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
The first two tasks took $\text{2:40 PM}-\text{1:00 PM}=100$ minutes. Thus, each task takes $100\div 2=50$ minutes. So the third task finishes at $\text{2:40 PM}+50$ minutes $=\fbox{\textbf{(B)}\; \te
3
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 $a$ be the number written two times, and $b$ the number written three times. Then $2a + 3b = 100$ . Plugging in $a = 28$ doesn't yield an integer for $b$ , so it must be that $b = 28$ , and we get
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第 4 题
综合
This problem is also problem number 5 on the 2015 AMC 10B, just with different names. This is the link. Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?
💡 解题思路
Let --- denote any of the 6 racers not named. Then the correct order looks like this:
5
第 5 题
行程问题
The Tigers beat the Sharks 2 out of the 3 times they played. They then played N more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for N ?
💡 解题思路
The ratio of the Shark's victories to games played is $\frac{1}{3}$ . For $N$ to be at its smallest, the Sharks must win all the subsequent games because $\frac{1}{3} < \frac{95}{100}$ . Then we can w
6
第 6 题
数论
Back in 1930, Tillie had to memorize her multiplication facts from 0 × 0 to 12 × 12 . The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?
💡 解题思路
There are a total of $(12+1) \times (12+1) = 169$ products (don't forget to include the 0's!), and a product is odd if and only if both its factors are odd. There are $6$ odd numbers between $0$ and $
7
第 7 题
几何·角度
A regular 15-gon has L lines of symmetry, and the smallest positive angle for which it has rotational symmetry is R degrees. What is L+R ?
💡 解题思路
From consideration of a smaller regular polygon with an odd number of sides (e.g. a pentagon), we see that the lines of symmetry go through a vertex of the polygon and bisect the opposite side. Hence
8
第 8 题
综合
What is the value of (625^{\log_5 2015})^{\frac{1}{4}} ?
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is \tfrac{1}{2} , independently of what has happened before. What is the probability that Larry wins the game?
💡 解题思路
If Larry wins, he either wins on the first move, or the third move, or the fifth move, etc. Let $W$ represent "player wins", and $L$ represent "player loses". Then the events corresponding to Larry wi
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第 10 题
几何·面积
How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?
💡 解题思路
Since we want non-congruent triangles that are neither isosceles nor equilateral, we can just list side lengths $(a,b,c)$ with $a<b<c$ . Furthermore, "positive area" tells us that $c < a + b$ and the
11
第 11 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
Clearly the line and the coordinate axes form a right triangle. Since the x-intercept and y-intercept are 5 and 12 respectively, 5 and 12 are two sides of the triangle that are not the hypotenuse, and
12
第 12 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 ? https://youtu.be/ba6w1OhXqOQ?t=423 ~ pi_is_3.14
💡 解题思路
The left-hand side of the equation can be factored as $(x-b)(x-a+x-c) = (x-b)(2x-(a+c))$ , from which it follows that the roots of the equation are $x=b$ , and $x=\tfrac{a+c}{2}$ . The sum of the root
13
第 13 题
几何·面积
Quadrilateral ABCD is inscribed in a circle with \angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4, and BC=6 . What is AC ?
💡 解题思路
$\angle ADB$ and $\angle ACB$ are both subtended by segment $AB$ , hence $\angle ACB = \angle ADB = 40^\circ$ . By considering $\triangle ABC$ , it follows that $\angle ABC = 180^\circ - (70^\circ + 4
14
第 14 题
几何·面积
A circle of radius 2 is centered at A . An equilateral triangle with side 4 has a vertex at A . What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?
💡 解题思路
The area of the circle is $\pi \cdot 2^2 = 4\pi$ , and the area of the triangle is $\frac{4^2 \cdot\sqrt{3}}{4} = 4\sqrt{3}$ . The difference between the area of the region that lies inside the circle
15
第 15 题
概率
At Rachelle's school, an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C in each of English and History. She thinks she has a \frac{1}{6} chance of getting an A in English, and a \tfrac{1}{4} chance of getting a B. In History, she has a \frac{1}{4} chance of getting an A, and a \frac{1}{3} chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5 ?
💡 解题思路
The probability that Rachelle gets a C in English is $1-\frac{1}{6}-\frac{1}{4} = \frac{7}{12}$ .
16
第 16 题
几何·面积
A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?
💡 解题思路
The distance from a corner to the center is 6, and from the corner to the top of the pyramid is 8, so the height is $\sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28} = 2\sqrt{7}$ .
17
第 17 题
概率
An unfair coin lands on heads with a probability of \tfrac{1}{4} . When tossed n>1 times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of n ?
💡 解题思路
When tossed $n$ times, the probability of getting exactly 2 heads and the rest tails is
18
第 18 题
数论
For every composite positive integer n , define r(n) to be the sum of the factors in the prime factorization of n . For example, r(50) = 12 because the prime factorization of 50 is 2 × 5^{2} , and 2 + 5 + 5 = 12 . What is the range of the function r , \{r(n): n is a composite positive integer\} ?
💡 解题思路
This problem becomes simple once we recognize that the domain of the function is $\{4, 6, 8, 9, 10, 12, 14, 15, \dots\}$ . By evaluating $r(4)$ to be $4$ , we can see that $\textbf{(E)}$ is incorrect.
19
第 19 题
几何·面积
In \triangle ABC , \angle C = 90^\circ and AB = 12 . Squares ABXY and CBWZ are constructed outside of the triangle. The points X , Y , Z , and W lie on a circle. What is the perimeter of the triangle?
For every positive integer n , let mod_5 (n) be the remainder obtained when n is divided by 5. Define a function f: \{0,1,2,3,\dots\} × \{0,1,2,3,4\} \to \{0,1,2,3,4\} recursively as follows: \[f(i,j) = \begin{cases}mod_5 (j+1) & if i = 0 and 0 \le j \le 4 , ; f(i-1,1) & if i \ge 1 and j = 0 , and ; f(i-1, f(i,j-1)) & if i \ge 1 and 1 \le j \le 4. \end{cases}\] What is f(2015,2) ?
💡 解题思路
Simply take some time to draw a table of values of $f(i,j)$ for the first few values of $i$ :
21
第 21 题
规律与数列
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 that 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
We can translate this wordy problem into this simple equation:
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第 22 题
计数
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
💡 解题思路
Consider shifting every person over three seats left after each person has gotten up and sat back down again, we are doing this because after this transformation we will only have to count the cases i
23
第 23 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
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第 24 题
几何·面积
Four circles, no two of which are congruent, have centers at A , B , C , and D , and points P and Q lie on all four circles. The radius of circle A is \tfrac{5}{8} times the radius of circle B , and the radius of circle C is \tfrac{5}{8} times the radius of circle D . Furthermore, AB = CD = 39 and PQ = 48 . Let R be the midpoint of \overline{PQ} . What is \overline{AR}+\overline{BR}+\overline{CR}+\overline{DR} ?
💡 解题思路
First, note that $PQ$ lies on the radical axis of any of the pairs of circles. Suppose that $O_1$ and $O_2$ are the centers of two circles $C_1$ and $C_2$ that intersect exactly at $P$ and $Q$ , with
25
第 25 题
几何·面积
A bee starts flying from point P_0 . She flies 1 inch due east to point P_1 . For j \ge 1 , once the bee reaches point P_j , she turns 30^{\circ} counterclockwise and then flies j+1 inches straight to point P_{j+1} . When the bee reaches P_{2015} she is exactly a √(b) + c √(d) inches away from P_0 , where a , b , c and d are positive integers and b and d are not divisible by the square of any prime. What is a+b+c+d ?
💡 解题思路
Let $x=e^{i\pi/6}$ , a $30^\circ$ counterclockwise rotation centered at the origin. Notice that $P_k$ on the complex plane is: