The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x . What is the value of b ?
💡 解题思路
$2x + 7 = 3 \iff x = -2$ , so we require $-2b-10 = -2 \iff -2b = 8 \iff b = \boxed{\textbf{(B) } -4}$ .
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第 3 题
几何·面积
A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle? (\mathrm {A}) \ \frac 14x^2 (\mathrm {B}) \ \frac 25x^2 (\mathrm {C})\ \frac 12x^2 (\mathrm {D}) \ x^2 (\mathrm {E})\ \frac 32x^2
💡 解题思路
Let $w$ be the width, so the length is $2w$ . By the Pythagorean Theorem , $w^2 + 4w^2 = x^2 \Longrightarrow \frac{x}{\sqrt{5}} = w$ . The area of the rectangle is $(w)(2w) = 2w^2 = 2\left(\frac{x}{\s
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第 4 题
行程问题
A store normally sells windows at \100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? (\mathrm {A}) \ 100 (\mathrm {B}) \ 200 (\mathrm {C})\ 300 (\mathrm {D}) \ 400 (\mathrm {E})\ 500$
💡 解题思路
For $n$ windows, the store offers a discount of $100 \cdot \left\lfloor\frac{n}{5}\right\rfloor$ ( floor function ). Dave receives a discount of $100 \cdot \left\lfloor \frac{7}{5}\right \rfloor = 100
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第 5 题
统计
The average (mean) of 20 numbers is 30 , and the average of 30 other numbers is 20 . What is the average of all 50 numbers? (\mathrm {A}) \ 23 (\mathrm {B}) \ 24 (\mathrm {C})\ 25 (\mathrm {D}) \ 10 (\mathrm {E})\ 27
💡 解题思路
The sum of the first $20$ numbers is $20 \cdot 30$ and the sum of the other $30$ numbers is $30\cdot 20$ . Hence the overall average is $\frac{20 \cdot 30 + 30 \cdot 20}{50} = 24 \ \mathrm{(B)}$ .
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第 6 题
行程问题
Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? (\mathrm {A}) \ 4 (\mathrm {B}) \ 5 (\mathrm {C})\ 6 (\mathrm {D}) \ 7 (\mathrm {E})\ 8
💡 解题思路
Let $D_J, D_M$ be the distances traveled by Josh and Mike, respectively, and let $r,t$ be the time and rate of Mike. Using $d = rt$ , we have that $D_M = rt$ and $D_J = \left(\frac{4}{5}r\right)\left(
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第 7 题
几何·面积
Square EFGH is inside square ABCD so that each side of EFGH can be extended to pass through a vertex of ABCD . Square ABCD has side length √(50) , E is between B and H , and BE = 1 . What is the area of the inner square EFGH ? [图] (\mathrm {A}) \ 25 (\mathrm {B}) \ 32 (\mathrm {C})\ 36 (\mathrm {D}) \ 40 (\mathrm {E})\ 42
💡 解题思路
Arguable the hardest part of this question is to visualize the diagram. Since each side of $EFGH$ can be extended to pass through a vertex of $ABCD$ , we realize that $EFGH$ must be tilted in such a f
8
第 8 题
数字运算
Let A,M , and C be digits with \[(100A+10M+C)(A+M+C) = 2005.\] What is A ? (\mathrm {A}) \ 1 (\mathrm {B}) \ 2 (\mathrm {C})\ 3 (\mathrm {D}) \ 4 (\mathrm {E})\ 5
💡 解题思路
Clearly the two quantities are both integers, so we check the prime factorization of $2005 = 5 \cdot 401$ . It is easy to see now that $(A,M,C) = (4,0,1)$ works, so the answer is $\mathrm{(D)}$ .
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第 9 题
方程
There are two values of a for which the equation 4x^2 + ax + 8x + 9 = 0 has only one solution for x . What is the sum of those values of a ? (\mathrm {A}) \ -16 (\mathrm {B}) \ -8 (\mathrm {C})\ 0 (\mathrm {D}) \ 8 (\mathrm {E})\ 20
💡 解题思路
https://youtu.be/3dfbWzOfJAI?t=222 ~AVM2023
10
第 10 题
立体几何
A wooden cube n units on a side is painted red on all six faces and then cut into n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n ? (\mathrm {A}) \ 3 (\mathrm {B}) \ 4 (\mathrm {C})\ 5 (\mathrm {D}) \ 6 (\mathrm {E})\ 7
💡 解题思路
There are $6n^3$ sides total on the unit cubes, and $6n^2$ are painted red.
11
第 11 题
统计
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? (\mathrm {A}) \ 41 (\mathrm {B}) \ 42 (\mathrm {C})\ 43 (\mathrm {D}) \ 44 (\mathrm {E})\ 45
💡 解题思路
Let the digits be $A, B, C$ so that $B = \frac {A + C}{2}$ . In order for this to be an integer, $A$ and $C$ have to have the same parity . There are $9$ possibilities for $A$ , and $5$ for $C$ . $B$
12
第 12 题
坐标几何
A line passes through A(1,1) and B(100,1000) . How many other points with integer coordinates are on the line and strictly between A and B ? (\mathrm {A}) \ 0 (\mathrm {B}) \ 2 (\mathrm {C})\ 3 (\mathrm {D}) \ 8 (\mathrm {E})\ 9
💡 解题思路
For convenience’s sake, we can transform $A$ to the origin and $B$ to $(99,999)$ (this does not change the problem). The line $AB$ has the equation $y = \frac{999}{99}x = \frac{111}{11}x$ . The coordi
13
第 13 题
规律与数列
In the five-sided star shown, the letters A , B , C , D and E are replaced by the numbers 3 , 5 , 6 , 7 and 9 , although not necessarily in that order. The sums of the numbers at the ends of the line segments \overline{AB} , \overline{BC} , \overline{CD} , \overline{DE} and \overline{EA} form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence? [图] (\mathrm {A}) \ 9 (\mathrm {B}) \ 10 (\mathrm {C})\ 11 (\mathrm {D}) \ 12 (\mathrm {E})\ 13
💡 解题思路
$(A+B) + (B+C) + (C+D) + (D+E) + (E+A) = 2(A+B+C+D+E)$ (i.e., each number is counted twice). The sum $A + B + C + D + E$ will always be $3 + 5 + 6 + 7 + 9 = 30$ , so the arithmetic sequence has a sum
14
第 14 题
概率
On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? (\mathrm {A}) \ \frac{5}{11} (\mathrm {B}) \ \frac{10}{21} (\mathrm {C})\ \frac{1}{2} (\mathrm {D}) \ \frac{11}{21} (\mathrm {E})\ \frac{6}{11}
💡 解题思路
There are $1 + 2 + 3 + 4 + 5 + 6 = 21$ dots total. Casework :
15
第 15 题
几何·面积
Let \overline{AB} be a diameter of a circle and C be a point on \overline{AB} with 2 · AC = BC . Let D and E be points on the circle such that \overline{DC} \perp \overline{AB} and \overline{DE} is a second diameter. What is the ratio of the area of \triangle DCE to the area of \triangle ABD ? [图] (\text {A}) \ \frac {1}{6} (\text {B}) \ \frac {1}{4} (\text {C})\ \frac {1}{3} (\text {D}) \ \frac {1}{2} (\text {E})\ \frac {2}{3}
💡 解题思路
Notice that the bases of both triangles are diameters of the circle. Hence the ratio of the areas is just the ratio of the heights of the triangles, or $\frac{CF}{CD}$ ( $F$ is the foot of the perpend
16
第 16 题
几何·面积
Three circles of radius s are drawn in the first quadrant of the xy -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the x -axis, and the third is tangent to the first circle and the y -axis. A circle of radius r > s is tangent to both axes and to the second and third circles. What is r/s ? [图] (\mathrm {A}) \ 5 (\mathrm {B}) \ 6 (\mathrm {C})\ 8 (\mathrm {D}) \ 9 (\mathrm {E})\ 10
💡 解题思路
Set $s =1$ so that we only have to find $r$ . Draw the segment between the center of the third circle and the large circle; this has length $r+1$ . We then draw the radius of the large circle that is
17
第 17 题
几何·角度
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1 . The cube is then cut in the same manner along the dashed lines shown in Figure 2 . This creates nine pieces. What is the volume of the piece that contains vertex W ? (\mathrm {A}) \ \frac{1}{12} (\mathrm {B}) \ \frac{1}{9} (\mathrm {C})\ \frac{1}{8} (\mathrm {D}) \ \frac{1}{6} (\mathrm {E})\ \frac{1}{4}
💡 解题思路
It is a pyramid with height $1$ and base area $\frac{1}{4}$ , so using the formula for the volume of a pyramid, $\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12} \Rightarrow \boxed
18
第 18 题
数论
Call a number "prime-looking" if it is composite but not divisible by 2 , 3 , or 5 . The three smallest prime-looking numbers are 49 , 77 , and 91 . There are 168 prime numbers less than 1000 . How many prime-looking numbers are there less than 1000 ? (\mathrm {A}) \ 100 (\mathrm {B}) \ 102 (\mathrm {C})\ 104 (\mathrm {D}) \ 106 (\mathrm {E})\ 108
💡 解题思路
The given states that there are $168$ prime numbers less than $1000$ , which is a fact we must somehow utilize. Since there seems to be no easy way to directly calculate the number of "prime-looking"
19
第 19 题
计数
A faulty car odometer proceeds from digit 3 to digit 5 , always skipping the digit 4 , regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050 . If the odometer now reads 002005 , how many miles has the car actually traveled? (\mathrm {A}) \ 1404 (\mathrm {B}) \ 1462 (\mathrm {C})\ 1604 (\mathrm {D}) \ 1605 (\mathrm {E})\ 1804
💡 解题思路
We find the number of numbers with a $4$ and subtract from $2005$ . Quick counting tells us that there are $200$ numbers with a 4 in the hundreds place, $200$ numbers with a 4 in the tens place, and $
20
第 20 题
函数
For each x in [0,1] , define \[f(x) = \begin{cases} 2x, if 0 ≤ x ≤ \frac{1}{2} ; 2-2x, if \frac{1}{2} < x ≤ 1. \end{cases}\] Let f^{[2]}(x) = f(f(x)) , and f^{[n + 1]}(x) = f^{[n]}(f(x)) for each integer n ≥ 2 . For how many values of x in [0,1] is f^{[2005]}(x) = 1/2 ? (\mathrm {A}) \ 0 (\mathrm {B}) \ 2005 (\mathrm {C})\ 4010 (\mathrm {D}) \ 2005^2 (\mathrm {E})\ 2^{2005}
💡 解题思路
For the two functions $f(x)=2x,0\le x\le \frac{1}{2}$ and $f(x)=2-2x,\frac{1}{2}\le x\le 1$ ,as long as $f(x)$ is between $0$ and $1$ , $x$ will be in the right domain, so we don't need to worry about
21
第 21 题
整数运算
How many ordered triples of integers (a,b,c) , with a ≥ 2 , b ≥ 1 , and c ≥ 0 , satisfy both \log_{a}b = c^{2005} and a + b + c = 2005 ? (A) \ 0 (B) \ 1 (C) \ 2 (D) \ 3 (E) \ 4
💡 解题思路
$a^{c^{2005}} = b$
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第 22 题
几何·面积
A rectangular box P is inscribed in a sphere of radius r . The surface area of P is 384, and the sum of the lengths of its 12 edges is 112 . What is r ? (A)\ 8 (B)\ 10 (C)\ 12 (D)\ 14 (E)\ 16
💡 解题思路
Box P has dimensions $l$ , $w$ , and $h$ . Its surface area is \[2lw+2lh+2wh=384,\] and the sum of all its edges is \[l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28.\]
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第 23 题
概率
Two distinct numbers a and b are chosen randomly from the set \{2, 2^2, 2^3, \ldots, 2^{25}\} . What is the probability that \log_{a}b is an integer? (A)\ \frac{2}{25} (B)\ \frac{31}{300} (C)\ \frac{13}{100} (D)\ \frac{7}{50} (E)\ \frac{1}{2}
💡 解题思路
Let $\log_{a}b = z$ , so $a^z = b$ . Define $a = 2^x$ , $b = 2^y$ ; then $\left(2^x\right)^z = 2^{xz}= 2^y$ , so $x \mid y$ . Here we can just make a table and count the number of values of $y$ per va
24
第 24 题
几何·角度
Let P(x)=(x-1)(x-2)(x-3) . For how many polynomials Q(x) does there exist a polynomial R(x) of degree 3 such that P(Q(x)) = P(x) · R(x) ? \mathrm {(A) } \ 19 \mathrm {(B) } \ 22 \mathrm {(C) } \ 24 \mathrm {(D) } \ 27 \mathrm {(E) } \ 32
💡 解题思路
We can write the problem as
25
第 25 题
几何·面积
Let S be the set of all points with coordinates (x,y,z) , where x , y , and z are each chosen from the set \{0,1,2\} . How many equilateral triangles all have their vertices in S ? (\mathrm {A}) \ 72 (\mathrm {B}) \ 76 (\mathrm {C})\ 80 (\mathrm {D}) \ 84 (\mathrm {E})\ 88
💡 解题思路
For this solution, we will just find as many equilateral triangles as possible, until it becomes intuitive that there are no more size of triangles left.
