed{\textbf{(A)} ~367}.$ [asy] /* Made by MRENTHUSIASM */ size(11.5cm); fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green); fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green); add(grid(15,15,linewidth(1.25))); int adj = 1; int curDown = 2; int curLeft = 4; int curUp = 6; int curRight = 8; label("$1$",(7.5,7.5)); for (int len = 3; len<=15; len+=2) { for (int i=1; i<=len-1; ++i) { label("$"+string(curDown)+"$",(7.5+adj,7.5+adj-i)); label("$"+string(curLeft)+"$",(7.5+adj-i,7.5-adj)); label("$"+string(curUp)+"$",(7.5-adj,7.5-adj+i)); label("$"+string(curRight)+"$",(7.5-adj+i,7.5+adj)); ++curDown; ++curLeft; ++curUp; ++curRight; } ++adj; curDown = len^2 + 1; curLeft = len^2 + len + 2; curUp = len^2 + 2*len + 3; curRight = len^2 + 3*len + 4; } [/asy] This solution is extremely time-consuming and error-prone. Please do not try it in a real competition unless you have no other options.

~MRENTHUSIASM ~Taco12

Video Solution by OmegaLearn (Using Pattern Finding)

https://youtu.be/bb4HB7pwO3Q

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/GYpAm8v1h-U?t=412

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=667

~Interstigation

See Also

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 10 Problems and Solutions

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