Difference between revisions of "2023 AMC 10A Problems/Problem 4"

(Solution 1)
(Solution 1)
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~zhenghua
 
~zhenghua
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==Solution 2==
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Say the chosen side is <math>a</math> and the other sides are <math>b,c,d</math>.
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By the Generalised Polygon Inequality, <math>a<b+c+d</math>. We also have <math>a+b+c+d=26\Rightarrow b+c+d=26-a</math>.
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Combining these two, we get <math>a<26-a\Rightarrow a<13</math>.
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The smallest length that satisfies this is <math>a=\boxed {\textbf{(D) 12}}</math>
  
 
== See Also ==
 
== See Also ==
 
{{AMC10 box|year=2023|ab=A|num-b=3|num-a=5}}
 
{{AMC10 box|year=2023|ab=A|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:01, 9 November 2023

Problem

A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?

$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13$

Solution 1

Let's use the triangle inequality. We know that for a triangle, the 2 shorter sides must always be longer than the longest side. Similarly for a convex quadrilateral, the shortest 3 sides must always be longer than the longest side. Thus, the answer is $\frac{26}{2}-1=13-1=\boxed {\textbf{(D) 12}}$

~zhenghua

Solution 2

Say the chosen side is $a$ and the other sides are $b,c,d$. By the Generalised Polygon Inequality, $a<b+c+d$. We also have $a+b+c+d=26\Rightarrow b+c+d=26-a$. Combining these two, we get $a<26-a\Rightarrow a<13$. The smallest length that satisfies this is $a=\boxed {\textbf{(D) 12}}$

See Also

2023 AMC 10A (ProblemsAnswer KeyResources)
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 AMC 10 Problems and Solutions

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