Makarla attended two meetings during her 9 -hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
💡 解题思路
The total number of minutes in her $9$ -hour work day is $9 \times 60 = 540.$ The total amount of time spend in meetings in minutes is $45 + 45 \times 2 = 135.$ The answer is then $\frac{135}{540}$ $=
3
第 3 题
综合
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
💡 解题思路
After you draw $4$ socks, you can have one of each color, so (according to the pigeonhole principle ), if you pull $\boxed{\textbf{(C)}\ 5}$ then you will be guaranteed a matching pair.
4
第 4 题
统计
For a real number x , define \heartsuit(x) to be the average of x and x^2 . What is \heartsuit(1)+\heartsuit(2)+\heartsuit(3) ?
💡 解题思路
The average of two numbers, $a$ and $b$ , is defined as $\frac{a+b}{2}$ . Thus the average of $x$ and $x^2$ would be $\frac{x(x+1)}{2}$ . With that said, we need to find the sum when we plug, $1$ , $2
5
第 5 题
整数运算
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
💡 解题思路
$31 \equiv 3 \pmod {7}$ so the week cannot start with Saturday, Sunday, Tuesday or Wednesday as that would result in an unequal number of Mondays and Wednesdays. Therefore, Monday, Thursday, and Frida
6
第 6 题
几何·面积
A circle is centered at O , \overline{AB} is a diameter and C is a point on the circle with \angle COB = 50^\circ . What is the degree measure of \angle CAB ?
💡 解题思路
Assuming we do not already know an inscribed angle is always half of its central angle, we will try a different approach. Since $O$ is the center, $OC$ and $OA$ are radii and they are congruent. Thus,
7
第 7 题
几何·面积
A triangle has side lengths 10 , 10 , and 12 . A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
💡 解题思路
The triangle is isosceles. The height of the triangle is therefore given by $h = \sqrt{10^2 - \left( \dfrac{12}{2} \right)^2} = \sqrt{64} = 8$
8
第 8 题
应用题
A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of \48 , and a group of 10th graders buys tickets costing a total of \64 . How many values for x are possible?
💡 解题思路
We find the greatest common factor of $48$ and $64$ to be $16$ . The number of factors of $16$ is $5$ which is the answer $(E)$ .
9
第 9 题
概率
Lucky Larry's teacher asked him to substitute numbers for a , b , c , d , and e in the expression a-(b-(c-(d+e))) and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a , b , c , and d were 1 , 2 , 3 , and 4 , respectively. What number did Larry substitute for e ?
💡 解题思路
We simply plug in the numbers \[1 - 2 - 3 - 4 + e = 1 - (2 - (3 - (4 + e)))\] \[-8 + e = -2 - e\] \[2e = 6\] \[e = 3 \;\;(D)\]
10
第 10 题
行程问题
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
💡 解题思路
Let $x$ be the time it is not raining, and $y$ be the time it is raining, in hours.
11
第 11 题
应用题
A shopper plans to purchase an item that has a listed price greater than \textdollar 100 and can use any one of the three coupons. Coupon A gives 15\% off the listed price, Coupon B gives \textdollar 30 off the listed price, and Coupon C gives 25\% off the amount by which the listed price exceeds \textdollar 100 . Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y - x ?
💡 解题思路
Let the listed price be $(100 + p)$ , where $p > 0$
12
第 12 题
综合
At the beginning of the school year, 50\% of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and 50\% answered "No." At the end of the school year, 70\% answered "Yes" and 30\% answered "No." Altogether, x\% of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of x ?
💡 解题思路
The minimum possible value would be $70 - 50 = 20\%$ . The maximum possible value would be $30 + 50 = 80\%$ . The difference is $80 - 20 = \boxed{\textbf{(D) }60}$ .
13
第 13 题
方程
What is the sum of all the solutions of x = |2x-|60-2x|| ?
💡 解题思路
We evaluate this in cases:
14
第 14 题
统计
The average of the numbers 1, 2, 3,·s, 98, 99, and x is 100x . What is x ?
💡 解题思路
We first sum the first $99$ numbers: $\frac{99(100)}{2}=99\cdot50$ . Then, we know that the sum of the series is $99\cdot50+x$ . There are $100$ terms, so we can divide this sum by $100$ and set it eq
15
第 15 题
数论
On a 50 -question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse’s total score on the contest was 99 . What is the maximum number of questions that Jesse could have answered correctly?
💡 解题思路
Let $a$ be the amount of questions Jesse answered correctly, $b$ be the amount of questions Jesse left blank, and $c$ be the amount of questions Jesse answered incorrectly. Since there were $50$ quest
16
第 16 题
几何·面积
A square of side length 1 and a circle of radius \dfrac{√(3)}{3} share the same center. What is the area inside the circle, but outside the square?
💡 解题思路
The radius of the circle is $\frac{\sqrt{3}}{3} = \sqrt{\frac{1}{3}}$ . Half the diagonal of the square is $\frac{\sqrt{1^2+1^2}}{2} = \frac{\sqrt{2}}{2} = \sqrt{\frac12}$ . We can see that the circle
17
第 17 题
统计
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37 th and 64 th , respectively. How many schools are in the city?
💡 解题思路
There are $x$ schools. This means that there are $3x$ people. Because no one's score was the same as another person's score, that means that there could only have been $1$ median score. This implies t
18
第 18 题
数论
Positive integers a , b , and c are randomly and independently selected with replacement from the set \{1, 2, 3,\dots, 2010\} . What is the probability that abc + ab + a is divisible by 3 ?
💡 解题思路
We group this into groups of $3$ , because $3|2010$ . This means that every residue class mod 3 has an equal probability.
19
第 19 题
几何·面积
A circle with center O has area 156π . Triangle ABC is equilateral, \overline{BC} is a chord on the circle, OA = 4√(3) , and point O is outside \triangle ABC . What is the side length of \triangle ABC ?
💡 解题思路
The formula for the area of a circle is $\pi r^2$ so the radius of this circle is $\sqrt{156}.$
20
第 20 题
几何·面积
Two circles lie outside regular hexagon ABCDEF . The first is tangent to \overline{AB} , and the second is tangent to \overline{DE} . Both are tangent to lines BC and FA . What is the ratio of the area of the second circle to that of the first circle?
💡 解题思路
A good diagram is very helpful.
21
第 21 题
数论
A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7 ?
💡 解题思路
View the palindrome as some number with form (decimal representation): $a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$ . But because the number is a palindrome, $a_3 = a_0, a_2 = a_1$ . Recombi
22
第 22 题
统计
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
💡 解题思路
We can count the total number of ways to distribute the candies (ignoring the restrictions), and then subtract the overcount to get the answer.
23
第 23 题
统计
The entries in a 3 × 3 array include all the digits from 1 through 9 , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
💡 解题思路
Observe that all tables must have 1s and 9s in the corners, 8s and 2s next to those corner squares, and 4-6 in the middle square. Also note that for each table, there exists a valid table diagonally s
24
第 24 题
规律与数列
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
💡 解题思路
Let $a,ar,ar^{2},ar^{3}$ be the quarterly scores for the Raiders. We know $r > 1$ because the sequence is said to be increasing. We also know that each of $a, ar, ar^2, ar^3$ is an integer. We start b
25
第 25 题
整数运算
Let a > 0 , and let P(x) be a polynomial with integer coefficients such that What is the smallest possible value of a ?
💡 解题思路
We observe that because $P(1) = P(3) = P(5) = P(7) = a$ , if we define a new polynomial $R(x)$ such that $R(x) = P(x) - a$ , $R(x)$ has roots when $P(x) = a$ ; namely, when $x=1,3,5,7$ .