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| − | {{duplicate|[[2024 AMC 12A Problems/Problem 2|2024 AMC 12A #2]] and [[2024 AMC 10A Problems/Problem 2|2024 AMC 10A #2]]}}
| + | #redirect[[2024 AMC 10A Problems/Problem 2]] |
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| − | ==Problem==
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| − | Define <math>\blacktriangledown(a) = \sqrt{a - 1}</math> and <math>\blacktriangle(a) = \sqrt{a + 1}</math> for all real numbers <math>a</math>. What is the value of <cmath>\frac{\blacktriangledown(20 + \blacktriangle(2024))}{\blacktriangledown(\blacktriangle(24))}~?</cmath>
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| − | <math>\textbf{(A)}~ 1 \qquad \textbf{(B)}~ 2 \qquad \textbf{(C)}~ 4 \qquad \textbf{(D)}~ 8 \qquad \textbf{(E)}~ 16</math>
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| − | ==Solution==
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| − | The value of the expression is <cmath>\frac{\sqrt{20+\sqrt{2024+1}-1}}{\sqrt{\sqrt{24+1}-1}}=\frac{\sqrt{20+\sqrt{2025}-1}}{\sqrt{\sqrt{25}-1}}=\frac{\sqrt{20+45-1}}{\sqrt{5-1}}=\frac{\sqrt{64}}{\sqrt{4}}=\frac{8}{2}=\boxed{\textbf{(C)}~4}.</cmath>
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| − | ==See also==
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| − | {{AMC12 box|year=2024|ab=A|num-b=1|num-a=3}}
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| − | {{AMC10 box|year=2024|ab=A|num-b=1|num-a=3}}
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| − | {{MAA Notice}}
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