2001 AMC 12 — Official Competition Problems (February 2024)
📅 2024 A 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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第 1 题
规律与数列
The sum of two numbers is S . Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
💡 解题思路
Suppose the two numbers are $a$ and $b$ , with $a+b=S$ . Then the desired sum is $2(a+3)+2(b+3)=2(a+b)+12=2S +12$ , which is answer $\boxed{\textbf{(E)}}$ .
2
第 2 题
规律与数列
Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n . For example, P(23) = 6 and S(23) = 5 . Suppose N is a two-digit number such that N = P(N)+S(N) . What is the units digit of N ? (A)\ 2 (B)\ 3 (C)\ 6 (D)\ 8 (E)\ 9
💡 解题思路
Denote $a$ and $b$ as the tens and units digit of $N$ , respectively. Then $N = 10a+b$ . It follows that $10a+b=ab+a+b$ , which implies that $9a=ab$ . Since $a\neq0$ , $b=9$ . So the units digit of $N
3
第 3 题
行程问题
The state income tax where Kristin lives is charged at the rate of p\% of the first \textdollar 28000 of annual income plus (p + 2)\% of any amount above \textdollar 28000 . Kristin noticed that the state income tax she paid amounted to (p + 0.25)\% of her annual income. What was her annual income?
💡 解题思路
Let $A$ , $T$ be Kristin's annual income and the income tax total, respectively. Notice that \begin{align*} T &= p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) \\ &= [p\%\cdot28000 + p\%\cdot(A - 28000)] +
4
第 4 题
统计
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5 . What is their sum?
💡 解题思路
Let $m$ be the mean of the three numbers. Then the least of the numbers is $m-10$ and the greatest is $m + 15$ . The middle of the three numbers is the median, $5$ . So $\dfrac{1}{3}[(m-10) + 5 + (m +
5
第 5 题
数论
What is the product of all positive odd integers less than 10000 ? (A)\ \dfrac{10000!}{(5000!)^2} (B)\ \dfrac{10000!}{2^{5000}} (C)\ \dfrac{9999!}{2^{5000}} (D)\ \dfrac{10000!}{2^{5000} · 5000!} (E)\ \dfrac{5000!}{2^{5000}} \[1(3)(5) ·s (9,999) = 9,999!!\] We can express the double factorial of an odd number (2n-1)!! in terms of standard factorials using the formula: \[(2n-1)!! = \frac{(2n)!}{(2n)!!} = \frac{(2n)!}{2^n(n!)}\] Setting 2n = 10,000 , we have n = 5,000 . Substituting these values into the formula gives: \[9,999!! = \frac{10,000!}{2^{5,000}(5,000!)}\]
💡 解题思路
If you did not see the pattern, then we may solve a easier problem.
6
第 6 题
数字运算
A telephone number has the form ABC-DEF-GHIJ , where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, A > B > C , D > E > F , and G > H > I > J . Furthermore, D , E , and F are consecutive even digits; G , H , I , and J are consecutive odd digits; and A + B + C = 9 . Find A .
💡 解题思路
We start by noting that there are $10$ letters, meaning there are $10$ digits in total. Listing them all out, we have $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ . Clearly, the most restrictive condition is the co
7
第 7 题
应用题
A charity sells 140 benefit tickets for a total of \2001$ . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
💡 解题思路
Let's multiply ticket costs by $2$ , then the half price becomes an integer, and the charity sold $140$ tickets worth a total of $4002$ dollars.
8
第 8 题
几何·面积
Which of the cones listed below can be formed from a 252^\circ sector of a circle of radius 10 by aligning the two straight sides? [图] (A) A cone with slant height of 10 and radius 6(B) A cone with height of 10 and radius 6(C) A cone with slant height of 10 and radius 7(D) A cone with height of 10 and radius 7(E) A cone with slant height of 10 and radius 8
Let f be a function satisfying f(xy) = \frac{f(x)}y for all positive real numbers x and y . If f(500) =3 , what is the value of f(600) ? (A)\ 1 (B)\ 2 (C)\ \frac52 (D)\ 3 (E)\ \frac{18}5
💡 解题思路
Letting $x = 500$ and $y = \dfrac{6}{5}$ in the given equation, we get $f(600) = f(500\cdot\frac{6}{5}) = \frac{3}{\left(\frac{6}{5}\right)} = \boxed{\textbf{(C) } \frac{5}{2}}$ .
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第 10 题
几何·面积
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to [图]
💡 解题思路
Consider any single tile:
11
第 11 题
概率
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
💡 解题思路
Imagine that we draw all the chips in random order, i.e., we do not stop when the last chip of a color is drawn. To draw out all the white chips first, the last chip left must be red, and all previous
12
第 12 题
数论
How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5 ?
💡 解题思路
Out of the numbers $1$ to $12$ four are divisible by $3$ and three by $4$ , counting $12$ twice. Hence $6$ out of these $12$ numbers are multiples of $3$ or $4$ .
13
第 13 题
方程
The parabola with equation p(x) = ax^2+bx+c and vertex (h,k) is reflected about the line y=k . This results in the parabola with equation q(x) = dx^2+ex+f . Which of the following equals a+b+c+d+e+f ? (A)\ 2b (B)\ 2c (C)\ 2a+2b (D)\ 2h (E)\ 2k
💡 解题思路
We write $p(x)$ as $a(x-h)^2+k$ (this is possible for any parabola). Then the reflection of $p(x)$ is $q(x) = -a(x-h)^2+k$ . Then we find $p(x) + q(x) = 2k$ . Since $p(1) = a+b+c$ and $q(1) = d+e+f$ ,
14
第 14 题
几何·面积
Given the nine-sided regular polygon A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9 , how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set \{A_1,A_2,\dots,A_9\} ? (A) 30 (B) 36 (C) 63 (D) 66 (E) 72
💡 解题思路
Each of the $\binom{9}{2} = 36$ pairs of vertices determines $2$ equilateral triangles — one facing towards the center, and one outwards — for a total of $2 \cdot 36 = 72$ triangles. However, the $3$
15
第 15 题
行程问题
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) (A) \frac {1}{2} \sqrt {3} (B) 1 (C) \sqrt {2} (D) \frac {3}{2} (E) 2
💡 解题思路
Given any path on the surface, we can unfold the surface into a plane to get a path of the same length in the plane. Consider the net of a tetrahedron in the picture below. A pair of opposite points i
16
第 16 题
规律与数列
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? (A) 8! (B) 2^8 · 8! (C) (8!)^2 (D) \frac {16!}{2^8} (E) 16!
💡 解题思路
Suppose the spider tries to put on all $2 \cdot 8 = 16$ items in a random order, so that each of the $16!$ possible permutations is equally probable. This means that for any fixed leg, the probability
17
第 17 题
概率
A point P is selected at random from the interior of the pentagon with vertices A = (0,2) , B = (4,0) , C = (2 π + 1, 0) , D = (2 π + 1,4) , and E=(0,4) . What is the probability that \angle APB is obtuse? (A) \frac {1}{5} (B) \frac {1}{4} (C) \frac {5}{16} (D) \frac {3}{8} (E) \frac {1}{2}
💡 解题思路
The angle $APB$ is obtuse if and only if $P$ lies inside the circle with diameter $AB$ . (This follows for example from the fact that the inscribed angle is half of the central angle for the same arc.
18
第 18 题
几何·面积
A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [图] (A) \frac {1}{3} (B) \frac {2}{5} (C) \frac {5}{12} (D) \frac {4}{9} (E) \frac {1}{2}
The polynomial p(x) = x^3+ax^2+bx+c has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The y -intercept of the graph of y=p(x) is 2. What is b ? (A)\ -11 (B)\ -10 (C)\ -9 (D)\ 1 (E)\ 5
💡 解题思路
We are given $c=2$ . So the product of the roots is $-c = -2$ by Vieta's formulas . These also tell us that $\frac{-a}{3}$ is the average of the zeros, so $\frac{-a}3=-2 \implies a = 6$ . We are also
20
第 20 题
几何·面积
Points A = (3,9) , B = (1,1) , C = (5,3) , and D=(a,b) lie in the first quadrant and are the vertices of quadrilateral ABCD . The quadrilateral formed by joining the midpoints of \overline{AB} , \overline{BC} , \overline{CD} , and \overline{DA} is a square. What is the sum of the coordinates of point D ? (A) 7 (B) 9 (C) 10 (D) 12 (E) 16
💡 解题思路
[asy] pair A=(3,9), B=(1,1), C=(5,3), D=(7,3); draw(A--B--C--D--cycle); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,N); label("$D$",D,E); pair AB = (A + B)/2, BC = (B + C)/2, CD = (C + D)/2, DA
21
第 21 题
整数运算
Four positive integers a , b , c , and d have a product of 8! and satisfy: \[\begin{array}{rl} ab + a + b & = 524 ; bc + b + c & = 146 ; cd + c + d & = 104 \end{array}\] What is a-d ? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12
💡 解题思路
Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
22
第 22 题
综合
💡 解题思路
Because we see that there are only lines and there is a rectangle, we can coordbash (place this figure on coordinates). Because this is a general figure, we can assume the sides are $7$ and $10$ (or a
23
第 23 题
几何·角度
A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? (A) \frac {1 + i \sqrt {11}}{2} (B) \frac {1 + i}{2} (C) \frac {1}{2} + i (D) 1 + \frac {i}{2} (E) \frac {1 + i \sqrt {13}}{2}
💡 解题思路
Let the polynomial be $P$ and let the two integer zeros be $z_1$ and $z_2$ . We can then write $P(x)=(x-z_1)(x-z_2)(x^2+ax+b)$ for some integers $a$ and $b$ .
24
第 24 题
几何·面积
In \triangle ABC , \angle ABC=45^\circ . Point D is on \overline{BC} so that 2· BD=CD and \angle DAB=15^\circ . Find \angle ACB.(A) 54^\circ (B) 60^\circ (C) 72^\circ (D) 75^\circ (E) 90^\circ
💡 解题思路
Draw a good diagram! Now, let's call $BD=t$ , so $DC=2t$ . Given the rather nice angles of $\angle ABD = 45^\circ$ and $\angle ADC = 60^\circ$ as you can see, let's do trig. Drop an altitude from $A$
25
第 25 题
规律与数列
Consider sequences of positive real numbers of the form x, 2000, y, \dots in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 2001 appear somewhere in the sequence? (A) 1 (B) 2 (C) 3 (D) 4 (E) more than 4
💡 解题思路
It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that $\forall$ (for all) $n>1:~ a_n = a_{n-1}a_{n+1} - 1$ . This can be r