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AMC数学竞赛真题2016年12B 18-19

2018-09-11 重点归纳

AMC12是针对高中学生的数学测验，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。其主要目的在于激发学生对数学的兴趣，参予AMC12的学生应该不难发现测验的问题都很具挑战性，但测验的题型都不会超过学生的学习范围。这项测验希望每个考生能从竞赛中享受数学。那么接下来跟随小编来看一下AMC12的官方真题以及官方解答吧：

AMC真题 18

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$

$\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

AMC真题解答

Consider the case when $x \geq 0$ , $y \geq 0$ . $x^2+y^2=x+y$ $(x - \frac{1}{2})^2+(y - \frac{1}{2})^2=\frac{1}{2}$ Notice the circle intersect the axes at points $(0, 1)$ and $(1, 0)$ . Find the area of this circle in the first quadrant. The area is made of a semi-circle with radius of $\frac{\sqrt{2}}{2}$ and a triangle: $A = \frac{\pi}{4} +\frac{1}{2}$ amc真题 Because of symmetry, the area is the same in all four quadrants. The answer is $\boxed{\textbf{(B)}\ \pi + 2}$

AMC真题 19

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?

$amc12数学竞赛$

AMC真题解答 1

By: dragonfly

We can solve this problem by listing it as an infinite geometric equation. We get that to have the same amount of tosses, they have a $\frac{1}{8}$ chance of getting all heads. Then the next probability is all of them getting tails and then on the second try, they all get heads. The probability of that happening is $\left(\frac{1}{8}\right)^2$ .We then get the geometric equation

$x=\frac{1}{8}+\left(\frac{1}{8}\right)^2+\left(\frac{1}{8}\right)^3...$

And then we find that $x$ equals to $\boxed{\textbf{(B)}\ \frac{1}{7}}$ because of the formula of the sum for an infinite series, $amc真题数学竞赛$ .

AMC真题解答 2

Call it a "win" if the boys all flip their coins the same number of times, and the probability that they win is $P$ . The probability that they win on their first flip is $\frac{1}{8}$ . If they don't win on their first flip, that means they all flipped tails (which also happens with probability $\frac{1}{8}$ ) and that their chances of winning have returned to what they were at the beginning. This covers all possible sequences of winning flips. So we have