Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1\!:\!00 PM and finishes the second task at 2\!:\!40 PM. When does she finish the third task?
💡 解题思路
Marie finishes $2$ tasks in $1$ hour and $40$ minutes. Therefore, one task should take $50$ minutes to finish. $50$ minutes after $2\!:\!40$ PM is $3\!:\!30$ PM, so our answer is $\boxed{\textbf{(B) }
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第 3 题
规律与数列
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100 , and one of the numbers is 28. What is the other number?
💡 解题思路
Let the first number be $x$ and the second be $y$ . We have $2x+3y=100$ . We are given one of the numbers is $28$ . If $x$ were to be $28$ , $y$ would not be an integer, thus $y=28$ . $2x+3(28)=100$ ,
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第 4 题
规律与数列
Four siblings ordered an extra large pizza. Alex ate \frac15 , Beth \frac13 , and Cyril \frac14 of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
💡 解题思路
Let the pizza have $60$ slices, since the least common multiple of $(5,3,4)=60$ . Therefore, Alex ate $\frac{1}{5}\times60=12$ slices, Beth ate $\frac{1}{3}\times60=20$ slices, and Cyril ate $\frac{1}
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第 5 题
整数运算
David, Hikmet, Jack, Marta, Rand, and Todd were in a 12 -person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6 th place. Who finished in 8 th place?
💡 解题思路
Because Marta was $6$ th, Jack was $5$ th, so Todd was $3$ rd. Thus, Rand was $2$ nd and the 8th place finisher was $\boxed{\mathbf{(B)}\ \mathrm{Hikmet}}$
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第 6 题
综合
Mahdi practices exactly one sport each day of the week. He runs three days a week but never on two consecutive days. On Monday he plays basketball and two days later golf. He swims and plays tennis, but he never plays tennis the day after running or swimming. Which day of the week does Mahdi swim?
💡 解题思路
Mahdi does basketball on Monday and golf on Wednesday. Since there are four consecutive days between Wednesday and Monday exclusive, he must run on Tuesday. Thus, he must also run on Thursday and Satu
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第 7 题
分数与比例
Consider the operation "minus the reciprocal of," defined by a\diamond b=a-\frac{1}{b} . What is ((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) ?
💡 解题思路
$1\diamond2=1-\dfrac{1}{2}=\dfrac{1}{2}$ , so $(1\diamond2)\diamond3=\dfrac{1}{2}\diamond3=\dfrac{1}{2}-\dfrac{1}{3}=\dfrac{1}{6}$ . Also, $2\diamond3=2-\dfrac{1}{3}=\dfrac{5}{3}$ , so $1\diamond(2\di
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第 8 题
时间问题
The letter F shown below is rotated 90^\circ clockwise around the origin, then reflected in the y -axis, and then rotated a half turn around the origin. What is the final image?
💡 解题思路
The first rotation moves the base of the $F$ to the negative $y$ -axis, and the stem to the positive $x$ -axis. The reflection then moves the stem to the negative $x$ -axis, with the base unchanged. T
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第 9 题
几何·面积
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius 3 and center (0,0) that lies in the first quadrant, the portion of the circle with radius \tfrac{3}{2} and center (0,\tfrac{3}{2}) that lies in the first quadrant, and the line segment from (0,0) to (3,0) . What is the area of the shark's fin falcata? [图]
💡 解题思路
The area of the shark's fin falcata is just the area of the quarter-circle minus the area of the semicircle. The quarter-circle has radius $3$ so it has area $\dfrac{9\pi}{4}$ . The semicircle has rad
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第 10 题
综合
💡 解题思路
Since $-5>-2015$ , the product must end with a $5$ .
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第 11 题
数论
Among the positive integers less than 100 , each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
💡 解题思路
The one digit prime numbers are $2$ , $3$ , $5$ , and $7$ . So there are a total of $4\cdot4=16$ ways to choose a two digit number with both digits as primes and $4$ ways to choose a one digit prime,
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第 12 题
几何·面积
For how many integers x is the point (x, -x) inside or on the circle of radius 10 centered at (5, 5) ? https://www.youtube.com/watch?v=BeD8xOvfzE0 ~Education, the Study of Everything=
💡 解题思路
The equation of the circle is $(x-5)^2+(y-5)^2=100$ . Plugging in the given conditions we have $(x-5)^2+(-x-5)^2 \leq 100$ . Expanding gives: $x^2-10x+25+x^2+10x+25\leq 100$ , which simplifies to $x^2
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第 13 题
几何·面积
The line 12x+5y=60 forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
💡 解题思路
We find the $x$ -intercepts and the $y$ -intercepts to find the intersections of the axes and the line. If $x=0$ , then $y=12$ . If $y$ is $0$ , then $x=5$ . Our three vertices are $(0,0)$ , $(5,0)$ ,
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第 14 题
方程
Let a , b , and c be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation (x-a)(x-b)+(x-b)(x-c)=0 ?
💡 解题思路
Expanding the equation and combining like terms results in $2x^2-(a+2b+c)x+(ab+bc)=0$ . By Vieta's formula the sum of the roots is $\dfrac{-[-(a+2b+c)]}{2}=\dfrac{a+2b+c}{2}$ . To maximize this expres
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第 15 题
综合
The town of Hamlet has 3 people for each horse, 4 sheep for each cow, and 3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
💡 解题思路
Let the amount of people be $p$ , horses be $h$ , sheep be $s$ , cows be $c$ , and ducks be $d$ . We know \[3p=h\] \[4s=c\] \[3d=p\] Then the total amount of people, horses, sheep, cows, and ducks may
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第 16 题
数论
Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10 , inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's? https://www.youtube.com/watch?v=vulB2z_PdRE&feature=youtu.be
💡 解题思路
We can solve this problem with a brute force approach.
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第 17 题
立体几何
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron? [图]
💡 解题思路
The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals $4$ and $5$ , for an area $A=10$ . The height $h$ , of one pyramid, is $
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第 18 题
概率
Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
💡 解题思路
We can simplify the problem first, then apply reasoning to the original problem. Let's say that there are $8$ coins. Shaded coins flip heads, and blank coins flip tails. So, after the first flip;
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第 19 题
几何·面积
In \triangle{ABC} , \angle{C} = 90^{\circ} and AB = 12 . Squares ABXY and ACWZ are constructed outside of the triangle. The points X, Y, Z , and W lie on a circle. What is the perimeter of the triangle?
💡 解题思路
The center of the circle lies on the intersection between the perpendicular bisectors of chords $ZW$ and $YX$ . Therefore we know the center of the circle must also be the midpoint of the hypotenuse.
20
第 20 题
立体几何
Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let s denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of s ?
💡 解题思路
Let $n$ be the number of steps. We have
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第 22 题
综合
In the figure shown below, ABCDE is a regular pentagon and AG=1 . What is FG + JH + CD ? [图]
💡 解题思路
[asy] pair A=(cos(pi/5)-sin(pi/10),cos(pi/10)+sin(pi/5)), B=(2*cos(pi/5)-sin(pi/10),cos(pi/10)), C=(1,0), D=(0,0), E1=(-sin(pi/10),cos(pi/10)); //(0,0) is a convenient point //E1 to prevent conflict w
23
第 23 题
分数与比例
Let n be a positive integer greater than 4 such that the decimal representation of n! ends in k zeros and the decimal representation of (2n)! ends in 3k zeros. Let s denote the sum of the four least possible values of n . What is the sum of the digits of s ?
💡 解题思路
A trailing zero requires a factor of two and a factor of five. Since factors of two occur more often than factors of five, we can focus on the factors of five. We make a chart of how many trailing zer
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第 24 题
坐标几何
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin p_0=(0,0) facing to the east and walks one unit, arriving at p_1=(1,0) . For n=1,2,3,\dots , right after arriving at the point p_n , if Aaron can turn 90^\circ left and walk one unit to an unvisited point p_{n+1} , he does that. Otherwise, he walks one unit straight ahead to reach p_{n+1} . Thus the sequence of points continues p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0) , and so on in a counterclockwise spiral pattern. What is p_{2015} ? [图]
💡 解题思路
The first thing we would do is track Aaron's footsteps:
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第 25 题
几何·面积
A rectangular box measures a × b × c , where a , b , and c are integers and 1≤ a ≤ b ≤ c . The volume and the surface area of the box are numerically equal. How many ordered triples (a,b,c) are possible?
💡 解题思路
We need \[abc = 2(ab+bc+ac) \quad \text{ or } \quad (a-2)bc = 2a(b+c).\] Since $a\le b$ and $a,b,c$ are all positive $,ac \le bc$ . From the first equation we get $abc \le 6bc$ . Thus $a\le 6$ . From