2018 AMC 10B — Official Competition Problems (February 2018)
📅 2018 B 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
Kate bakes a 20 -inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
💡 解题思路
The area of the pan is $20\cdot18=360$ . Since the area of each piece is $2\cdot2=4$ , there are $\frac{360}{4} = \boxed{\textbf{(A) } 90}$ pieces.
2
第 2 题
统计
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
💡 解题思路
Suppose that Sam's average speed during the last $30$ minutes was $x$ mph.
3
第 3 题
行程问题
In the expression (\underline{ }×\underline{ })+(\underline{ }×\underline{ }) each blank is to be filled in with one of the digits 1,2,3, or 4, with each digit being used once. How many different values can be obtained?
💡 解题思路
We have $\binom{4}{2}$ ways to choose the pairs, and we have $2!$ ways for the values to be rearranged, hence $\frac{6}{2}=\boxed{\textbf{(B) }3}$
4
第 4 题
几何·面积
A three-dimensional rectangular box with dimensions X , Y , and Z has faces whose surface areas are 24 , 24 , 48 , 48 , 72 , and 72 square units. What is X + Y + Z ?
💡 解题思路
Let $X$ be the length of the shortest dimension and $Z$ be the length of the longest dimension. Thus, $XY = 24$ , $YZ = 72$ , and $XZ = 48$ . Divide the first two equations to get $\frac{Z}{X} = 3$ .
5
第 5 题
数论
How many subsets of \{2,3,4,5,6,7,8,9\} contain at least one prime number?
💡 解题思路
We use complementary counting, or
6
第 6 题
概率
A box contains 5 chips, numbered 1 , 2 , 3 , 4 , and 5 . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4 . What is the probability that 3 draws are required?
💡 解题思路
Notice that the only four ways such that $3$ draws are required are $1,2$ ; $1,3$ ; $2,1$ ; and $3,1$ . Notice that each of those cases has a $\frac{1}{5} \cdot \frac{1}{4}$ chance, so the answer is $
7
第 7 题
几何·面积
In the figure below, N congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let A be the combined area of the small semicircles and B be the area of the region inside the large semicircle but outside the semicircles. The ratio A:B is 1:18 . What is N ? [图]
💡 解题思路
Use the answer choices and calculate them. The one that works is $\bold{\boxed{\text{(D)19}}}$ .
8
第 8 题
综合
Sara makes a staircase out of toothpicks as shown: [图] This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
💡 解题思路
Notice that the sequence of numbers of toothpicks from the 1st step to the 3rd step is $4, 10, 18$
9
第 9 题
概率
The faces of each of 7 standard dice are labeled with the integers from 1 to 6 . Let p be the probabilities that when all 7 dice are rolled, the sum of the numbers on the top faces is 10 . What other sum occurs with the same probability as p ?
💡 解题思路
The number 10 can be achieved by \( 4+6 \), \( 5+5 \), \( 6+4 \), in which there must exist at least 5 number ones.
10
第 10 题
几何·角度
In the rectangular parallelepiped shown, AB = 3 , BC = 1 , and CG = 2 . Point M is the midpoint of \overline{FG} . What is the volume of the rectangular pyramid with base BCHE and apex M ? [图]
💡 解题思路
Consider the cross-sectional plane and label its area $b$ . Note that the volume of the triangular prism that encloses the pyramid is $\frac{bh}{2}=3$ , and we want the rectangular pyramid that shares
11
第 11 题
数论
Which of the following expressions is never a prime number when p is a prime number?
💡 解题思路
Each expression is in the form $p^2 + n$ .
12
第 12 题
几何·面积
Line segment \overline{AB} is a diameter of a circle with AB = 24 . Point C , not equal to A or B , lies on the circle. As point C moves around the circle, the centroid (center of mass) of \triangle ABC traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
💡 解题思路
By the Inscribed Angle Theorem, $\triangle ABC$ is a right triangle with $\angle C=90^{\circ}.$ So, its circumcenter is the midpoint of $\overline{AB},$ and its median from $C$ is half as long as $\ov
13
第 13 题
数论
How many of the first 2018 numbers in the sequence 101, 1001, 10001, 100001, \dots are divisible by 101 ?
💡 解题思路
The number $10^n+1$ is divisible by 101 if and only if $10^n\equiv -1\pmod{101}$ . We note that $(10,10^2,10^3,10^4)\equiv (10,-1,-10,1)\pmod{101}$ , so the powers of 10 are 4-periodic mod 101.
14
第 14 题
统计
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
💡 解题思路
To minimize the number of distinct values, we want to maximize the number of times a number appears. So, we could have $223$ numbers appear $9$ times, $1$ number appear once, and the mode appear $10$
15
第 15 题
几何·面积
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point A in the figure on the right. The box has base length w and height h . What is the area of the sheet of wrapping paper? [图]
💡 解题思路
Consider one-quarter of the image (the wrapping paper is divided up into $4$ congruent squares). The length of each dotted line is $h$ . The area of the rectangle that is $w$ by $h$ is $wh$ . The comb
16
第 16 题
数论
Let a_1,a_2,\dots,a_{2018} be a strictly increasing sequence of positive integers such that \[a_1+a_2+·s+a_{2018}=2018^{2018}.\] What is the remainder when a_1^3+a_2^3+·s+a_{2018}^3 is divided by 6 ?
💡 解题思路
Verify that $a^3 \equiv a \pmod{6}$ manually for all $a\in \mathbb{Z}/6\mathbb{Z}$ . We check: $0^3 \equiv 0 \pmod{6}$ , $1^3 \equiv 1 \pmod{6}$ , $2^3 \equiv 8 \equiv 2 \pmod{6}$ , $3^3 \equiv 27 \eq
17
第 17 题
几何·面积
In rectangle PQRS , PQ=8 and QR=6 . Points A and B lie on \overline{PQ} , points C and D lie on \overline{QR} , points E and F lie on \overline{RS} , and points G and H lie on \overline{SP} so that AP=BQ<4 and the convex octagon ABCDEFGH is equilateral. The length of a side of this octagon can be expressed in the form k+m√(n) , where k , m , and n are integers and n is not divisible by the square of any prime. What is k+m+n ?
💡 解题思路
Let $AP=BQ=x$ . Then $AB=8-2x$ .
18
第 18 题
统计
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for the trip?
💡 解题思路
We can begin to put this into cases. Let's call the pairs $a$ , $b$ and $c$ , and assume that a member of pair $a$ is sitting in the leftmost seat of the second row. We can have the following cases th
19
第 19 题
数论
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
💡 解题思路
Suppose that Chloe is $c$ years old today, so Joey is $c+1$ years old today. After $n$ years, Chloe and Zoe will be $n+c$ and $n+1$ years old, respectively. We are given that \[\frac{n+c}{n+1}=1+\frac
20
第 20 题
函数
A function f is defined recursively by f(1)=f(2)=1 and \[f(n)=f(n-1)-f(n-2)+n\] for all integers n ≥ 3 . What is f(2018) ?
💡 解题思路
For all integers $n \geq 7,$ note that \begin{align*} f(n)&=f(n-1)-f(n-2)+n \\ &=[f(n-2)-f(n-3)+n-1]-f(n-2)+n \\ &=-f(n-3)+2n-1 \\ &=-[f(n-4)-f(n-5)+n-3]+2n-1 \\ &=-f(n-4)+f(n-5)+n+2 \\ &=-[f(n-5)-f(n
21
第 21 题
数字运算
Mary chose an even 4 -digit number n . She wrote down all the divisors of n in increasing order from left to right: 1,2,\ldots,\dfrac{n}{2},n . At some moment Mary wrote 323 as a divisor of n . What is the smallest possible value of the next divisor written to the right of 323 ?
💡 解题思路
Let $d$ be the next divisor written to the right of $323.$
22
第 22 题
几何·面积
Real numbers x and y are chosen independently and uniformly at random from the interval [0,1] . Which of the following numbers is closest to the probability that x,y, and 1 are the side lengths of an obtuse triangle?
💡 解题思路
The Pythagorean Inequality tells us that in an obtuse triangle, $a^{2} + b^{2} c$ . So, we have two inequalities: \[x^2 + y^2 1\] The first equation is $\frac14$ of a circle with radius $1$ , and th
23
第 23 题
数论
How many ordered pairs (a, b) of positive integers satisfy the equation \[a· b + 63 = 20· lcm(a, b) + 12·gcd(a,b),\] where gcd(a,b) denotes the greatest common divisor of a and b , and lcm(a,b) denotes their least common multiple?
💡 解题思路
Let $x = \text{lcm}(a, b)$ , and $y = \text{gcd}(a, b)$ . Therefore, $a\cdot b = \text{lcm}(a, b)\cdot \text{gcd}(a, b) = x\cdot y$ . Thus, the equation becomes
24
第 24 题
几何·面积
Let ABCDEF be a regular hexagon with side length 1 . Denote by X , Y , and Z the midpoints of sides \overline {AB} , \overline{CD} , and \overline{EF} , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of \triangle ACE and \triangle XYZ ? [图] ~MRENTHUSIASM
💡 解题思路
[asy] /* Made by MRENTHUSIASM */ size(200); draw(polygon(6)); pair A, B, C, D, E, F, X, Y, Z, M, N, O, P, Q, R; A = dir(120); B = dir(60); C = dir(0); D = dir(300); E = dir(240); F = dir(180); X = mid
25
第 25 题
方程
Let \lfloor x \rfloor denote the greatest integer less than or equal to x . How many real numbers x satisfy the equation x^2 + 10,000\lfloor x \rfloor = 10,000x ?
💡 解题思路
This rewrites itself to $x^2=10,000\{x\}$ where $\lfloor x \rfloor + \{x\} = x$ .