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【同步首發】2018AMC10A官方真題及答案

Category: AMC美國數學競賽, 國際競賽, 福利干貨, 競賽資料 Date: 2018年2月9日上午10:26

2018AMC10A真題

答案解析請參考文末

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Problem 1

What is the value of $\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$ $\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8$

Problem 2

Liliane has? $50\%$ ?more soda than Jacqueline, and Alice has? $25\%$ ?more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?

$\textbf{(A)}$ ?Liliane has? $20\%$ ?more soda than Alice.

$\textbf{(B)}$ ?Liliane has? $25\%$ ?more soda than Alice.

$\textbf{(C)}$ ?Liliane has? $45\%$ ?more soda than Alice.

$\textbf{(D)}$ ?Liliane has? $75\%$ ?more soda than Alice.

$\textbf{(E)}$ ?Liliane has? $100\%$ ?more soda than Alice.

Problem 3

A unit of blood expires after? $10!=10\cdot 9 \cdot 8 \cdots 1$ ?seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$

Problem 4

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

Problem 5

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let? $d$ ?be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of? $d$ ?

$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$

Problem 6

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that? $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?

$\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600$

Problem 7

For how many (not necessarily positive) integer values of? $n$ ?is the value of? $4000\cdot \left(\tfrac{2}{5}\right)^n$ ?an integer?

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }6 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad$

Problem 8

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?

$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

Problem 9

All of the triangles in the diagram below are similar to iscoceles triangle? $ABC$ , in which? $AB=AC$ . Each of the 7 smallest triangles has area 1, and? $\triangle ABC$ ?has area 40. What is the area of trapezoid? $DBCE$ ?

$[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0)); draw((10,10)--(50,10)); label("$B$",(0,0),SW); label("$C$",(60,0),SE); label("$E$",(50,10),E); label("$D$",(10,10),W); label("$A$",(30,30),N); draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10)); draw((15,15)--(45,15)); [/asy]$

$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24$

Problem 10

Suppose that real number? $x$ ?satisfies $\sqrt{49-x^2}-\sqrt{25-x^2}=3$ . What is the value of? $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?

$\textbf{(A) }8 \qquad \textbf{(B) }\sqrt{33}+8\qquad \textbf{(C) }9 \qquad \textbf{(D) }2\sqrt{10}+4 \qquad \textbf{(E) }12 \qquad$

Problem 11

When? $7$ ?fair standard? $6$ -sided die are thrown, the probability that the sum of the numbers on the top faces is? $10$ ?can be written as $\frac{n}{6^{7}}$ , where? $n$ ?is a positive integer. What is? $n$ ?

$\textbf{(A) }42\qquad \textbf{(B) }49\qquad \textbf{(C) }56\qquad \textbf{(D) }63\qquad \textbf{(E) }84\qquad$

Problem 12

How many ordered pairs of real numbers? $(x,y)$ ?satisfy the following system of equations? $x+3y=3$ $||x|-|y||=1$

Problem 13

A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point? $A$ ?falls on point? $B$ . What is the length in inches of the crease?

$[asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray); [/asy]$

$\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2$

Problem 14

What is the greatest integer less than or equal to $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$

$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$

Problem 15

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points? $A$ ?and? $B$ , as shown in the diagram. The distance? $AB$ ?can be written in the form? $\tfrac{m}{n}$ , where? $m$ ?and? $n$ ?are relatively prime positive integers. What is? $m+n$ ?

$[asy] draw(circle((0,0),13)); draw(circle((5,-6.2),5)); draw(circle((-5,-6.2),5)); label("$B$", (9.5,-9.5), S); label("$A$", (-9.5,-9.5), S); [/asy]$

$\textbf{(A) } 21 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 69 \qquad \textbf{(E) } 93$

Problem 16

Right triangle? $ABC$ ?has leg lengths? $AB=20$ ?and? $BC=21$ . Including? $\overline{AB}$ ?and? $\overline{BC}$ , how many line segments with integer length can be drawn from vertex? $B$ ?to a point on hypotenuse? $\overline{AC}$ ?

$\textbf{(A) }5 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }15 \qquad$

Problem 17

Let? $S$ ?be a set of 6 integers taken from? $\{1,2,\dots,12\}$ ?with the property that if? $a$ ?and? $b$ ?are elements of? $S$ ?with? $a<b$ , then? $b$ ?is not a multiple of? $a$ . What is the least possible values of an element in? $S?$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Problem 18

How many nonnegative integers can be written in the form $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$ where? $a_i\in \{-1,0,1\}$ ?for? $0\le i \le 7$ ?

$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048$

Problem 19

A number? $m$ ?is randomly selected from the set? $\{11,13,15,17,19\}$ , and a number? $n$ ?is randomly selected from? $\{1999,2000,2001,\ldots,2018\}$ . What is the probability that? $m^n$ ?has a units digit of? $1$ ?

$\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5}$

Problem 20

A scanning code consists of a? $7 \times 7$ ?grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of? $49$ ?squares. A scanning code is called? $symmetric$ ?if its look does not change when the entire square is rotated by a multiple of? $90 ^{\circ}$ ?counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?

$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

Problem 21

Which of the following describes the set of values of? $a$ ?for which the curves? $x^2+y^2=a^2$ ?and? $y=x^2-a$ ?in the real? $xy$ -plane intersect at exactly? $3$ ?points?

$\textbf{(A) }a=\frac14 \qquad \textbf{(B) }\frac14 < a < \frac12 \qquad \textbf{(C) }a>\frac14 \qquad \textbf{(D) }a=\frac12 \qquad \textbf{(E) }a>\frac12 \qquad$

Problem 22

Let? $a, b, c,$ ?and? $d$ ?be positive integers such that? $\gcd(a, b)=24$ ,? $\gcd(b, c)=36$ ,? $\gcd(c, d)=54$ , and? $70<\gcd(d, a)<100$ . Which of the following must be a divisor of? $a$ ?

$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

Problem 23

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square? $S$ ?so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from? $S$ ?to the hypotenuse is 2 units. What fraction of the field is planted?

$[asy] draw((0,0)--(4,0)--(0,3)--(0,0)); draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); label("$4$", (2,0), N); label("$3$", (0,1.5), E); label("$2$", (.8,1), E); label("$S$", (0,0), NE); draw((0.3,0.3)--(1.4,1.9), dashed); [/asy]$

$\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75}$

Problem 24

Triangle? $ABC$ ?with? $AB=50$ ?and? $AC=10$ ?has area? $120$ . Let? $D$ ?be the midpoint of? $\overline{AB}$ , and let? $E$ ?be the midpoint of? $\overline{AC}$ . The angle bisector of? $\angle BAC$ ?intersects? $\overline{DE}$ ?and? $\overline{BC}$ ?at? $F$ ?and? $G$ , respectively. What is the area of quadrilateral? $FDBG$ ?

$\textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad$

Problem 25

For a positive integer? $n$ ?and nonzero digits? $a$ ,? $b$ , and? $c$ , let? $A_n$ ?be the? $n$ -digit integer each of whose digits is equal to? $a$ ; let? $B_n$ ?be the? $n$ -digit integer each of whose digits is equal to? $b$ , and let? $C_n$ ?be the? $2n$ -digit (not? $n$ -digit) integer each of whose digits is equal to? $c$ . What is the greatest possible value of? $a + b + c$ ?for which there are at least two values of? $n$ ?such that? $C_n - B_n = A_n^2$ ?

$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$

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