The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches? [图]
💡 解题思路
We can see that there are 4 whole squares, since the area of each square will be 1, 4 * 1 = 4. Next, there are 12 half squares, and 2 half squares are 1 whole square, so 12/2 = 6 whole squares. The ar
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第 2 题
分数与比例
Consider these two operations: a ◆ b = a^2 - b^2 ; a ★ b = (a - b)^2 What is the output of (5 ◆ 3) ★ 6?
💡 解题思路
We can find a general solution to any $((a \, \blacklozenge \, b) \, \bigstar \, c)$ . \[((a \, \blacklozenge \, b) \, \bigstar \, c)\] \[=((a^2-b^2) \, \bigstar \, c)\] \[=(a^2-b^2-c)^2\] \[=a^4+b^4-
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第 3 题
计数
When three positive integers a , b , and c are multiplied together, their product is 100 . Suppose a < b < c . In how many ways can the numbers be chosen?
💡 解题思路
The positive divisors of $100$ are \[1,2,4,5,10,20,25,50,100.\] It is clear that $10\leq c\leq50,$ so we apply casework to $c:$
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第 4 题
综合
The letter M in the figure below is first reflected over the line q and then reflected over the line p . What is the resulting image? [图]
💡 解题思路
When M is first reflected over the line $q,$ we obtain the following diagram: [asy] /* Made by MRENTHUSIASM */ usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotat
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第 5 题
规律与数列
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned 6 years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is 30 years. How many years older than Bella is Anna?
💡 解题思路
Five years ago, Bella was $6$ years old, and the kitten was $0$ years old.
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第 6 题
行程问题
Three positive integers are equally spaced on a number line. The middle number is 15, and the largest number is 4 times the smallest number. What is the smallest of these three numbers?
💡 解题思路
Let the smallest number be $x.$ It follows that the largest number is $4x.$
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第 7 题
行程问题
When the World Wide Web first became popular in the 1990 s, download speeds reached a maximum of about 56 kilobits per second. Approximately how many minutes would the download of a 4.2 -megabyte song have taken at that speed? (Note that there are 8000 kilobits in a megabyte.)
💡 解题思路
Notice that the number of kilobits in this song is $4.2 \cdot 8000 = 8 \cdot 7 \cdot 6 \cdot 100.$
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第 8 题
综合
What is the value of \[\frac{1}{3}·\frac{2}{4}·\frac{3}{5}·s\frac{18}{20}·\frac{19}{21}·\frac{20}{22}?\]
💡 解题思路
Note that common factors (from $3$ to $20,$ inclusive) of the numerator and the denominator cancel. Therefore, the original expression becomes \[\frac{1}{\cancel{3}}\cdot\frac{2}{\cancel{4}}\cdot\frac
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第 9 题
几何·角度
A cup of boiling water ( 212^{\circ}F ) is placed to cool in a room whose temperature remains constant at 68^{\circ}F . Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?
💡 解题思路
Initially, the difference between the water temperature and the room temperature is $212-68=144$ degrees Fahrenheit.
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第 10 题
坐标几何
One sunny day, Ling decided to take a hike in the mountains. She left her house at 8 \textsc{am} , drove at a constant speed of 45 miles per hour, and arrived at the hiking trail at 10 \textsc{am} . After hiking for 3 hours, Ling drove home at a constant speed of 60 miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip? [图]
💡 解题思路
Therefore, the answer is $\boxed{\textbf{(E)}}.$
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第 11 题
行程问题
Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating 3 inches of pasta from the middle of one piece. In the end, he has 10 pieces of pasta whose total length is 17 inches. How long, in inches, was the piece of pasta he started with?
💡 解题思路
Let's say that the first strand of pasta he had is x. The first time he takes a bite of this strand will make it 2 pieces. This is x - 3. The second time he takes the bites of EACH strand will become
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第 12 题
综合
💡 解题思路
First, we calculate that there are a total of $4\cdot4=16$ possibilities. Now, we list all of two-digit perfect squares. $64$ and $81$ are the only ones that can be made using the spinner. Consequentl
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第 13 题
规律与数列
How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice another, and the sum of the two numbers is 28 .”
💡 解题思路
Let $m$ and $n$ be positive integers such that $m>n$ and $m+n=28.$ It follows that $m=2n+d$ for some positive integer $d.$ We wish to find the number of possible values for $d.$
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第 14 题
统计
In how many ways can the letters in BEEKEEPER be rearranged so that two or more E s do not appear together?
💡 解题思路
All valid arrangements of the letters must be of the form \[\textbf{E\underline{\hspace{3mm}}E\underline{\hspace{3mm}}E\underline{\hspace{3mm}}E\underline{\hspace{3mm}}E}.\] The problem is equivalent
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第 15 题
应用题
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce? [图]
💡 解题思路
By the answer choices, we can disregard the points that do not have integer weights. As a result, we obtain the following diagram:
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第 16 题
统计
Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?
💡 解题思路
Note that the sum of the first two numbers is $21\cdot2=42,$ the sum of the middle two numbers is $26\cdot2=52,$ and the sum of the last two numbers is $30\cdot2=60.$
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第 17 题
数论
If n is an even positive integer, the \emph{double factorial} notation n!! represents the product of all the even integers from 2 to n . For example, 8!! = 2 · 4 · 6 · 8 . What is the units digit of the following sum? \[2!! + 4!! + 6!! + ·s + 2018!! + 2020!! + 2022!!\]
💡 解题思路
Notice that once $n>8,$ the units digit of $n!!$ will be $0$ because there will be a factor of $10.$ Thus, we only need to calculate the units digit of \[2!!+4!!+6!!+8!! = 2+8+48+48\cdot8.\] We only c
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第 18 题
几何·面积
The midpoints of the four sides of a rectangle are (-3,0), (2,0), (5,4), and (0,4). What is the area of the rectangle?
💡 解题思路
The midpoints of the four sides of every rectangle are the vertices of a rhombus whose area is half the area of the rectangle: Note that the diagonals of the rhombus have the same lengths as the sides
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第 19 题
统计
Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores. [图] Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85 . What is the minimum number of students who received extra points? (Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)
💡 解题思路
We notice that $13$ students have scores under $85$ currently, and only $5$ have scores over $85$ . We find the median of these two numbers, getting:
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第 20 题
综合
💡 解题思路
The sum of the numbers in each row is $12$ . Consider the second row. In order for the sum of the numbers in this row to equal $12$ , the two shaded numbers must add up to $13$ : [asy] unitsize(0.5cm)
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第 21 题
分数与比例
Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
💡 解题思路
Let $x$ be the number of shots that Candace made in the first half, and let $y$ be the number of shots Candace made in the second half. Since Candace and Steph took the same number of attempts, with a
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第 22 题
行程问题
A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?
💡 解题思路
Initially, suppose that the bus is at Stop $0$ (starting point) and Zia is at Stop $3.$
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第 23 题
几何·面积
A \triangle or \bigcirc is placed in each of the nine squares in a 3 -by- 3 grid. Shown below is a sample configuration with three \triangle s in a line. [图] How many configurations will have three \triangle s in a line and three \bigcirc s in a line? ==Sol ~wamofan
💡 解题思路
We will only consider cases where the three identical symbols are the same column, but at the end we shall double our answer as the same holds true for rows. There are $3$ ways to choose a column with
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第 24 题
几何·面积
The figure below shows a polygon ABCDEFGH , consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that AH = EF = 8 and GH = 14 . What is the volume of the prism? [图]
💡 解题思路
While imagining the folding, $\overline{AB}$ goes on $\overline{BC},$ $\overline{AH}$ goes on $\overline{CI},$ and $\overline{EF}$ goes on $\overline{FG}.$ So, $BJ=CI=8$ and $FG=BC=8.$ Also, $\overlin
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第 25 题
概率
A cricket randomly hops between 4 leaves, on each turn hopping to one of the other 3 leaves with equal probability. After 4 hops what is the probability that the cricket has returned to the leaf where it started?
💡 解题思路
Let $A$ denote the leaf where the cricket starts and $B$ denote one of the other $3$ leaves. Note that: