2005B 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
应用题
A scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two for 1 dollar. What was their profit, in dollars?
A positive number x has the property that x\% of x is 4 . What is x ?
💡 解题思路
Since $x\%$ means $0.01x$ , the statement " $x\% \text{ of } x \text{ is 4}$ " can be rewritten as " $0.01x \cdot x = 4$ ":
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第 3 题
分数与比例
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
💡 解题思路
Let $m =$ Brianna's money. We have $\frac15 m = \frac13 (\mbox{CDs}) \Rightarrow \frac35 m = (\mbox{CDs})$ . Thus, the money left over is $m-\frac35m = \frac25m$ , so the answer is $\boxed{\textbf{(C)
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第 4 题
综合
At the beginning of the school year, Lisa's goal was to earn an A on at least 80\% of her 50 quizzes for the year. She earned an A on 22 of the first 30 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A ?
💡 解题思路
Lisa's goal was to get an $A$ on $80\% \cdot 50 = 40$ quizzes. She already has $A$ 's on $22$ quizzes, so she needs to get $A$ 's on $40-22=18$ more. There are $50-30=20$ quizzes left, so she can affo
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第 5 题
几何·面积
An 8 -foot by 10 -foot bathroom floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? [图]
💡 解题思路
There are $80$ tiles. Each tile has $[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$ shaded. Thus:
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第 6 题
几何·面积
In \triangle ABC , we have AC=BC=7 and AB=2 . Suppose that D is a point on line AB such that B lies between A and D and CD=8 . What is BD ?
💡 解题思路
Draw height $CH$ (Perpendicular line from point C to line AD). We have that $BH=1$ . By the Pythagorean Theorem , $CH=\sqrt{48}$ . Since $CD=8$ , $HD=\sqrt{8^2-48}=\sqrt{16}=4$ , and $BD=HD-1$ , so $B
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第 7 题
几何·面积
What is the area enclosed by the graph of |3x|+|4y|=12 ? (A)\ 6 (B)\ 12 (C)\ 16 (D)\ 24 (E)\ 25
💡 解题思路
If we get rid of the absolute values, we are left with the following 4 equations (using the logic that if $|a|=b$ , then $a$ is either $b$ or $-b$ ):
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第 8 题
函数
For how many values of a is it true that the line y = x + a passes through the vertex of the parabola y = x^2 + a^2 ? (A)\ 0 (B)\ 1 (C)\ 2 (D)\ 10 (E)\ infinitely many
💡 解题思路
We see that the vertex of the quadratic function $y = x^2 + a^2$ is $(0,\,a^2)$ . The y-intercept of the line $y = x + a$ is $(0,\,a)$ . We want to find the values (if any) such that $a=a^2$ . Solving
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第 9 题
统计
On a certain math exam, 10\% of the students got 70 points, 25\% got 80 points, 20\% got 85 points, 15\% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?
💡 解题思路
To begin, we see that the remaining $30\%$ of the students got $95$ points. Assume that there are $20$ students; we see that $2$ students got $70$ points, $5$ students got $80$ points, $4$ students go
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第 10 题
规律与数列
The first term of a sequence is 2005 . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the {2005}^{th} term of the sequence? (A) 29 (B) 55 (C) 85 (D) 133 (E) 250
💡 解题思路
Performing this operation several times yields the results of $133$ for the second term, $55$ for the third term, and $250$ for the fourth term. The sum of the cubes of the digits of $250$ equal $133$
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第 11 题
概率
An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is 20$ or more?
💡 解题思路
The only way to get a total of \$ $20$ or more is if you pick a twenty and another bill, or if you pick both tens. There are a total of $\dbinom{8}{2}=\dfrac{8\times7}{2\times1}=28$ ways to choose $2$
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第 12 题
方程
The quadratic equation x^2+mx+n has roots twice those of x^2+px+m , and none of m,n, and p is zero. What is the value of n/p ?
💡 解题思路
Let $x^2 + px + m = 0$ have roots $a$ and $b$ . Then
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第 13 题
综合
Suppose that 4^{x_1}=5 , 5^{x_2}=6 , 6^{x_3}=7 , ... , 127^{x_{124}}=128 . What is x_1x_2...x_{124} ? (A)\ {{{2}}} (B)\ {{{\frac{5}{2}}}} (C)\ {{{3}}} (D)\ {{{\frac{7}{2}}}} (E)\ {{{4}}}
💡 解题思路
We see that we can re-write $4^{x_1}=5$ , $5^{x_2}=6$ , $6^{x_3}=7$ , ... , $127^{x_{124}}=128$ as $\left(...\left(\left(\left(4^{x_1}\right)^{x_2}\right)^{x_3}\right)...\right)^{x_{124}}=128$ by usin
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第 14 题
几何·面积
A circle having center (0,k) , with k>6 , is tangent to the lines y=x , y=-x and y=6 . What is the radius of this circle? (A)\ 6√(2)-6 (B)\ 6 (C)\ 6√(2) (D)\ 12 (E)\ 6+6√(2)
💡 解题思路
Let $R$ be the radius of the circle. Draw the two radii that meet the points of tangency to the lines $y = \pm x$ . We can see that a square is formed by the origin, two tangency points, and the cente
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第 15 题
规律与数列
The sum of four two-digit numbers is 221 . None of the eight digits is 0 and no two of them are the same. Which of the following is not included among the eight digits? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 5
💡 解题思路
$221$ can be written as the sum of four two-digit numbers, let's say $\overline{ae}$ , $\overline{bf}$ , $\overline{cg}$ , and $\overline{dh}$ . Then $221= 10(a+b+c+d)+(e+f+g+h)$ . The last digit of $
16
第 16 题
坐标几何
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? \mathrm {(A)}\ √(2) \mathrm {(B)}\ √(3) \mathrm {(C)}\ 1+√(2) \mathrm {(D)}\ 1+√(3) \mathrm {(E)}\ 3
💡 解题思路
The eight spheres are formed by shifting spheres of radius $1$ and center $(0, 0, 0)$ $\pm 1$ in the $x, y, z$ directions. Hence, the centers of the spheres are $(\pm 1, \pm 1, \pm 1)$ . For a sphere
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第 17 题
分数与比例
How many distinct four-tuples (a,b,c,d) of rational numbers are there with \[a·\log_{10}2+b·\log_{10}3+c·\log_{10}5+d·\log_{10}7=2005?\] (A)\ 0 (B)\ 1 (C)\ 17 (D)\ 2004 (E)\ infinitely many
💡 解题思路
Using the laws of logarithms , the given equation becomes
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第 18 题
几何·面积
Let A(2,2) and B(7,7) be points in the plane. Define R as the region in the first quadrant consisting of those points C such that \triangle ABC is an acute triangle. What is the closest integer to the area of the region R ? (A)\ 25 (B)\ 39 (C)\ 51 (D)\ 60 (E)\ 80
💡 解题思路
[asy] Label f; f.p=fontsize(6); xaxis(-1,15,Ticks(f, 2.0)); yaxis(-1,15,Ticks(f, 2.0)); pair A = MP("A",(2,2),SW), B = MP("B",(7,7),NE); D(A--B); filldraw((0,4)--(4,0)--(14,0)--(0,14)--cycle,gray); fi
19
第 19 题
数字运算
Let x and y be two-digit integers such that y is obtained by reversing the digits of x . The integers x and y satisfy x^{2}-y^{2}=m^{2} for some positive integer m . What is x+y+m ? (A)\ 88 (B)\ 112 (C)\ 116 (D)\ 144 (E)\ 154
💡 解题思路
let $x=10a+b$ , then $y=10b+a$ where $a$ and $b$ are nonzero digits.
20
第 20 题
综合
Let a,b,c,d,e,f,g and h be distinct elements in the set \{-7,-5,-3,-2,2,4,6,13\}. What is the minimum possible value of (a+b+c+d)^{2}+(e+f+g+h)^{2}?(A)\ 30 (B)\ 32 (C)\ 34 (D)\ 40 (E)\ 50
💡 解题思路
The sum of the set is $-7-5-3-2+2+4+6+13=8$ , so if we could have the sum in each set of parenthesis be $4$ then the minimum value would be $2(4^2)=32$ . Considering the set of four terms containing $
21
第 21 题
整数运算
A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n ? (A)\ {{{0}}} (B)\ {{{1}}} (C)\ {{{2}}} (D)\ {{{3}}} (E)\ {{{4}}}
💡 解题思路
We may let $n = 7^k \cdot m$ , where $m$ is not divisible by 7. Using the fact that the number of divisors function $d(n)$ is multiplicative, we have $d(n) = d(7^k)d(m) = (k+1)d(m) = 60$ . Also, $d(7n
22
第 22 题
规律与数列
A sequence of complex numbers z_{0}, z_{1}, z_{2}, ... is defined by the rule \[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\] where \overline {z_{n}} is the complex conjugate of z_{n} and i^{2}=-1 . Suppose that |z_{0}|=1 and z_{2005}=1 . How many possible values are there for z_{0} ?
💡 解题思路
Since $|z_0|=1$ , let $z_0=e^{i\theta_0}$ , where $\theta_0$ is an argument of $z_0$ . We will prove by induction that $z_n=e^{i\theta_n}$ , where $\theta_n=2^n(\theta_0+\frac{\pi}{2})-\frac{\pi}{2}$
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第 23 题
综合
Let S be the set of ordered triples (x,y,z) of real numbers for which \[\log_{10}(x+y) = z and \log_{10}(x^{2}+y^{2}) = z+1.\] There are real numbers a and b such that for all ordered triples (x,y,z) in S we have x^{3}+y^{3}=a · 10^{3z} + b · 10^{2z}. What is the value of a+b?
💡 解题思路
Let $x + y = s$ and $x^2 + y^2 = t$ . Then, $\log(s)=z$ implies $\log(10s) = z+1= \log(t)$ ,so $t=10s$ . Therefore, $x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})$ . Since $s = 10^z$ , w
24
第 24 题
几何·面积
All three vertices of an equilateral triangle are on the parabola y = x^2 , and one of its sides has a slope of 2 . The x -coordinates of the three vertices have a sum of m/n , where m and n are relatively prime positive integers. What is the value of m + n ? (A)\ {{{14}}} (B)\ {{{15}}} (C)\ {{{16}}} (D)\ {{{17}}} (E)\ {{{18}}}
💡 解题思路
Let the three points be at $A = (x_1, x_1^2)$ , $B = (x_2, x_2^2)$ , and $C = (x_3, x_3^2)$ , such that the slope between the first two is $2$ , and $A$ is the point with the least $y$ -coordinate.
25
第 25 题
概率
Six ants simultaneously stand on the six vertices of a regular octahedron , with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability . What is the probability that no two ants arrive at the same vertex? (A)\ \frac {5}{256} (B)\ \frac {21}{1024} (C)\ \frac {11}{512} (D)\ \frac {23}{1024} (E)\ \frac {3}{128}
💡 解题思路
We approach this problem by counting the number of ways ants can do their desired migration, and then multiply this number by the probability that each case occurs.