Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
💡 解题思路
Cagney can frost one in $20$ seconds, and Lacey can frost one in $30$ seconds. Working together, they can frost one in $\frac{20\cdot30}{20+30} = \frac{600}{50} = 12$ seconds. In $300$ seconds ( $5$ m
3
第 3 题
行程问题
A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold n grams of clay. What is n ?
💡 解题思路
The first box has volume $2\times3\times5=30\text{ cm}^3$ , and the second has volume $(2\times2)\times(3\times3)\times(5)=180\text{ cm}^3$ . The second has a volume that is $6$ times greater, so it h
4
第 4 题
分数与比例
In a bag of marbles, \frac{3}{5} of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
💡 解题思路
Assume that there are 5 total marbles in the bag. The actual number does not matter, since all we care about is the ratios, and the only operation performed on the marbles in the bag is doubling.
5
第 5 题
行程问题
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 280 pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?
💡 解题思路
So let the number of blueberries be $b,$ the number of raspberries be $r,$ the number of grapes be $g,$ and finally the number of cherries be $c.$
6
第 6 题
规律与数列
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
💡 解题思路
Let the three numbers be equal to $a$ , $b$ , and $c$ . We can now write three equations:
7
第 7 题
几何·面积
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
💡 解题思路
Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series),
8
第 8 题
统计
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
💡 解题思路
The iterative average of any 5 integers $a,b,c,d,e$ is defined as:
9
第 9 题
数论
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
💡 解题思路
In this solution we refer to moving to the left as decreasing the year or date number and moving to the right as increasing the year or date number. Every non-leap year we move to the right results in
10
第 10 题
几何·面积
A triangle has area 30 , one side of length 10 , and the median to that side of length 9 . Let \theta be the acute angle formed by that side and the median. What is \sin{\theta} ?
💡 解题思路
$AB$ is the side of length $10$ , and $CD$ is the median of length $9$ . The altitude of $C$ to $AB$ is $6$ because the 0.5(altitude)(base)=Area of the triangle. $\theta$ is $\angle CDE$ . To find $\s
11
第 11 题
概率
Alex, Mel, and Chelsea play a game that has 6 rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is \frac{1}{2} , and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?
💡 解题思路
If $m$ is the probability Mel wins and $c$ is the probability Chelsea wins, $m=2c$ and $m+c=\frac12$ . From this we get $m=\frac13$ and $c=\frac16$ . For Alex to win three, Mel to win two, and Chelsea
12
第 12 题
几何·面积
A square region ABCD is externally tangent to the circle with equation x^2+y^2=1 at the point (0,1) on the side CD . Vertices A and B are on the circle with equation x^2+y^2=4 . What is the side length of this square?
💡 解题思路
The circles have radii of $1$ and $2$ . Draw the triangle shown in the figure above and write expressions in terms of $s$ (length of the side of the square) for the sides of the triangle. Because $AO$
13
第 13 题
计数
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
💡 解题思路
Let Paula work at a rate of $p$ , the two helpers work at a combined rate of $h$ , and the time it takes to eat lunch be $L$ , where $p$ and $h$ are in house/hours and L is in hours. Then the labor on
14
第 14 题
几何·面积
The closed curve in the figure is made up of 9 congruent circular arcs each of length \frac{2π}{3} , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? [图]
A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 90 ^{\circ} clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
💡 解题思路
First, look for invariants. The center, unaffected by rotation, must be black. So automatically, the chance is less than $\frac{1}{2}.$ Note that a $90^{\circ}$ rotation requires that black squares be
16
第 16 题
几何·面积
Circle C_1 has its center O lying on circle C_2 . The two circles meet at X and Y . Point Z in the exterior of C_1 lies on circle C_2 and XZ=13 , OZ=11 , and YZ=7 . What is the radius of circle C_1 ? [图]
💡 解题思路
Let $r$ denote the radius of circle $C_1$ . Note that quadrilateral $ZYOX$ is cyclic. By Ptolemy's Theorem, we have $11XY=13r+7r$ and $XY=20r/11$ . Let $t$ be the measure of angle $YOX$ . Since $YO=OX
17
第 17 题
数论
Let S be a subset of \{1,2,3,\dots,30\} with the property that no pair of distinct elements in S has a sum divisible by 5 . What is the largest possible size of S ?
💡 解题思路
Of the integers from $1$ to $30$ , there are six each of $0,1,2,3,4\ (\text{mod}\ 5)$ . We can create several rules to follow for the elements in subset $S$ . No element can be $1\ (\text{mod}\ 5)$ if
18
第 18 题
几何·面积
Triangle ABC has AB=27 , AC=26 , and BC=25 . Let I be the intersection of the internal angle bisectors of \triangle ABC . What is BI ?
💡 解题思路
Inscribe circle $C$ of radius $r$ inside triangle $ABC$ so that it meets $AB$ at $Q$ , $BC$ at $R$ , and $AC$ at $S$ . Note that angle bisectors of triangle $ABC$ are concurrent at the center $O$ (als
19
第 19 题
计数
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? (A)\ 60 (B)\ 170 (C)\ 290 (D)\ 320 (E)\ 660
💡 解题思路
Note that if $n$ is the number of friends each person has, then $n$ can be any integer from $1$ to $4$ , inclusive, truly.
20
第 20 题
综合
Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)·s (x^{1024}+1024)\] The coefficient of x^{2012} is equal to 2^a . What is a ?
💡 解题思路
Every term in the expansion of the product is formed by taking one term from each factor and multiplying them all together. Therefore, we pick a power of $x$ or a power of $2$ from each factor.
21
第 21 题
整数运算
Let a , b , and c be positive integers with a\geb\gec such that a^2-b^2-c^2+ab=2011 and a^2+3b^2+3c^2-3ab-2ac-2bc=-1997 . What is a ?
💡 解题思路
Add the two equations.
22
第 22 题
立体几何
Distinct planes p_1,p_2,....,p_k intersect the interior of a cube Q . Let S be the union of the faces of Q and let P =\bigcup_{j=1}^{k}p_{j} . The intersection of P and S consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of Q . What is the difference between the maximum and minimum possible values of k ?
💡 解题思路
We need two different kinds of planes that only intersect $Q$ at the mentioned segments (we call them traces in this solution). These will be all the possible $p_j$ 's.
23
第 23 题
几何·面积
Let S be the square one of whose diagonals has endpoints (1/10,7/10) and (-1/10,-7/10) . A point v=(x,y) is chosen uniformly at random over all pairs of real numbers x and y such that 0 \le x \le 2012 and 0\le y\le 2012 . Let T(v) be a translated copy of S centered at v . What is the probability that the square region determined by T(v) contains exactly two points with integer coefficients in its interior?
💡 解题思路
The unit square's diagonal has a length of $\sqrt{0.2^2 + 1.4^2} = \sqrt{2}$ . Because $S$ square is not parallel to the axis, the two points must be adjacent.
24
第 24 题
规律与数列
Let \{a_k\}_{k=1}^{2011} be the sequence of real numbers defined by a_1=0.201,a_2=(0.2011)^{a_1},a_3=(0.20101)^{a_2},a_4=(0.201011)^{a_3} , and in general, \[a_k=\begin{cases}(0.\underbrace{20101·s 0101}_{k+2 digits})^{a_{k-1}} if k is odd, ; (0.\underbrace{20101·s 01011}_{k+2 digits})^{a_{k-1}} if k is even.\end{cases}\] Rearranging the numbers in the sequence \{a_k\}_{k=1}^{2011} in decreasing order produces a new sequence \{b_k\}_{k=1}^{2011} . What is the sum of all integers k , 1\le k \le 2011 , such that a_k=b_k?
💡 解题思路
First, we must understand two important bull functions: $f(x) = b^x$ for $0 0$ (increasing power function for positive $x$ ). $f(x)$ is used to establish inequalities when we change the exponent and
25
第 25 题
分数与比例
Let f(x)=|2\{x\}-1| where \{x\} denotes the fractional part of x . The number n is the smallest positive integer such that the equation \[nf(xf(x))=x\] has at least 2012 real solutions. What is n ? Note: the fractional part of x is a real number y=\{x\} such that 0\le y<1 and x-y is an integer.
💡 解题思路
Our goal is to determine how many times the graph of $nf(xf(x))=x$ intersects the graph of $y=x$ . (Conversely, we can also divide the equation by $n$ to get $f(xf(x))=\frac{x}{n}$ and look at the gra