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AMC10每日一题（2001年真题#14）

2018-06-04
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　　AMC不但是美国顶尖数学人才的人才库，更为学校提供了解申请入学者在数学科目上的学习成就与表现评估。AMC成功地为许多学生因测验成绩优良而进入理想学校。藉由设计严谨的试题，达到激发应试者解决问题的能力，培养对数学的兴趣。试题由简至难兼具，使任何程度的学生都能感受到挑战，还可以筛选出特有天赋者。AMC10是针对高中一年级及初中三年级学生的数学测验，25题选择题、考试时间75分钟;包含演算概念理解的数学题型。AMC10的测验允许使用计算器(工程用计算器除外)。AMC10的主要目的是在刺激学生对数学的兴趣并且透过以选择题方式来开发学生对数学的才能;测验题型范围由容易到困难。参予AMC10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。今天课窝小编为大家整理了AMC10真题练习，希望考生们认真阅读，能够对你的考试有所帮助。

2001 AMC 10 竞赛试题/第15题

Problem

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $10$ and altitude $12$ , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

$\textbf{(A)}\ \frac{8}3\qquad\textbf{(B)}\ \frac{30}{11}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ \frac{25}{8}\qquad\textbf{(E)}\ \frac{7}{2}$

Solution

Let the diameter of the cylinder be $2r$ . Examining the cross section of the cone and cylinder, we find two similar triangles. Hence, $\frac{12-2r}{12}=\frac{2r}{10}$ which we solve to find $r=\frac{30}{11}$ . Our answer is $\boxed{\textbf{(B)}\ \frac{30}{11}}$ .

2001 AMC 10 竞赛试题/第16题

Problem

In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$ , $w$ , $x$ , $y$ , and $z$ . Find $y + z$ .

$\textbf{(A)}\ 43 \qquad \textbf{(B)}\ 44 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 47$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$w$",(2.5,2.5));[/asy]

Solutions

Solution 1

We know that $y+z=2v$ , so we could find one variable rather than two.

$v+24+w=43+v$

$24+w=43$

$w=19$

$44+x=24+x+z$ $z=20$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$20$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]