Difference between revisions of "2023 AMC 8 Problems/Problem 16"
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==Problem== | ==Problem== | ||
− | |||
− | < | + | The letters <math>P, Q,</math> and <math>R</math> are entered into a <math>20\times20</math> table according to the pattern shown below. How many <math>P</math>s, <math>Q</math>s, and <math>R</math>s will appear in the completed table? |
− | + | <asy> | |
− | + | /* Made by MRENTHUSIASM */ | |
− | + | size(125); | |
− | |||
− | P | ||
− | |||
− | |||
− | |||
− | |||
+ | for (int y = 0; y<=5; ++y) { | ||
+ | for (int x = 0; x<=5; ++x) { | ||
+ | draw((x,0)--(x,6),mediumgrey); | ||
+ | draw((0,y)--(6,y),mediumgrey); | ||
+ | } | ||
+ | } | ||
+ | label("$P$",(0.5,0.5)); | ||
+ | label("$Q$",(1.5,0.5)); | ||
+ | label("$R$",(2.5,0.5)); | ||
+ | label("$P$",(3.5,0.5)); | ||
+ | label("$Q$",(4.5,0.5)); | ||
+ | |||
+ | label("$Q$",(0.5,1.5)); | ||
+ | label("$R$",(1.5,1.5)); | ||
+ | label("$P$",(2.5,1.5)); | ||
+ | label("$Q$",(3.5,1.5)); | ||
+ | label("$R$",(4.5,1.5)); | ||
+ | |||
+ | label("$R$",(0.5,2.5)); | ||
+ | label("$P$",(1.5,2.5)); | ||
+ | label("$Q$",(2.5,2.5)); | ||
+ | label("$R$",(3.5,2.5)); | ||
+ | label("$P$",(4.5,2.5)); | ||
+ | |||
+ | label("$P$",(0.5,3.5)); | ||
+ | label("$Q$",(1.5,3.5)); | ||
+ | label("$R$",(2.5,3.5)); | ||
+ | label("$P$",(3.5,3.5)); | ||
+ | label("$Q$",(4.5,3.5)); | ||
+ | |||
+ | label("$Q$",(0.5,4.5)); | ||
+ | label("$R$",(1.5,4.5)); | ||
+ | label("$P$",(2.5,4.5)); | ||
+ | label("$Q$",(3.5,4.5)); | ||
+ | label("$R$",(4.5,4.5)); | ||
+ | |||
+ | label("$\vdots$",(0.5,5.5)); | ||
+ | label("$\vdots$",(1.5,5.5)); | ||
+ | label("$\vdots$",(2.5,5.5)); | ||
+ | label("$\vdots$",(3.5,5.5)); | ||
+ | label("$\vdots$",(4.5,5.5)); | ||
+ | |||
+ | label("$\cdots$",(5.5,0.5)); | ||
+ | label("$\cdots$",(5.5,1.5)); | ||
+ | label("$\cdots$",(5.5,2.5)); | ||
+ | label("$\cdots$",(5.5,3.5)); | ||
+ | label("$\cdots$",(5.5,4.5)); | ||
+ | |||
+ | label("$\cdot$",(5.3,5.3)); | ||
+ | label("$\cdot$",(5.45,5.45)); | ||
+ | label("$\cdot$",(5.6,5.6)); | ||
+ | </asy> | ||
<math>\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math> | <math>\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math> | ||
Revision as of 04:49, 27 January 2023
Contents
Problem
The letters and
are entered into a
table according to the pattern shown below. How many
s,
s, and
s will appear in the completed table?
Solution 1
In our grid we can see there are
,
and
of the letters P, Q and R’s respectively. We can see our pattern between each is
,
,
for the P, Q and R’s respectively. This such pattern will follow in our bigger example, so we can see that the only answer choice which satisfies this condition is
(Note: you could also "cheese" this problem by listing out all of the letters horizontally in a single line and looking at the repeating pattern.)
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat
Solution 2
We think about which letter is in the diagonal with of a letter. We find that it is
. The rest of the grid with the P's and R's is symmetrical, so therefore, the answer is
.
Solution by ILoveMath31415926535
Solution 3
Notice that rows and
are the same, for any
. Additionally, rows
,
, and
collectively contain the same number of
s,
s, and
s, because the letters are just substituted for one another. Therefore, the number of
s,
s, and
s in the first
rows is
. The first row has
,
, and
, and the second row has
,
, and
. Adding these up, we obtain
.
~mathboy100
Animated Video Solution
~Star League (https://starleague.us)
Video Solution by OmegaLearn (Using Cyclic Patterns)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=3990
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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.