Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?
💡 解题思路
Letting $x$ be the third side, then by the triangle inequality, $20-15 < x < 20+15$ , or $5 < x < 35$ . Therefore the perimeter must be greater than 40 but less than 70. 72 is not in this range, so $\
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第 3 题
统计
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80 . After he graded Payton's test, the test average became 81 . What was Payton's score on the test?
💡 解题思路
If the average of the first $14$ peoples' scores was $80$ , then the sum of all of their tests is $14 \cdot 80 = 1120$ . When Payton's score was added, the sum of all of the scores became $15 \cdot 81
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第 4 题
分数与比例
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller number?
💡 解题思路
Let $a$ be the bigger number and $b$ be the smaller.
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第 5 题
整数运算
Sreshtha needs to estimate the quantity \frac{a}{b} - c , where a, b, and c are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of \frac{a}{b} - c ?
💡 解题思路
To maximize our estimate, we want to maximize $\frac{a}{b}$ and minimize $c$ , because both terms are positive values. Therefore we round $c$ down. To maximize $\frac{a}{b}$ , round $a$ up and $b$ dow
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第 6 题
分数与比例
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be 2 : 1 ?
💡 解题思路
This problem can be converted to a system of equations. Let $p$ be Pete's current age and $c$ be Claire's current age.
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第 7 题
立体几何
Two right circular cylinders have the same volume. The radius of the second cylinder is 10\% more than the radius of the first. What is the relationship between the heights of the two cylinders?
💡 解题思路
Let the radius of the first cylinder be $r_1$ and the radius of the second cylinder be $r_2$ . Also, let the height of the first cylinder be $h_1$ and the height of the second cylinder be $h_2$ . We a
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第 8 题
几何·面积
The ratio of the length to the width of a rectangle is 4 : 3 . If the rectangle has diagonal of length d , then the area may be expressed as kd^2 for some constant k . What is k ?
💡 解题思路
Let the rectangle have length $4x$ and width $3x$ . Then by $3-4-5$ triangles (or the Pythagorean Theorem), we have $d = 5x$ , and so $x = \dfrac{d}{5}$ . Hence, the area of the rectangle is $3x \cdot
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第 9 题
概率
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?
💡 解题思路
If Cheryl gets two marbles of the same color, then Claudia and Carol must take all four marbles of the two other colors. The probability of this happening, given that Cheryl has two marbles of a certa
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第 10 题
整数运算
Integers x and y with x>y>0 satisfy x+y+xy=80 . What is x ?
💡 解题思路
Use SFFT to get $(x+1)(y+1)=81$ . The terms $(x+1)$ and $(y+1)$ must be factors of $81$ , which include $1, 3, 9, 27, 81$ . Because $x > y$ , $x+1$ is equal to $27$ or $81$ . But if $x+1=81$ , then $y
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第 11 题
几何·面积
On a sheet of paper, Isabella draws a circle of radius 2 , a circle of radius 3 , and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly k \ge 0 lines. How many different values of k are possible?
💡 解题思路
Isabella can get $0$ lines if the circles are concentric, $1$ if internally tangent, $2$ if overlapping, $3$ if externally tangent, and $4$ if non-overlapping and not externally tangent. There are $\b
12
第 12 题
几何·面积
The parabolas y=ax^2 - 2 and y=4 - bx^2 intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area 12 . What is a+b ?
💡 解题思路
Clearly, the parabolas must intersect the x-axis at the same two points. Their distance multiplied by $4 - (-2)$ (the distance between the y-intercepts), all divided by 2 is equal to 12, the area of t
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第 13 题
逻辑推理
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?
💡 解题思路
We can eliminate answer choices $\textbf{(A)}$ and $\textbf{(B)}$ because there are an even number of scores, so if one is false, the other must be false too. Answer choice $\textbf{(C)}$ must be true
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第 14 题
综合
What is the value of a for which \frac{1}{\log_2a} + \frac{1}{\log_3a} + \frac{1}{\log_4a} = 1 ?
💡 解题思路
We use the change of base formula to show that \[\log_a b = \dfrac{\log_b b}{\log_b a} = \dfrac{1}{\log_b a}.\] Thus, our equation becomes \[\log_a 2 + \log_a 3 + \log_a 4 = 1,\] which becomes after c
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第 15 题
分数与比例
What is the minimum number of digits to the right of the decimal point needed to express the fraction \frac{123456789}{2^{26}· 5^4} as a decimal?
💡 解题思路
Multiply the numerator and denominator of the fraction by $5^{22}$ (which is the same as multiplying by 1) to give $\frac{5^{22} \cdot 123456789}{10^{26}}$ . Now, instead of thinking about this as a f
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第 16 题
立体几何
Tetrahedron ABCD has AB=5 , AC=3 , BC=4 , BD=4 , AD=3 , and CD=\tfrac{12}5\sqrt2 . What is the volume of the tetrahedron?
💡 解题思路
Drop altitudes of triangle $ABC$ and triangle $ABD$ down from $C$ and $D$ , respectively. Both will hit the same point; let this point be $T$ . Because both triangle $ABC$ and triangle $ABD$ are 3-4-5
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第 17 题
概率
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
💡 解题思路
We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by $2^8 = 256$ at the end. We casework on how many people are standing.
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第 18 题
规律与数列
The zeros of the function f(x) = x^2-ax+2a are integers. What is the sum of the possible values of a ?
💡 解题思路
The problem asks us to find the sum of every integer value of $a$ such that the roots of $x^2 - ax + 2a = 0$ are both integers.
19
第 19 题
几何·面积
For some positive integers p , there is a quadrilateral ABCD with positive integer side lengths, perimeter p , right angles at B and C , AB=2 , and CD=AD . How many different values of p<2015 are possible?
💡 解题思路
Let $BC = x$ and $CD = AD = y$ be positive integers. Drop a perpendicular from $A$ to $CD$ to show that, using the Pythagorean Theorem, that \[x^2 + (y - 2)^2 = y^2.\] Simplifying yields $x^2 - 4y + 4
20
第 20 题
几何·面积
Isosceles triangles T and T' are not congruent but have the same area and the same perimeter. The sides of T have lengths 5 , 5 , and 8 , while those of T' have lengths a , a , and b . Which of the following numbers is closest to b ?
💡 解题思路
The area of $T$ is $\dfrac{1}{2} \cdot 8 \cdot 3 = 12$ and the perimeter is 18.
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第 21 题
几何·面积
A circle of radius r passes through both foci of, and exactly four points on, the ellipse with equation x^2+16y^2=16. The set of all possible values of r is an interval [a,b). What is a+b?
💡 解题思路
We can graph the ellipse by seeing that the center is $(0, 0)$ and finding the ellipse's intercepts. The points where the ellipse intersects the coordinate axes are $(0, 1), (0, -1), (4, 0)$ , and $(-
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第 22 题
数论
For each positive integer n , let S(n) be the number of sequences of length n consisting solely of the letters A and B , with no more than three A s in a row and no more than three B s in a row. What is the remainder when S(2015) is divided by 12 ?
💡 解题思路
One method of approach is to find a recurrence for $S(n)$ .
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第 23 题
几何·面积
Let S be a square of side length 1. Two points are chosen independently at random on the sides of S . The probability that the straight-line distance between the points is at least \frac12 is \frac{a-bπ}{c} , where a,b, and c are positive integers and gcd(a,b,c) = 1 . What is a+b+c ?
💡 解题思路
Divide the boundary of the square into halves, thereby forming 8 segments. Without loss of generality, let the first point $A$ be in the bottom-left segment. Then, it is easy to see that any point in
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第 24 题
分数与比例
Rational numbers a and b are chosen at random among all rational numbers in the interval [0,2) that can be written as fractions \frac{n}{d} where n and d are integers with 1 \le d \le 5 . What is the probability that \[(cos(aπ)+isin(bπ))^4\] is a real number?
💡 解题思路
Let $\cos(a\pi) = x$ and $\sin(b\pi) = y$ . Consider the binomial expansion of the expression: \[x^4 + 4ix^{3}y - 6x^{2}y^{2} - 4ixy^3 + y^4.\]
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第 25 题
几何·面积
A collection of circles in the upper half-plane, all tangent to the x -axis, is constructed in layers as follows. Layer L_0 consists of two circles of radii 70^2 and 73^2 that are externally tangent. For k \ge 1 , the circles in \bigcup_{j=0}^{k-1}L_j are ordered according to their points of tangency with the x -axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer L_k consists of the 2^{k-1} circles constructed in this way. Let S=\bigcup_{j=0}^{6}L_j , and for every circle C denote by r(C) its radius. What is \[\sum_{C\in S} \frac{1}{√(r(C))}?\] [图]
💡 解题思路
Let us start with the two circles in $L_0$ and the circle in $L_1$ . Let the larger circle in $L_0$ be named circle $X$ with radius $x$ and the smaller be named circle $Y$ with radius $y$ . Also let t