Josanna's test scores to date are 90, 80, 70, 60, and 85 . Her goal is to raise here test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
💡 解题思路
The average of her current scores is $77$ . To raise it $3$ points, she needs an average of $80$ , and so after her $6$ tests, a sum of $480$ . Her current sum is $385$ , so she needs a $480 - 385 = \
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第 3 题
几何·面积
At a store, when a length is reported as x inches that means the length is at least x - 0.5 inches and at most x + 0.5 inches. Suppose the dimensions of a rectangular tile are reported as 2 inches by 3 inches. In square inches, what is the minimum area for the rectangle?
💡 解题思路
The minimum dimensions of the rectangle are $1.5$ inches by $2.5$ inches. The minimum area is $1.5\times2.5=\boxed{\mathrm{(A) \ } 3.75}$ square inches.
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第 4 题
应用题
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where A < B . How many dollars must LeRoy give to Bernardo so that they share the costs equally?
💡 解题思路
The difference in how much LeRoy and Bernardo paid is $B-A$ . To share the costs equally, LeRoy must give Bernardo half of the difference, which is $\boxed{\textbf{(C) } \;\frac{B-A}{2}}$
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第 5 题
数字运算
In multiplying two positive integers a and b , Ron reversed the digits of the two-digit number a . His erroneous product was 161 . What is the correct value of the product of a and b ?
💡 解题思路
We have $161 = 7 \cdot 23.$ Since $a$ has two digits, the factors must be $23$ and $7,$ so $a = 32$ and $b = 7.$ Then, $ab = 7 \times 32 = \boxed{\mathrm{\textbf{(E)}\ } 224}.$
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第 6 题
综合
On Halloween Casper ate \frac{1}{3} of his candies and then gave 2 candies to his brother. The next day he ate \frac{1}{3} of his remaining candies and then gave 4 candies to his sister. On the third day he ate his final 8 candies. How many candies did Casper have at the beginning?
💡 解题思路
Let $x$ represent the amount of candies Casper had at the beginning.
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第 7 题
几何·面积
The sum of two angles of a triangle is \frac{6}{5} of a right angle, and one of these two angles is 30^{\circ} larger than the other. What is the degree measure of the largest angle in the triangle?
💡 解题思路
The sum of two angles in a triangle is $\frac{6}{5}$ of a right angle $\longrightarrow \frac{6}{5} \times 90 = 108$
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第 8 题
逻辑推理
At a certain beach if it is at least 80^{\circ} F and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10? (A)\ The temperature was cooler than 80^{\circ} F and it was not sunny.(B)\ The temperature was cooler than 80^{\circ} F or it was not sunny.(C)\ If the temperature was at least 80^{\circ} F, then it was sunny.(D)\ If the temperature was cooler than 80^{\circ} F, then it was sunny.(E)\ If the temperature was cooler than 80^{\circ} F, then it was not sunny.
💡 解题思路
The beach not being crowded only means that it is not both hot and sunny. Equivalently, it is either cool or cloudy, which means $\boxed{\textbf{(B)}}$ is correct.
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第 9 题
几何·面积
The area of \triangleEBD is one third of the area of \triangleABC . Segment DE is perpendicular to segment AB . What is BD ?
💡 解题思路
$\triangle ABC \sim \triangle EBD$ by AA Similarity. Therefore $DE = \frac{3}{4} BD$ . Find the areas of the triangles. \[\triangle ABC: 3 \times 4 \times \frac{1}{2} = 6\] \[\triangle EBD: BD \times
10
第 10 题
分数与比例
Consider the set of numbers \{1, 10, 10^2, 10^3, \ldots, 10^{10}\} . The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
💡 解题思路
The requested ratio is \[\dfrac{10^{10}}{10^9 + 10^8 + \ldots + 10 + 1}.\] Using the formula for a geometric series, we have \[10^9 + 10^8 + \ldots + 10 + 1 = \dfrac{10^{10} - 1}{10 - 1} = \dfrac{10^{
11
第 11 题
计数
There are 52 people in a room. what is the largest value of n such that the statement "At least n people in this room have birthdays falling in the same month" is always true?
💡 解题思路
Pretend you have $52$ people you want to place in $12$ boxes, because there are $12$ months in a year. By the Pigeonhole Principle , one box must have at least $\left\lceil \frac{52}{12} \right\rceil$
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第 12 题
几何·面积
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
💡 解题思路
Let $s$ be Keiko's speed in meters per second, $a$ be the length of the straight parts of the track, $b$ be the radius of the smaller circles, and $b+6$ be the radius of the larger circles. The length
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第 13 题
概率
Two real numbers are selected independently at random from the interval [-20, 10] . What is the probability that the product of those numbers is greater than zero?
💡 解题思路
For the product of two numbers to be greater than zero, they either have to both be negative or both be positive. The interval for a positive number is $\frac{1}{3}$ of the total interval, and the int
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第 14 题
几何·面积
A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot?
💡 解题思路
Let the sides of the rectangular parking lot be $a$ and $b$ . Then $a^2 + b^2 = 625$ and $ab = 168$ . Add the two equations together, then factor. \begin{align*} a^2 + 2ab + b^2 &= 625 + 168 \times 2\
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第 15 题
分数与比例
Let @ denote the "averaged with" operation: a @ b = \frac{a+b}{2} . Which of the following distributive laws hold for all numbers x, y, and z ? \[I. x @ (y + z) = (x @ y) + (x @ z)\] \[II. x + (y @ z) = (x + y) @ (x + z)\] \[III. x @ (y @ z) = (x @ y) @ (x @ z)\]
💡 解题思路
Simplify each operation and see which ones hold true.
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第 16 题
几何·面积
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
💡 解题思路
If the side lengths of the dart board and the side lengths of the center square are all $\sqrt{2},$ then the side length of the legs of the triangles are $1$ .
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第 17 题
几何·面积
In the given circle, the diameter \overline{EB} is parallel to \overline{DC} , and \overline{AB} is parallel to \overline{ED} . The angles AEB and ABE are in the ratio 4 : 5 . What is the degree measure of angle BCD ?
💡 解题思路
We can let $\angle AEB$ be $4x$ and $\angle ABE$ be $5x$ because they are in the ratio $4 : 5$ . When an inscribed angle contains the diameter , the inscribed angle is a right angle . Therefore by tri
18
第 18 题
几何·面积
Rectangle ABCD has AB = 6 and BC = 3 . Point M is chosen on side AB so that \angle AMD = \angle CMD . What is the degree measure of \angle AMD ?
💡 解题思路
It is given that $\angle AMD \sim \angle CMD$ . Since $\angle AMD$ and $\angle CDM$ are alternate interior angles and $\overline{AB} \parallel \overline{DC}$ , $\angle AMD \cong \angle CDM \longrighta
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第 19 题
方程
What is the product of all the roots of the equation \[√(5 | x | + 8) = √(x^2 - 16).\]
💡 解题思路
First, square both sides, and isolate the absolute value. \begin{align*} 5|x|+8&=x^2-16\\ 5|x|&=x^2-24\\ |x|&=\frac{x^2-24}{5}. \\ \end{align*} Solve for the absolute value and factor.
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第 20 题
几何·面积
Rhombus ABCD has side length 2 and \angle B = 120 °. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R ?
💡 解题思路
Suppose that $P$ is a point in the rhombus $ABCD$ and let $\ell_{BC}$ be the perpendicular bisector of $\overline{BC}$ . Then $PB < PC$ if and only if $P$ is on the same side of $\ell_{BC}$ as $B$ . T
21
第 21 题
规律与数列
Brian writes down four integers w > x > y > z whose sum is 44 . The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9 . What is the sum of the possible values for w ?
💡 解题思路
The largest difference, $9,$ must be between $w$ and $z.$
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第 22 题
几何·面积
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
💡 解题思路
It is often easier to first draw a diagram for such a problem.
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第 23 题
数字运算
What is the hundreds digit of 2011^{2011}?
💡 解题思路
Since $2011 \equiv 11 \pmod{1000},$ we know that $2011^{2011} \equiv 11^{2011} \pmod{1000}.$
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第 24 题
坐标几何
A lattice point in an xy -coordinate system is any point (x, y) where both x and y are integers. The graph of y = mx +2 passes through no lattice point with 0 < x \le 100 for all m such that \frac{1}{2} < m < a . What is the maximum possible value of a ?
💡 解题思路
For $y=mx+2$ to not pass through any lattice points with $0<x\leq 100$ is the same as saying that $mx\notin\mathbb Z$ for $x\in\{1,2,\dots,100\}$ , or in other words, $m$ is not expressible as a ratio
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第 25 题
几何·面积
Let T_1 be a triangle with side lengths 2011, 2012, and 2013 . For n \ge 1 , if T_n = \triangle ABC and D, E, and F are the points of tangency of the incircle of \triangle ABC to the sides AB, BC , and AC, respectively, then T_{n+1} is a triangle with side lengths AD, BE, and CF, if it exists. What is the perimeter of the last triangle in the sequence ( T_n ) ?
💡 解题思路
By constructing the bisectors of each angle and the perpendicular radii of the incircle the triangle consists of 3 kites.