Difference between revisions of "2023 AMC 8 Problems/Problem 4"
Themathguyd (talk | contribs) |
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curRight = len^2 + 3*len + 4; | curRight = len^2 + 3*len + 4; | ||
} | } | ||
+ | |||
+ | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
+ | </asy> | ||
+ | Note that given time constraint, its better to only count from perfect squares (in pink). | ||
+ | <asy> | ||
+ | /* Grid Made by MRENTHUSIASM */ | ||
+ | /* Squares pattern solution input by TheMathGuyd */ | ||
+ | size(175); | ||
+ | |||
+ | void ds(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); | ||
+ | } | ||
+ | |||
+ | void ps(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,pink+opacity(0.3)); | ||
+ | } | ||
+ | real ts = 0.5; | ||
+ | |||
+ | ds((0.5,4.5));label("$\mathbf{39}$",(0.5,4.5)); | ||
+ | ds((1.5,3.5));label("$\mathbf{19}$",(1.5,3.5)); | ||
+ | ds((3.5,1.5));label("$\mathbf{23}$",(3.5,1.5)); | ||
+ | ds((4.5,0.5));label("$\mathbf{47}$",(4.5,0.5)); | ||
+ | |||
+ | ps((3.5,3.5));label("$1$",(3.5,3.5)); | ||
+ | ps((4.5,2.5));label("$9$",(4.5,2.5)); | ||
+ | ps((5.5,1.5));label("$25$",(5.5,1.5)); | ||
+ | ps((6.5,0.5));label("$49$",(6.5,0.5)); | ||
+ | ps((3.5,4.5));label("$4$",(3.5,4.5)); | ||
+ | ps((2.5,5.5));label("$16$",(2.5,5.5)); | ||
+ | ps((1.5,6.5));label("$36$",(1.5,6.5)); | ||
+ | label(scale(ts)*"$\leftarrow$",(1,6),NE); | ||
+ | label(scale(ts)*"$+1$",(1,6),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(1,6),SW); | ||
+ | label(scale(ts)*"$+2$",(1,5),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(1,5),SW); | ||
+ | label(scale(ts)*"$+3$",(1,4),NW); | ||
+ | label(scale(ts)*"$+1$",(2,5),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(2,5),SW); | ||
+ | label(scale(ts)*"$+2$",(2,4),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(2,4),SW); | ||
+ | label(scale(ts)*"$+3$",(2,3),NW); | ||
+ | |||
+ | label(scale(ts)*"$\leftarrow$",(5,1),NE); | ||
+ | label(scale(ts)*"$-1$",(5,1),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(4,1),NE); | ||
+ | label(scale(ts)*"$-2$",(4,1),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(6,0),NE); | ||
+ | label(scale(ts)*"$-1$",(6,0),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(5,0),NE); | ||
+ | label(scale(ts)*"$-2$",(5,0),NW); | ||
+ | |||
+ | add(grid(7,7,grey+linewidth(1.25))); //USES OLYMPIAD.ASY | ||
draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
Line 95: | Line 147: | ||
From the four numbers that appear in the shaded squares, <math>\boxed{\textbf{(D)}\ 3}</math> of them are prime: <math>19,23,</math> and <math>47.</math> | From the four numbers that appear in the shaded squares, <math>\boxed{\textbf{(D)}\ 3}</math> of them are prime: <math>19,23,</math> and <math>47.</math> | ||
− | ~MathFun1000, MRENTHUSIASM | + | ~MathFun1000, MRENTHUSIASM, TheMathGuyd |
+ | |||
+ | |||
==Video Solution by Magic Square== | ==Video Solution by Magic Square== |
Revision as of 12:45, 27 January 2023
Contents
Problem
The numbers from to
are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number
How many of these four numbers are prime?
Solution
We fill out the grid, as shown below:
Note that given time constraint, its better to only count from perfect squares (in pink).
From the four numbers that appear in the shaded squares,
of them are prime:
and
~MathFun1000, MRENTHUSIASM, TheMathGuyd
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=5392
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=EcrktBc8zrM
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.