A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
💡 解题思路
It is easy to see that the ferry boat takes $6$ trips total. The total number of people taken to the island is
3
第 3 题
几何·面积
Rectangle ABCD , pictured below, shares 50\% of its area with square EFGH . Square EFGH shares 20\% of its area with rectangle ABCD . What is \frac{AB}{AD} ?
💡 解题思路
If we shift $A$ to coincide with $E$ , and add new horizontal lines to divide $EFGH$ into five equal parts:
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第 4 题
逻辑推理
If x<0 , then which of the following must be positive?
💡 解题思路
$x$ is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use $-1$ .
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第 5 题
计数
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next n shots are bullseyes she will be guaranteed victory. What is the minimum value for n ?
💡 解题思路
Let $k$ be the number of points Chelsea currently has. In order to guarantee victory, we must consider the possibility that the opponent scores the maximum amount of points by getting only bullseyes.
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第 6 题
规律与数列
A palindrome , such as 83438 , is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x ?
💡 解题思路
$x$ is at most $999$ , so $x+32$ is at most $1031$ . The minimum value of $x+32$ is $1000$ . However, the only palindrome between $1000$ and $1032$ is $1001$ , which means that $x+32$ must be $1001$ .
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第 7 题
统计
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
💡 解题思路
The water tower holds $\frac{100000}{0.1} = 1000000$ times more water than Logan's miniature. The volume of a sphere is: $V=\dfrac{4}{3}\pi r^3$ . Since we are comparing the heights (m), we should com
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第 8 题
几何·面积
Triangle ABC has AB=2 · AC . Let D and E be on \overline{AB} and \overline{BC} , respectively, such that \angle BAE = \angle ACD . Let F be the intersection of segments AE and CD , and suppose that \triangle CFE is equilateral. What is \angle ACB ?
💡 解题思路
Let $\angle BAE = \angle ACD = x$ .
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第 9 题
几何·面积
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
💡 解题思路
Imagine making the cuts one at a time. The first cut removes a box $2\times 2\times 3$ . The second cut removes two boxes, each of dimensions $2\times 2\times 0.5$ , and the third cut does the same as
10
第 10 题
规律与数列
The first four terms of an arithmetic sequence are p , 9 , 3p-q , and 3p+q . What is the 2010^th term of this sequence?
💡 解题思路
$3p-q$ and $3p+q$ are consecutive terms, so the common difference is $(3p+q)-(3p-q) = 2q$ .
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第 11 题
方程
The solution of the equation 7^{x+7} = 8^x can be expressed in the form x = \log_b 7^7 . What is b ?
💡 解题思路
This problem is quickly solved with knowledge of the laws of exponents and logarithms.
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第 12 题
计数
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements. Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs?
💡 解题思路
Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.
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第 13 题
坐标几何
For how many integer values of k do the graphs of x^2+y^2=k^2 and xy = k not intersect?
💡 解题思路
The image below shows the two curves for $k=4$ . The blue curve is $x^2+y^2=k^2$ , which is clearly a circle with radius $k$ , and the red curve is a part of the curve $xy=k$ .
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第 14 题
几何·面积
Nondegenerate \triangle ABC has integer side lengths, \overline{BD} is an angle bisector, AD = 3 , and DC=8 . What is the smallest possible value of the perimeter?
💡 解题思路
By the Angle Bisector Theorem , we know that $\frac{AB}{BC} = \frac{3}{8}$ . If we use the lowest possible integer values for $AB$ and $BC$ (the lengths of $AD$ and $DC$ , respectively), then $AB + BC
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第 15 题
概率
A coin is altered so that the probability that it lands on heads is less than \frac{1}{2} and when the coin is flipped four times, the probability of an equal number of heads and tails is \frac{1}{6} . What is the probability that the coin lands on heads?
💡 解题思路
Let $x$ be the probability of flipping heads. It follows that the probability of flipping tails is $1-x$ .
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第 16 题
概率
Bernardo randomly picks 3 distinct numbers from the set \{1,2,3,...,7,8,9\} and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set \{1,2,3,...,6,7,8\} and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
💡 解题思路
We can solve this by breaking the problem down into $2$ cases and adding up the probabilities.
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第 17 题
几何·面积
Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r . The area of \triangle ACE is 70\% of the area of the hexagon. What is the sum of all possible values of r ?
💡 解题思路
It is clear that $\triangle ACE$ is an equilateral triangle. From the Law of Cosines on $\triangle ABC$ , we get that $AC^2 = r^2+1^2-2r\cos{\frac{2\pi}{3}} = r^2+r+1$ . Therefore, the area of $\trian
18
第 18 题
几何·面积
A 16-step path is to go from (-4,-4) to (4,4) with each step increasing either the x -coordinate or the y -coordinate by 1. How many such paths stay outside or on the boundary of the square -2 \le x \le 2 , -2 \le y \le 2 at each step?
💡 解题思路
Each path must go through either the second or the fourth quadrant. Each path that goes through the second quadrant must pass through exactly one of the points $(-4,4)$ , $(-3,3)$ , and $(-2,2)$ .
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第 19 题
概率
Each of 2010 boxes in a line contains a single red marble, and for 1 \le k \le 2010 , the box in the kth position also contains k white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let P(n) be the probability that Isabella stops after drawing exactly n marbles. What is the smallest value of n for which P(n) < \frac{1}{2010} ?
💡 解题思路
The probability of drawing a white marble from box $k$ is $\frac{k}{k + 1}$ , and the probability of drawing a red marble from box $k$ is $\frac{1}{k+1}$ .
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第 20 题
规律与数列
Arithmetic sequences (a_n) and (b_n) have integer terms with a_1=b_1=1<a_2 \le b_2 and a_n b_n = 2010 for some n . What is the largest possible value of n ?
💡 解题思路
Since $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1$ , we can write the terms of each sequence as
21
第 21 题
坐标几何
The graph of y=x^6-10x^5+29x^4-4x^3+ax^2 lies above the line y=bx+c except at three values of x , where the graph and the line intersect. What is the largest of these values?
💡 解题思路
The $x$ values in which $y=x^6-10x^5+29x^4-4x^3+ax^2$ intersect at $y=bx+c$ are the same as the zeros of $y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c$ .
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第 22 题
函数
What is the minimum value of f(x)=|x-1| + |2x-1| + |3x-1| + ·s + |119x - 1 | ?
💡 解题思路
If we graph each term separately, we will notice that all of the zeros occur at $\frac{1}{m}$ , where $m$ is any integer from $1$ to $119$ , inclusive: $|mx-1|=0\implies mx=1\implies x=\frac{1}{m}$ .
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第 23 题
数论
The number obtained from the last two nonzero digits of 90! is equal to n . What is n ? Let P be the result of dividing 90! by tens such that P is not divisible by 10 . We want to consider P \mod 100 . But because 100 is not prime, and because P is obviously divisible by 4 (if in doubt, look at the answer choices), we only need to consider P \mod 25 . However, 25 is a very particular number. 1 · 2 · 3 · 4 \equiv -1 (mod 25) , and so is 6 · 7 · 8 · 9 . How can we group terms to take advantage of this fact? There might be a problem when you cancel out the 10 s from 90! . One method is to cancel out a factor of 2 from an existing number along with a factor of 5 . But this might prove cumbersome, as the grouping method will not be as effective. Instead, take advantage of inverses in modular arithmetic. Just leave the negative powers of 2 in a "storage base," and take care of the other terms first. Then, use Fermat's Little Theorem to solve for the power of 2 . Video Solution: https://youtu.be/30CamkkifHM?t=766
💡 解题思路
We will use the fact that for any integer $n$ , \begin{align*}(5n+1)(5n+2)(5n+3)(5n+4)&=[(5n+4)(5n+1)][(5n+2)(5n+3)]\\ &=(25n^2+25n+4)(25n^2+25n+6)\equiv 4\cdot 6\\ &=24\pmod{25}\equiv -1\pmod{25}.\en
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第 24 题
函数
Let f(x) = \log_{10} (\sin(π x) · \sin(2 π x) · \sin (3 π x) ·s \sin(8 π x)) . The intersection of the domain of f(x) with the interval [0,1] is a union of n disjoint open intervals. What is n ?
💡 解题思路
The question asks for the number of disjoint open intervals, which means we need to find the number of disjoint intervals such that the function is defined within them.
25
第 25 题
几何·面积
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
💡 解题思路
It should first be noted that given any quadrilateral of fixed side lengths, there is exactly one way to manipulate the angles so that the quadrilateral becomes cyclic.