Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2021 AMC 10B Problems/Problem 17: Difference between revisions

← Older editNewer edit →
MRENTHUSIASM (talk | contribs)
editor
9,986 edits
MRENTHUSIASM (talk | contribs)
editor
9,986 edits
All solutions to this problem are incredibly similar (repetitive). So, I cleaned up this page by keeping a comprehensive solution. I will give credit to all users who ever contributed to the written solutions. Let me know if you are unhappy with this edit
Line 13: Line 13:
<math>\textbf{(E) }\text{Tyrone was given card 7.}</math>
<math>\textbf{(E) }\text{Tyrone was given card 7.}</math>


==Solution 1==
==Solution==
 
Oscar must be given 3 and 1, so we rule out <math>\textbf{(A) \ }</math> and <math>\textbf{(B) \ }</math>. If Tyrone had card 7, then he would also have card 9, and then Kim must have 10 and 7 so we rule out <math>\textbf{(E) \ }</math>. If Aditi was given card 4, then she would have card 3, which Oscar already had. So the answer is <math>\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}</math>
 
~smarty101 and smartypantsno_3
 
==Solution 2==
Oscar must be given 3 and 1. Aditi cannot be given 3 or 1, so she must have 2 and 5. Similarly, Ravon cannot be given 1, 2, 3, or 5, so he must have 4 and 7, and the answer is <math>\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}</math>.
 
-SmileKat32
 
==Solution 3 (Comprehensive but Unnecessary)==
By observations, we consider the scores from lowest to highest. We make the following logical deduction:
By observations, we consider the scores from lowest to highest. We make the following logical deduction:
<cmath>\begin{align*}
<cmath>\begin{align*}
\text{Oscar's score is 4.} &\implies \text{Oscar is given cards 1 and 3.} \\
\text{Oscar's score is 4.} &\implies \text{Oscar is given cards 1 and 3.} \\
&\implies \text{Aditi is given cards 2 and 5.} \\
&\implies \text{Aditi is given cards 2 and 5.} \\
&\implies \text{Ravon is given cards 4 and 7.} \hspace{13mm} (*) \\
&\implies \text{Ravon is given cards 4 and 7.} && (\bigstar) \\
&\implies \text{Tyrone is given cards 6 and 10.} \\
&\implies \text{Tyrone is given cards 6 and 10.} \\
&\implies \text{Kim is given cards 8 and 9.}
&\implies \text{Kim is given cards 8 and 9.}
Line 35: Line 24:
Therefore, the answer is <math>\boxed{\textbf{(C) }\text{Ravon was given card 4.}}</math>
Therefore, the answer is <math>\boxed{\textbf{(C) }\text{Ravon was given card 4.}}</math>


Certainly, if we read the answer choices sooner, then we can stop at <math>(*)</math> and pick <math>\textbf{(C)}.</math>
Certainly, if we read the answer choices sooner, then we can stop at <math>(\bigstar)</math> and pick <math>\textbf{(C)}.</math>


~MRENTHUSIASM
~smarty101 ~smartypantsno_3 ~SmileKat32 ~MRENTHUSIASM


== Video Solution by OmegaLearn (Using logical deduction) ==
== Video Solution by OmegaLearn (Using logical deduction) ==

Revision as of 21:19, 7 September 2021

Problem

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--$11,$ Oscar--$4,$ Aditi--$7,$ Tyrone--$16,$ Kim--$17.$ Which of the following statements is true?

$\textbf{(A) }\text{Ravon was given card 3.}$

$\textbf{(B) }\text{Aditi was given card 3.}$

$\textbf{(C) }\text{Ravon was given card 4.}$

$\textbf{(D) }\text{Aditi was given card 4.}$

$\textbf{(E) }\text{Tyrone was given card 7.}$

Solution

By observations, we consider the scores from lowest to highest. We make the following logical deduction: \begin{align*} \text{Oscar's score is 4.} &\implies \text{Oscar is given cards 1 and 3.} \\ &\implies \text{Aditi is given cards 2 and 5.} \\ &\implies \text{Ravon is given cards 4 and 7.} && (\bigstar) \\ &\implies \text{Tyrone is given cards 6 and 10.} \\ &\implies \text{Kim is given cards 8 and 9.} \end{align*} Therefore, the answer is $\boxed{\textbf{(C) }\text{Ravon was given card 4.}}$

Certainly, if we read the answer choices sooner, then we can stop at $(\bigstar)$ and pick $\textbf{(C)}.$

~smarty101 ~smartypantsno_3 ~SmileKat32 ~MRENTHUSIASM

Video Solution by OmegaLearn (Using logical deduction)

https://youtu.be/zO0EuKPXuT0

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/FV9AnyERgJQ?t=284

~IceMatrix

Video Solution by Interstigation

https://youtu.be/8BPKs24eyes

~Interstigation

See Also

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_10B_Problems/Problem_17&oldid=161836"