2018 AMC 10A — Official Competition Problems (February 2018)
📅 2018 A 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
综合
What is the value of \[(((2+1)^{-1}+1)^{-1}+1)^{-1}+1?\]
💡 解题思路
For all nonzero numbers $a,$ recall that $a^{-1}=\frac1a$ is the reciprocal of $a.$
2
第 2 题
综合
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 Alice have? (A) Liliane has 20\% more soda than Alice. (B) Liliane has 25\% more soda than Alice. (C) Liliane has 45\% more soda than Alice. (D) Liliane has 75\% more soda than Alice. (E) Liliane has 100\% more soda than Alice.
💡 解题思路
Let's assume that Jacqueline has $1$ gallon(s) of soda. Then Alice has $1.25$ gallons and Liliane has $1.5$ gallons. Doing division, we find out that $\frac{1.5}{1.25}=1.2$ , which means that Liliane
3
第 3 题
综合
A unit of blood expires after 10!=10· 9 · 8 ·s 1 seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
💡 解题思路
The problem says there are $10! = 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ seconds. Convert $10!$ seconds to minutes by dividing by $60$ : $9\cdot 8\cdot 7\cdot 5\cdot 4\cdot
4
第 4 题
计数
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.)
💡 解题思路
We must place the classes into the periods such that no two classes are in the same period or in consecutive periods.
5
第 5 题
行程问题
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 ?
💡 解题思路
For each of the false statements, we identify its corresponding true statement. Note that:
6
第 6 题
综合
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?
💡 解题思路
If $65\%$ of the votes were likes, then $35\%$ of the votes were dislikes. $65\%-35\%=30\%$ , so $90$ votes is $30\%$ of the total number of votes. Doing quick arithmetic shows that the answer is $\bo
7
第 7 题
整数运算
For how many (not necessarily positive) integer values of n is the value of 4000· (\tfrac{2}{5})^n an integer?
💡 解题思路
Note that \[4000\cdot \left(\frac{2}{5}\right)^n=\left(2^5\cdot5^3\right)\cdot \left(2\cdot5^{-1}\right)^n=2^{5+n}\cdot5^{3-n}.\] Since this expression is an integer, we need:
8
第 8 题
概率
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?
💡 解题思路
Let $x$ be the number of $5$ -cent coins that Joe has. Therefore, he must have $(x+3) \ 10$ -cent coins and $(23-(x+3)-x) \ 25$ -cent coins. Since the total value of his collection is $320$ cents, we
9
第 9 题
几何·面积
All of the triangles in the diagram below are similar to isosceles 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 ? [图]
💡 解题思路
Let $x$ be the area of $ADE$ . Note that $x$ is comprised of the $7$ small isosceles triangles and a triangle similar to $ADE$ with side length ratio $3:4$ (so an area ratio of $9:16$ ). Thus, we have
10
第 10 题
综合
Suppose that real number x satisfies \[√(49-x^2)-√(25-x^2)=3\] What is the value of √(49-x^2)+√(25-x^2) ?
💡 解题思路
We let $a=\sqrt{49-x^2}+\sqrt{25-x^2}$ ; in other words, we want to find $a$ . We know that $a\cdot3=\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\cdot\left(\sqrt{49-x^2}-\sqrt{25-x^2}\right)=\left(\sqrt{4
11
第 11 题
概率
When 7 fair standard 6 -sided dice 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 ?
💡 解题思路
Add possibilities. There are $3$ ways to sum to $10$ , listed below.
12
第 12 题
综合
💡 解题思路
We can solve this by graphing the equations. The second equation looks challenging to graph, but start by graphing it in the first quadrant only (which is easy since the inner absolute value signs can
What is the greatest integer less than or equal to \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
💡 解题思路
We write \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\frac{3^{96}}{3^{96}+2^{96}}\cdot\frac{3^{100}}{3^{96}}+\frac{2^{96}}{3^{96}+2^{96}}\cdot\frac{2^{100}}{2^{96}}=\frac{3^{96}}{3^{96}+2^{96}}\cdot 81+\f
15
第 15 题
几何·面积
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 ? [图]
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} ?
💡 解题思路
[asy] unitsize(4); pair A, B, C, E, P; A=(-20, 0); B=origin; C=(0,21); E=(-21, 20); P=extension(B,E, A, C); draw(A--B--C--cycle); draw(B--P); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); d
17
第 17 题
数论
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 value of an element in S ?
💡 解题思路
We start with $2$ because $1$ is not an answer choice. We would have to include every odd number except $1$ to fill out the set, but then $3$ and $9$ would violate the rule, so that won't work.
18
第 18 题
整数运算
How many nonnegative integers can be written in the form \[a_7·3^7+a_6·3^6+a_5·3^5+a_4·3^4+a_3·3^3+a_2·3^2+a_1·3^1+a_0·3^0,\] where a_i\in \{-1,0,1\} for 0\le i \le 7 ?
💡 解题思路
This looks like balanced ternary, in which all the integers with absolute values less than $\frac{3^n}{2}$ are represented in $n$ digits. There are 8 digits. Plugging in 8 into the formula for the bal
19
第 19 题
概率
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 ?
💡 解题思路
Since we only care about the units digit, our set $\{11,13,15,17,19 \}$ can be turned into $\{1,3,5,7,9 \}$ . Call this set $A$ and call $\{1999, 2000, 2001, \cdots , 2018 \}$ set $B$ . Let's do casew
20
第 20 题
几何·面积
A scanning code consists of a 7 × 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 \textit{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?
💡 解题思路
Draw a $7 \times 7$ square.
21
第 21 题
综合
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?
💡 解题思路
Substituting $y=x^2-a$ into $x^2+y^2=a^2$ , we get \[x^2+(x^2-a)^2=a^2 \implies x^2+x^4-2ax^2=0 \implies x^2(x^2-(2a-1))=0\] Since this is a quartic, there are $4$ total roots (counting multiplicity).
22
第 22 题
数论
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 ? (gcd means greatest common factor)
💡 解题思路
The GCD information tells us that $24$ divides $a$ , both $24$ and $36$ divide $b$ , both $36$ and $54$ divide $c$ , and $54$ divides $d$ . Note that we have the prime factorizations: \begin{align*} 2
23
第 23 题
几何·面积
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? [图]
💡 解题思路
Note that the hypotenuse of the field is $5,$ and the area of the field is $6.$ Let $x$ be the side-length of square $S.$
24
第 24 题
几何·面积
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 ?
💡 解题思路
Let $BC = a$ , $BG = x$ , $GC = y$ , and the length of the perpendicular from $BC$ through $A$ be $h$ . By angle bisector theorem, we have that \[\frac{50}{x} = \frac{10}{y},\] where $y = -x+a$ . Ther
25
第 25 题
数字运算
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 ?
💡 解题思路
By geometric series, we have \begin{alignat*}{8} A_n&=a\bigl(\phantom{ }\underbrace{111\cdots1}_{n\text{ digits}}\phantom{ }\bigr)&&=a\left(1+10+10^2+\cdots+10^{n-1}\right)&&=a\cdot\frac{10^n-1}{9}, \