$(-1)^n=1$ if n is even and $-1$ if n is odd. So we have
2
第 2 题
综合
For real numbers x and y , define x\spadesuit y = (x + y)(x - y) . What is 3\spadesuit(4\spadesuit 5) ? \text {(A) } - 72 \text {(B) } - 27 \text {(C) } - 24 \text {(D) } 24 \text {(E) } 72
💡 解题思路
$4\spadesuit 5=-9$
3
第 3 题
综合
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? \text {(A) } 10 \text {(B) } 14 \text {(C) } 17 \text {(D) } 20 \text {(E) } 24
💡 解题思路
If the Cougars won by a margin of 14 points, then the Panthers' score would be half of (34-14). That's 10 $\Rightarrow \boxed{\text{(A)}}$ .
4
第 4 题
分数与比例
Mary is about to pay for five items at the grocery store. The prices of the items are 7.99 , 4.99 , 2.99 , 1.99 , and 0.99 . Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the 20.00 that she will receive in change? \text {(A) } 5 \text {(B) } 10 \text {(C) } 15 \text {(D) } 20 \text {(E) } 25
💡 解题思路
The total price of the items is $(8-.01)+(5-.01)+(3-.01)+(2-.01)+(1-.01)=19-.05=18.95$
5
第 5 题
行程问题
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? \text {(A) } 30 \text {(B) } 50 \text {(C) } 60 \text {(D) } 90 \text {(E) } 120
💡 解题思路
The speed that Bob is catching up to John is $5-3=2$ miles per hour. Since Bob is one mile behind John, it will take $\frac{1}{2} \Rightarrow \text{(A)}$ of an hour to catch up to John.
6
第 6 题
综合
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? \text {(A) } 129 \text {(B) } 137 \text {(C) } 174 \text {(D) } 223 \text {(E) } 411
💡 解题思路
Francesca makes a total of $100+100+400=600$ grams of lemonade, and in those $600$ grams, there are $25$ calories from the lemon juice and $386$ calories from the sugar, for a total of $25+386=411$ ca
7
第 7 题
统计
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? \text {(A) } 4 \text {(B) } 12 \text {(C) } 16 \text {(D) } 24 \text {(E) } 48
💡 解题思路
First, we seat the children.
8
第 8 题
方程
The lines x = \frac 14y + a and y = \frac 14x + b intersect at the point (1,2) . What is a + b ? \text {(A) } 0 \text {(B) } \frac 34 \text {(C) } 1 \text {(D) } 2 \text {(E) } \frac 94
💡 解题思路
$4x-4a=\frac{1}{4}x+b$
9
第 9 题
数字运算
How many even three-digit integers have the property that their digits, all read from left to right, are in strictly increasing order? \text {(A) } 21 \text {(B) } 34 \text {(C) } 51 \text {(D) } 72 \text {(E) } 150
💡 解题思路
Let the integer have digits $a$ , $b$ , and $c$ , read left to right. Because $1 \leq a<b<c$ , none of the digits can be zero and $c$ cannot be 2. If $c=4$ , then $a$ and $b$ must each be chosen from
10
第 10 题
几何·面积
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? \text {(A) } 43 \text {(B) } 44 \text {(C) } 45 \text {(D) } 46 \text {(E) } 47
💡 解题思路
If the second size has length x, then the first side has length 3x, and we have the third side which has length 15. By the triangle inequality, we have: \[\\ x+15>3x \Rightarrow 2x<15 \Rightarrow x<7.
11
第 11 题
分数与比例
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? \text {(A) } \frac 67 \text {(B) } \frac {13}{14} \text {(C) } 1 \text {(D) } \frac {14}{13} \text {(E) } \frac 76
💡 解题思路
Joe has 2 ounces of cream, as stated in the problem.
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第 12 题
函数
The parabola y=ax^2+bx+c has vertex (p,p) and y -intercept (0,-p) , where p\ne 0 . What is b ? \text {(A) } -p \text {(B) } 0 \text {(C) } 2 \text {(D) } 4 \text {(E) } p
💡 解题思路
Substituting $(0,-p)$ , we find that $y = -p = a(0)^2 + b(0) + c = c$ , so our parabola is $y = ax^2 + bx - p$ .
13
第 13 题
几何·面积
Rhombus ABCD is similar to rhombus BFDE . The area of rhombus ABCD is 24, and \angle BAD = 60^\circ . What is the area of rhombus BFDE ? [图] \textrm{(A) } 6 \textrm{(B) } 4\sqrt {3} \textrm{(C) } 8 \textrm{(D) } 9 \textrm{(E) } 6\sqrt {3}
💡 解题思路
The ratio of any length on $ABCD$ to a corresponding length on $BFDE^2$ is equal to the ratio of their areas. Since $\angle BAD=60$ , $\triangle ADB$ and $\triangle DBC$ are equilateral. $DB$ , which
14
第 14 题
规律与数列
Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4 cents per glob and J blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is 2.53 . Assume that B , J and N are all positive integers with N>1 . What is the cost of the jam Elmo uses to make the sandwiches? (A)\ 1.05 (B)\ 1.25 (C)\ 1.45 (D)\ 1.65 (E)\ 1.85
💡 解题思路
From the given, we know that
15
第 15 题
几何·面积
Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B are on the circle centered at O , and points C and D are on the circle centered at P , such that \overline{AD} and \overline{BC} are common external tangents to the circles. What is the area of hexagon AOBCPD ? [图]
💡 解题思路
Draw the altitude from $O$ onto $DP$ and call the point $H$ . Because $\angle OAD$ and $\angle ADP$ are right angles due to being tangent to the circles, and the altitude creates $\angle OHD$ as a rig
16
第 16 题
几何·面积
Regular hexagon ABCDEF has vertices A and C at (0,0) and (7,1) , respectively. What is its area? (A)\ 20\sqrt {3} (B)\ 22\sqrt {3} (C)\ 25\sqrt {3} (D)\ 27\sqrt {3} (E)\ 50
💡 解题思路
To find the area of the regular hexagon, we only need to calculate the side length. a distance of $\sqrt{7^2+1^2} = \sqrt{50} = 5\sqrt{2}$ apart. Half of this distance is the length of the longer leg
17
第 17 题
分数与比例
For a particular peculiar pair of dice, the probabilities of rolling 1 , 2 , 3 , 4 , 5 and 6 on each die are in the ratio 1:2:3:4:5:6 . What is the probability of rolling a total of 7 on the two dice? (A)\ \frac 4{63} (B)\ \frac 18 (C)\ \frac 8{63} (D)\ \frac 16 (E)\ \frac 27
💡 解题思路
The probability of getting an $x$ on one of these dice is $\frac{x}{21}$ .
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第 18 题
综合
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? (A)\ 120 (B)\ 121 (C)\ 221 (D)\ 230 (E)\ 231
💡 解题思路
Let the starting point be $(0,0)$ . After $10$ steps we can only be in locations $(x,y)$ where $|x|+|y|\leq 10$ . Additionally, each step changes the parity of exactly one coordinate. Hence after $10$
19
第 19 题
数论
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? (A)\ 4 (B)\ 5 (C)\ 6 (D)\ 7 (E)\ 8
💡 解题思路
First, The number of the plate is divisible by $9$ and in the form of $aabb$ , $abba$ or $abab$ .
20
第 20 题
概率
Let x be chosen at random from the interval (0,1) . What is the probability that \lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0 ? Here \lfloor x\rfloor denotes the greatest integer that is less than or equal to x . (A)\ \frac 18 (B)\ \frac 3{20} (C)\ \frac 16 (D)\ \frac 15 (E)\ \frac 14
💡 解题思路
Let $k$ be an arbitrary integer. For which $x$ do we have $\lfloor\log_{10}4x\rfloor = \lfloor\log_{10}x\rfloor = k$ ?
21
第 21 题
几何·面积
Rectangle ABCD has area 2006 . An ellipse with area 2006π passes through A and C and has foci at B and D . What is the perimeter of the rectangle? (The area of an ellipse is abπ where 2a and 2b are the lengths of the axes.) (A)\ \frac {16\sqrt {2006}}{π} (B)\ \frac {1003}4 (C)\ 8\sqrt {1003} (D)\ 6\sqrt {2006} (E)\ \frac {32\sqrt {1003}}π
💡 解题思路
[asy] size(7cm); real l=10, w=7, ang=asin(w/sqrt(l*l+w*w))*180/pi; draw((-l,-w)--(l,-w)--(l,w)--(-l,w)--cycle); draw(rotate(ang)*ellipse((0,0),2*l+2*w,l*w*2/sqrt(l^2+w^2))); draw(rotate(ang)*((0,0)--(
22
第 22 题
数论
Suppose a , b and c are positive integers with a+b+c=2006 , and a!b!c!=m· 10^n , where m and n are integers and m is not divisible by 10 . What is the smallest possible value of n ? (A)\ 489 (B)\ 492 (C)\ 495 (D)\ 498 (E)\ 501
💡 解题思路
The power of $10$ for any factorial is given by the well-known algorithm \[\left\lfloor \frac n{5}\right\rfloor + \left\lfloor \frac n{25}\right\rfloor + \left\lfloor \frac n{125}\right\rfloor + \cdot
23
第 23 题
几何·面积
Isosceles \triangle ABC has a right angle at C . Point P is inside \triangle ABC , such that PA=11 , PB=7 , and PC=6 . Legs \overline{AC} and \overline{BC} have length s=√(a+b\sqrt{2)} , where a and b are positive integers. What is a+b ? [图] (A)\ 85 (B)\ 91 (C)\ 108 (D)\ 121 (E)\ 127
💡 解题思路
[asy] pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P
24
第 24 题
几何·面积
Let S be the set of all point (x,y) in the coordinate plane such that 0 \le x \le \frac{π}{2} and 0 \le y \le \frac{π}{2} . What is the area of the subset of S for which \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\] (A)\ \dfrac{π^2}{9} (B)\ \dfrac{π^2}{8} (C)\ \dfrac{π^2}{6} (D)\ \dfrac{3π^2}{16} (E)\ \dfrac{2π^2}{9}
💡 解题思路
We start out by solving the equality first. \begin{align*} \sin^2x - \sin x \sin y + \sin^2y &= \frac34 \\ \sin x &= \frac{\sin y \pm \sqrt{\sin^2 y - 4 ( \sin^2y - \frac34 ) }}{2} \\ &= \frac{\sin y
25
第 25 题
规律与数列
A sequence a_1,a_2,\dots of non-negative integers is defined by the rule a_{n+2}=|a_{n+1}-a_n| for n≥ 1 . If a_1=999 , a_2<999 and a_{2006}=1 , how many different values of a_2 are possible? (A)\ 165 (B)\ 324 (C)\ 495 (D)\ 499 (E)\ 660
💡 解题思路
We say the sequence $(a_n)$ completes at $i$ if $i$ is the minimal positive integer such that $a_i = a_{i + 1} = 1$ . Otherwise, we say $(a_n)$ does not complete.