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Math Challenges for high school students

(3 Questions)
34 viewed last edited 3 years ago
These challenges can be tackled by applying what you have learned in high-school
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What is the sum of all solutions of x = \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{2x}}}} ? Hint: This be written as x = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}
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Monty Hall Problem: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Does switching giving you an advantange? Hint: Imagine a game show that has a 1000 doors and only one door has a car behind it. After you pick a door, the host opens 998 doors showing goats behind all of them. Would you switch then?
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If \log_2 (3) = x then what is \log_{12} (24) ?
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by Mahesh Godavarti