The arithmetic mean of the nine numbers in the set \{9, 99, 999, 9999, \ldots, 999999999\} is a 9 -digit number M , all of whose digits are distinct. The number M doesn't contain the digit (A)\ 0 (B)\ 2 (C)\ 4 (D)\ 6 (E)\ 8
💡 解题思路
We wish to find $\frac{9+99+\cdots +999999999}{9}$ , or $\frac{9(1+11+111+\cdots +111111111)}{9}=123456789$ . This doesn't have the digit 0, so the answer is $\boxed{\mathrm{(A)}\ 0}$
4
第 4 题
方程
What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x=4 ? (A)\ 0 (B)\ 1 (C)\ 10 (D)\ 11 (E)\ 12
💡 解题思路
By the distributive property,
5
第 5 题
几何·面积
Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. (A) \ 3π (B) \ 4π (C) \ 6π (D) \ 9π (E) \ 12π
💡 解题思路
A line going through the centers of the two smaller circles also goes through the diameter. The length of this line within the circle is $3+3+2+2=10.$ Because this is the length of the larger circle's
6
第 6 题
数论
For how many positive integers n is n^2 - 3n + 2 a prime number? (A)\ none (B)\ one (C)\ two (D)\ more\ than\ two,\ but\ finitely\ many (E)\ infinitely\ many
💡 解题思路
Factoring, we get $n^2 - 3n + 2 = (n-2)(n-1)$ . Either $n-1$ or $n-2$ is odd, and the other is even. Their product must yield an even number. The only prime that is even is $2$ , which is when $n$ is
7
第 7 题
逻辑推理
Let n be a positive integer such that \frac 12 + \frac 13 + \frac 17 + \frac 1n is an integer. Which of the following statements is not true: (A)\ 2\ divides\ n (B)\ 3\ divides\ n (C)\ 6\ divides\ n (D)\ 7\ divides\ n (E)\ n > 84
Suppose July of year N has five Mondays. Which of the following must occur five times in the August of year N ? (Note: Both months have 31 days.) \textrm{(A)}\ Monday \textrm{(B)}\ Tuesday \textrm{(C)}\ Wednesday \textrm{(D)}\ Thursday \textrm{(E)}\ Friday
💡 解题思路
If there are five Mondays, there are only three possibilities for their dates: $(1,8,15,22,29)$ , $(2,9,16,23,30)$ , and $(3,10,17,24,31)$ .
9
第 9 题
统计
Using the letters A , M , O , S , and U , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" USAMO occupies position (A) \ 112 (B) \ 113 (C) \ 114 (D) \ 115 (E) \ 116
💡 解题思路
There are $4!\cdot 4$ "words" beginning with each of the first four letters alphabetically. From there, there are $3!\cdot 3$ with $U$ as the first letter and each of the first three letters alphabeti
10
第 10 题
方程
Suppose that a and b are nonzero real numbers, and that the equation x^2 + ax + b = 0 has solutions a and b . Then the pair (a,b) is (A)\ (-2,1) (B)\ (-1,2) (C)\ (1,-2) (D)\ (2,-1) (E)\ (4,4)
💡 解题思路
Since $(x-a)(x-b) = x^2 - (a+b)x + ab = x^2 + ax + b = 0$ , it follows by comparing coefficients that $-a - b = a$ and that $ab = b$ . Since $b$ is nonzero, $a = 1$ , and $-1 - b = 1 \Longrightarrow b
11
第 11 题
几何·面积
The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares ? (A)\ 50 (B)\ 77 (C)\ 110 (D)\ 149 (E)\ 194
💡 解题思路
Let the three consecutive positive integers be $a-1$ , $a$ , and $a+1$ . Since the mean is $a$ , the sum of the integers is $3a$ . So $8$ times the sum is just $24a$ . With this, we now know that $a(a
12
第 12 题
方程
For which of the following values of k does the equation \frac{x-1}{x-2} = \frac{x-k}{x-6} have no solution for x ?
💡 解题思路
The domain over which we solve the equation is $\mathbb{R} \setminus \{2,6\}$ .
13
第 13 题
综合
Find the value(s) of x such that 8xy - 12y + 2x - 3 = 0 is true for all values of y .
The number 25^{64}· 64^{25} is the square of a positive integer N . In decimal representation, the sum of the digits of N is (A) \ 7 (B) \ 14 (C) \ 21 (D) \ 28 (E) \ 35
💡 解题思路
Taking the root, we get $N=\sqrt{25^{64}\cdot 64^{25}}=5^{64}\cdot 8^{25}$ .
15
第 15 题
数论
The positive integers A, B, A-B, and A+B are all prime numbers. The sum of these four primes is (A)\ even (B)\ divisible\ by\ 3 (C)\ divisible\ by\ 5 (D)\ divisible\ by\ 7 (E)\ prime
💡 解题思路
Since $A-B$ and $A+B$ must have the same parity , and since there is only one even prime number, it follows that $A-B$ and $A+B$ are both odd. Thus one of $A, B$ is odd and the other even. Since $A+B
16
第 16 题
几何·面积
For how many integers n is \dfrac n{20-n} the square of an integer? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 10
💡 解题思路
Let $x^2 = \frac{n}{20-n}$ , with $x \ge 0$ (note that the solutions $x < 0$ do not give any additional solutions for $n$ ). Then rewriting, $n = \frac{20x^2}{x^2 + 1}$ . Since $\text{gcd}(x^2, x^2 +
17
第 17 题
几何·面积
A regular octagon ABCDEFGH has sides of length two. Find the area of \triangle ADG .
💡 解题思路
[asy] unitsize(1cm); defaultpen(0.8); pair[] A = new pair[8]; A[0]=(0,0); for (int i=1; i<8; ++i) A[i] = A[i-1] + 2*dir(45*(i-1)); draw( A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle ); label(
18
第 18 题
几何·面积
Four distinct circles are drawn in a plane . What is the maximum number of points where at least two of the circles intersect? (A)\ 8 (B)\ 9 (C)\ 10 (D)\ 12 (E)\ 16
💡 解题思路
For any given pair of circles, they can intersect at most $2$ times. Since there are ${4\choose 2} = 6$ pairs of circles, the maximum number of possible intersections is $6 \cdot 2 = 12$ . We can cons
19
第 19 题
规律与数列
Suppose that \{a_n\} is an arithmetic sequence with \[a_1+a_2+·s+a_{100}=100 and a_{101}+a_{102}+·s+a_{200}=200.\] What is the value of a_2 - a_1 ?(A) \ 0.0001 (B) \ 0.001 (C) \ 0.01 (D) \ 0.1 (E) \ 1
💡 解题思路
We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...
20
第 20 题
综合
Let a , b , and c be real numbers such that a-7b+8c=4 and 8a+4b-c=7 . Then a^2-b^2+c^2 is (A)\ 0 (B)\ 1 (C)\ 4 (D)\ 7 (E)\ 8
💡 解题思路
Rearranging, we get $a+8c=7b+4$ and $8a-c=7-4b$
21
第 21 题
几何·面积
Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? (A)\ Andy (B)\ Beth (C)\ Carlos (D)\ Andy\ and \ Carlos\ tie\ for\ first. (E)\ All\ three\ tie.
💡 解题思路
We say Andy's lawn has an area of $x$ . Beth's lawn thus has an area of $\frac{x}{2}$ , and Carlos's lawn has an area of $\frac{x}{3}$ .
22
第 22 题
几何·面积
Let \triangle XOY be a right-angled triangle with m\angle XOY = 90^{\circ} . Let M and N be the midpoints of legs OX and OY , respectively. Given that XN = 19 and YM = 22 , find XY . (A)\ 24 (B)\ 26 (C)\ 28 (D)\ 30 (E)\ 32
💡 解题思路
Let $OM = x$ , $ON = y$ . By the Pythagorean Theorem on $\triangle XON, MOY$ respectively, \begin{align*} (2x)^2 + y^2 &= 19^2\\ x^2 + (2y)^2 &= 22^2\end{align*}
23
第 23 题
规律与数列
Let \{a_k\} be a sequence of integers such that a_1=1 and a_{m+n}=a_m+a_n+mn, for all positive integers m and n. Then a_{12} is (A) \ 45 (B) \ 56 (C) \ 67 (D) \ 78 (E) \ 89
💡 解题思路
When $m=1$ , $a_{n+1}=1+a_n+n$ . Hence, \[a_{2}=1+a_1+1\] \[a_{3}=1+a_2+2\] \[a_{4}=1+a_3+3\] \[\dots\] \[a_{12}=1+a_{11}+11\] Adding these equations up, we have that $a_{12}=12+(1+2+3+...+11)=\boxed{
24
第 24 题
几何·面积
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom? (A) \ 5 (B) \ 6 (C) \ 7.5 (D) \ 10 (E) \ 15 https://www.youtube.com/watch?v=H7K81Z3hvfo ~MathKatana
💡 解题思路
We can let this circle represent the ferris wheel with center $O,$ and $C$ represent the desired point $10$ feet above the bottom. Draw a diagram like the one above. We find out $\triangle OBC$ is a $
25
第 25 题
统计
When 15 is appended to a list of integers, the mean is increased by 2 . When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1 . How many integers were in the original list? (A) \ 4 (B) \ 5 (C) \ 6 (D) \ 7 (E) \ 8
💡 解题思路
Let $x$ be the sum of the integers and $y$ be the number of elements in the list. Then we get the equations $\dfrac{x+15}{y+1}=\dfrac{x}{y}+2$ and $\dfrac{x+15+1}{y+1+1}=\dfrac{x+16}{y+2}=\frac{x}{y}+