2013B 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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1
第 1 题
统计
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was 3 . In degrees, what was the low temperature in Lincoln that day?
💡 解题思路
Let $L$ be the low temperature. The high temperature is $L+16$ . The average is $\frac{L+(L+16)}{2}=3$ . Solving for $L$ , we get $L=\boxed{\textbf{(C)} \ -5}$
2
第 2 题
几何·面积
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr. Green's steps is 2 feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
💡 解题思路
Since each step is $2$ feet, his garden is $30$ by $40$ feet. Thus, the area of $30(40) = 1200$ square feet. Since he is expecting $\frac{1}{2}$ of a pound per square foot, the total amount of potatoe
3
第 3 题
计数
When counting from 3 to 201 , 53 is the 51^{st} number counted. When counting backwards from 201 to 3 , 53 is the n^{th} number counted. What is n ?
💡 解题思路
Note that $n$ is equal to the number of integers between $53$ and $201$ , inclusive. Thus, $n=201-53+1=\boxed{\textbf{(D)}\ 149}$
4
第 4 题
综合
💡 解题思路
Let Ray and Tom drive 40 miles. Ray's car would require $\frac{40}{40}=1$ gallon of gas and Tom's car would require $\frac{40}{10}=4$ gallons of gas. They would have driven a total of $40+40=80$ miles
5
第 5 题
统计
The average age of 33 fifth-graders is 11 . The average age of 55 of their parents is 33 . What is the average age of all of these parents and fifth-graders?
💡 解题思路
The sum of the ages of the fifth graders is $33 * 11$ , while the sum of the ages of the parents is $55 * 33$ . Therefore, the total sum of all their ages must be $2178$ , and given $33 + 55 = 88$ peo
6
第 6 题
方程
Real numbers x and y satisfy the equation x^2+y^2=10x-6y-34 . What is x+y ? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 6 (E)\ 8
💡 解题思路
If we move every term dependent on $x$ or $y$ to the LHS, we get $x^2 - 10x + y^2 + 6y = -34$ . Adding $34$ to both sides, we have $x^2 - 10x + y^2 + 6y + 34 = 0$ . We can split the $34$ into $25$ and
7
第 7 题
计数
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying ``1" , so Blair follows by saying ``1, 2" . Jo then says ``1, 2, 3" , and so on. What is the 53^{rd} number said?
💡 解题思路
We notice that the number of numbers said is incremented by one each time; that is, Jo says one number, then Blair says two numbers, then Jo says three numbers, etc. Thus, after nine "turns", $1+2+3+4
8
第 8 题
几何·面积
Line l_1 has equation 3x - 2y = 1 and goes through A = (-1, -2) . Line l_2 has equation y = 1 and meets line l_1 at point B . Line l_3 has positive slope, goes through point A , and meets l_2 at point C . The area of \triangle ABC is 3 . What is the slope of l_3 ?
💡 解题思路
Line $l_1$ has the equation $y=3x/2-1/2$ when rearranged. Substituting $1$ for $y$ , we find that line $l_2$ will meet this line at point $(1,1)$ , which is point $B$ . We call $\overline{BC}$ the bas
9
第 9 题
几何·面积
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12! ?
💡 解题思路
Looking at the prime numbers under $12$ , we see that there are $\left\lfloor\frac{12}{2}\right\rfloor+\left\lfloor\frac{12}{2^2}\right\rfloor+\left\lfloor\frac{12}{2^3}\right\rfloor=6+3+1=10$ factors
10
第 10 题
综合
Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
💡 解题思路
If Alex goes to the red booth 3 times, then goes to the blue booth once, Alex can exchange 6 red tokens for 4 silver tokens and one red token. Similarly, if Alex goes to the blue booth 2 times, then g
11
第 11 题
规律与数列
Two bees start at the same spot and fly at the same rate in the following directions. Bee A travels 1 foot north, then 1 foot east, then 1 foot upwards, and then continues to repeat this pattern. Bee B travels 1 foot south, then 1 foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly 10 feet away from each other?
💡 解题思路
Let A and B begin at $(0,0,0)$ . In $6$ steps, $A$ will have done his route twice, ending up at $(2,2,2)$ , and $B$ will have done his route three times, ending at $(-3,-3,0)$ . Their distance is $\sq
12
第 12 题
综合
Cities A , B , C , D , and E are connected by roads \widetilde{AB} , \widetilde{AD} , \widetilde{AE} , \widetilde{BC} , \widetilde{BD} , \widetilde{CD} , and \widetilde{DE} . How many different routes are there from A to B that use each road exactly once? (Such a route will necessarily visit some cities more than once.) [图]
💡 解题思路
Note that cities $C$ and $E$ can be removed when counting paths because if a path goes in to $C$ or $E$ , there is only one possible path to take out of cities $C$ or $E$ . So the diagram is as follow
13
第 13 题
几何·面积
The internal angles of quadrilateral ABCD form an arithmetic progression. Triangles ABD and DCB are similar with \angle DBA = \angle DCB and \angle ADB = \angle CBD . Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of ABCD ?
💡 解题思路
Since the angles of Quadrilateral $ABCD$ form an arithmetic sequence, we can assign each angle with the value $a$ , $a+d$ , $a+2d$ , and $a+3d$ . Also, since these angles form an arithmetic progressio
14
第 14 题
规律与数列
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is N . What is the smallest possible value of N ?
💡 解题思路
Let the first two terms of the first sequence be $x_{1}$ and $x_{2}$ and the first two of the second sequence be $y_{1}$ and $y_{2}$ . Computing the seventh term, we see that $5x_{1} + 8x_{2} = 5y_{1}
15
第 15 题
整数运算
The number 2013 is expressed in the form where a_1 \ge a_2 \ge ·s \ge a_m and b_1 \ge b_2 \ge ·s \ge b_n are positive integers and a_1 + b_1 is as small as possible. What is |a_1 - b_1| ?
💡 解题思路
The prime factorization of $2013$ is $61\cdot11\cdot3$ . To have a factor of $61$ in the numerator and to minimize $a_1,$ $a_1$ must equal $61$ . Now we notice that there can be no prime $p$ which is
16
第 16 题
几何·面积
Let ABCDE be an equiangular convex pentagon of perimeter 1 . The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let s be the perimeter of this star. What is the difference between the maximum and the minimum possible values of s ?
💡 解题思路
The five pointed star can be thought of as five triangles sitting on the five sides of the pentagon. Because the pentagon is equiangular, each of its angles has measure $\frac{180^\circ (5-2)}{5}=108^
17
第 17 题
综合
Let a,b, and c be real numbers such that \[a+b+c=2, and\] \[a^2+b^2+c^2=12\] What is the difference between the maximum and minimum possible values of c ? (A) 2 (B) \frac{10}{3} (C) 4 (D) \frac{16}{3} (E) \frac{20}{3}
💡 解题思路
Note that $a+b= 2-c$ . Now, by Cauchy-Schwarz , we have that $(a^2+b^2) \ge \frac{(2-c)^2}{2}$ . Therefore, we have that $\frac{(2-c)^2}{2}+c^2 \le 12$ . We then find the roots of $c$ that satisfy equ
18
第 18 题
概率
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with 2013 coins and when the game starts with 2014 coins? (A) Barbara will win with 2013 coins and Jenna will win with 2014 coins. (B) Jenna will win with 2013 coins, and whoever goes first will win with 2014 coins. (C) Barbara will win with 2013 coins, and whoever goes second will win with 2014 coins. (D) Jenna will win with 2013 coins, and Barbara will win with 2014 coins. (E) Whoever goes first will win with 2013 coins, and whoever goes second will win with 2014 coins.
💡 解题思路
We split into 2 cases: 2013 coins, and 2014 coins.
19
第 19 题
几何·面积
In triangle ABC , AB=13 , BC=14 , and CA=15 . Distinct points D , E , and F lie on segments \overline{BC} , \overline{CA} , and \overline{DE} , respectively, such that \overline{AD}\perp\overline{BC} , \overline{DE}\perp\overline{AC} , and \overline{AF}\perp\overline{BF} . The length of segment \overline{DF} can be written as \frac{m}{n} , where m and n are relatively prime positive integers. What is m+n ? [图]
💡 解题思路
Since $\angle{AFB}=\angle{ADB}=90^{\circ}$ , quadrilateral $ABDF$ is cyclic. It follows that $\angle{ADE}=\angle{ABF}$ , so $\triangle ABF \sim \triangle ADE$ are similar. In addition, $\triangle ADE
20
第 20 题
综合
For 135^\circ < x < 180^\circ , points P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x) and S =(\tan x, \tan^2 x) are the vertices of a trapezoid. What is \sin(2x) ?
💡 解题思路
Let $f,g,h,$ and $j$ be $\cos{x}, \cot{x}, \sin{x},$ and $\tan{x}$ , respectively. Then, we have four points $(f,f^2),(g,g^2),(h,h^2),(j,j^2)$ , and a pair of lines each connecting two points must be
21
第 21 题
坐标几何
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form y=ax+b with a and b integers such that a\in \{-2,-1,0,1,2\} and b\in \{-3,-2,-1,1,2,3\} . No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
💡 解题思路
Being on two parabolas means having the same distance from the common focus and both directrices. In particular, you have to be on an angle bisector of the directrices, and clearly on the same "side"
22
第 22 题
方程
Let m>1 and n>1 be integers. Suppose that the product of the solutions for x of the equation \[8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0\] is the smallest possible integer. What is m+n ?
💡 解题思路
Rearranging logs, the original equation becomes
23
第 23 题
计数
Bernardo chooses a three-digit positive integer N and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer S . For example, if N = 749 , Bernardo writes the numbers 10,\!444 and 3,\!245 , and LeRoy obtains the sum S = 13,\!689 . For how many choices of N are the two rightmost digits of S , in order, the same as those of 2N ?
💡 解题思路
First, we can examine the units digits of the number base 5 and base 6 and eliminate some possibilities.
24
第 24 题
几何·面积
Let ABC be a triangle where M is the midpoint of \overline{AC} , and \overline{CN} is the angle bisector of \angle{ACB} with N on \overline{AB} . Let X be the intersection of the median \overline{BM} and the bisector \overline{CN} . In addition \triangle BXN is equilateral with AC=2 . What is BN^2 ?
💡 解题思路
Let $BN=x$ and $NA=y$ . From the conditions, let's deduct some convenient conditions that seem sufficient to solve the problem.
25
第 25 题
整数运算
Let G be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+·s+c_2z^2+c_1z+50,\] where c_1,c_2,·s, c_{n-1} are integers and P(z) has distinct roots of the form a+ib with a and b integers. How many polynomials are in G ?
💡 解题思路
If we factor into irreducible polynomials (in $\mathbb{Q}[x]$ ), each factor $f_i$ has exponent $1$ in the factorization and degree at most $2$ (since the $a+bi$ with $b\ne0$ come in conjugate pairs w