2019 AMC 10B — Official Competition Problems (February 2019)
📅 2019 B 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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题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
分数与比例
Alicia had two containers. The first was \tfrac{5}{6} full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was \tfrac{3}{4} full of water. What is the ratio of the volume of the first container to the volume of the second container?
💡 解题思路
Let the first jar's volume be $A$ and the second's be $B$ . It is given that $\frac{5}{6}A=\frac{3}{4}B$ . We find that $\frac{A}{B}=\frac{\left(\frac{3}{4}\right)}{\left(\frac{5}{6}\right)}=\boxed{\t
2
第 2 题
数论
Consider the statement, "If n is not prime, then n-2 is prime." Which of the following values of n is a counterexample to this statement?
💡 解题思路
Since a counterexample must be a value of $n$ which is not prime, $n$ must be composite, so we eliminate $\text{A}$ and $\text{C}$ . Now we subtract $2$ from the remaining answer choices, and we see t
3
第 3 题
综合
In a high school with 500 students, 40\% of the seniors play a musical instrument, while 30\% of the non-seniors do not play a musical instrument. In all, 46.8\% of the students do not play a musical instrument. How many non-seniors play a musical instrument?
💡 解题思路
$60\%$ of seniors do not play a musical instrument. If we denote $x$ as the number of seniors, then \[\frac{3}{5}x + \frac{3}{10}\cdot(500-x) = \frac{468}{1000}\cdot500\]
4
第 4 题
坐标几何
All lines with equation ax+by=c such that a,b,c form an arithmetic progression pass through a common point. What are the coordinates of that point?
💡 解题思路
If all lines satisfy the condition, then we can just plug in values for $a$ , $b$ , and $c$ that form an arithmetic progression. Let's use $a=1$ , $b=2$ , $c=3$ , and $a=1$ , $b=3$ , $c=5$ . Then the
5
第 5 题
几何·面积
Triangle ABC lies in the first quadrant. Points A , B , and C are reflected across the line y=x to points A' , B' , and C' , respectively. Assume that none of the vertices of the triangle lie on the line y=x . Which of the following statements is not always true? (A) Triangle A'B'C' lies in the first quadrant. (B) Triangles ABC and A'B'C' have the same area. (C) The slope of line AA' is -1 . (D) The slopes of lines AA' and CC' are the same. (E) Lines AB and A'B' are perpendicular to each other.
💡 解题思路
Let's analyze all of the options separately.
6
第 6 题
规律与数列
There is a positive integer n such that (n+1)! + (n+2)! = n! · 440 . What is the sum of the digits of n ?
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or n pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of n ?
💡 解题思路
If he has enough money to buy $12$ pieces of red candy, $14$ pieces of green candy, and $15$ pieces of blue candy, then the smallest amount of money he could have is $\text{lcm}{(12,14,15)} = 420$ cen
8
第 8 题
几何·面积
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? [图]
💡 解题思路
We notice that the square can be split into $4$ congruent smaller squares, with the altitude of the equilateral triangle being the side of this smaller square. Note that area of shaded part in a quart
9
第 9 题
统计
The function f is defined by \[f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|\] for all real numbers x , where \lfloor r \rfloor denotes the greatest integer less than or equal to the real number r . What is the range of f ?
💡 解题思路
There are four cases we need to consider here.
10
第 10 题
几何·面积
In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of \triangle ABC is 50 units and the area of \triangle ABC is 100 square units?
💡 解题思路
Notice that whatever point we pick for $C$ , $AB$ will be the base of the triangle. Without loss of generality, let points $A$ and $B$ be $(0,0)$ and $(10,0)$ , since for any other combination of poin
11
第 11 题
分数与比例
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9:1 , and the ratio of blue to green marbles in Jar 2 is 8:1 . There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2 ?
💡 解题思路
Our ratios are $9:1$ in \( J_1 \) and $8:1$ in \( J_2 \).
12
第 12 题
规律与数列
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019 ?
💡 解题思路
Observe that $2019_{10} = 5613_7$ . To maximize the sum of the digits, we want as many $6$ s as possible (since $6$ is the highest value in base $7$ ), and this will occur with either of the numbers $
13
第 13 题
统计
What is the sum of all real numbers x for which the median of the numbers 4,6,8,17, and x is equal to the mean of those five numbers?
💡 解题思路
The mean is $\frac{4+6+8+17+x}{5}=\frac{35+x}{5}$ .
14
第 14 题
数字运算
The base-ten representation for 19! is 121,6T5,100,40M,832,H00 , where T , M , and H denote digits that are not given. What is T+M+H ?
💡 解题思路
We can figure out $H = 0$ by noticing that $19!$ will end with $3$ zeroes, as there are three factors of $5$ in its prime factorization, so there would be 3 powers of 10 meaning it will end in 3 zeros
15
第 15 题
几何·面积
Right triangles T_1 and T_2 , have areas of 1 and 2, respectively. A side of T_1 is congruent to a side of T_2 , and a different side of T_1 is congruent to a different side of T_2 . What is the square of the product of the lengths of the other (third) sides of T_1 and T_2 ?
💡 解题思路
First of all, let the two sides which are congruent be $x$ and $y$ , where $y > x$ . The only way that the conditions of the problem can be satisfied is if $x$ is the shorter leg of $T_{2}$ and the lo
16
第 16 题
几何·面积
In \triangle ABC with a right angle at C , point D lies in the interior of \overline{AB} and point E lies in the interior of \overline{BC} so that AC=CD,DE=EB, and the ratio AC:DE=4:3 . What is the ratio AD:DB?
💡 解题思路
Without loss of generality, let $AC = CD = 4$ and $DE = EB = 3$ . Let $\angle A = \alpha$ and $\angle B = \beta = 90^{\circ} - \alpha$ . As $\triangle ACD$ and $\triangle DEB$ are isosceles, $\angle A
17
第 17 题
综合
💡 解题思路
We see that the total probability will eventually sum to 1 from the infinite geometric series sum formula. Now, if the green ball goes into the \( k=1 \) bin with a \( 1/2 \) chance, then the red ball
18
第 18 题
行程问题
Henry decides one morning to do a workout, and he walks \tfrac{3}{4} of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks \tfrac{3}{4} of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks \tfrac{3}{4} of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked \tfrac{3}{4} of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point A kilometers from home and a point B kilometers from home. What is |A-B| ?
💡 解题思路
Let the two points that Henry walks in between be $P$ and $Q$ , with $P$ being closer to home. As given in the problem statement, the distances of the points $P$ and $Q$ from his home are $A$ and $B$
19
第 19 题
整数运算
Let S be the set of all positive integer divisors of 100,000. How many numbers are the product of two distinct elements of S?
💡 解题思路
The prime factorization of $100,000$ is $2^5 \cdot 5^5$ . Thus, we choose two numbers $2^a5^b$ and $2^c5^d$ where $0 \le a,b,c,d \le 5$ and $(a,b) \neq (c,d)$ , whose product is $2^{a+c}5^{b+d}$ , whe
20
第 20 题
几何·面积
As shown in the figure, line segment \overline{AD} is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1,\overarc{AEB},\overarc{BFC}, and \overarc{CGD}, have their diameters on \overline{AD}, lie in the same halfplane determined by line AD , and are tangent to line EG at E,F, and G, respectively. A circle of radius 2 has its center on F. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}·π-√(c)+d,\] where a,b,c, and d are positive integers and a and b are relatively prime. What is a+b+c+d ? [图]
💡 解题思路
This solution is essentially the same, but is much more clear and easier to understand, than Solution 2. I will TeX this at a later time, but if anyone wants to help in the mean time, please do -- tha
21
第 21 题
概率
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head？
💡 解题思路
We first want to find out which sequences of coin flips satisfy the given condition. For Debra to see the second tail before the second head, her first flip can't be heads, as that would mean she woul
22
第 22 题
概率
Raashan, Sylvia, and Ted play the following game. Each starts with \1 . A bell rings every 15 seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives \1 to that player. What is the probability that after the bell has rung 2019 times, each player will have \1 ? (For example, Raashan and Ted may each decide to give \1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have \0 , Sylvia will have \2 , and Ted will have \1 , and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their \1 to, and the holdings will be the same at the end of the second round.)
💡 解题思路
On the first turn, each player starts off with $\$1$ . Each turn after that, there are only two possibilities: either everyone stays at $\$1$ , which we will write as $(1-1-1)$ , or the distribution o
23
第 23 题
几何·面积
Points A=(6,13) and B=(12,11) lie on circle \omega in the plane. Suppose that the tangent lines to \omega at A and B intersect at a point on the x -axis. What is the area of \omega ?
💡 解题思路
First, observe that the two tangent lines are of identical length. Therefore, supposing that the point of intersection is $(x, 0)$ , the Pythagorean Theorem gives $\sqrt{(x-6)^2 + 13^2} = \sqrt{(x-12)
24
第 24 题
规律与数列
Define a sequence recursively by x_0=5 and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers n. Let m be the least positive integer such that \[x_m≤ 4+\frac{1}{2^{20}}.\] In which of the following intervals does m lie?
💡 解题思路
We first prove that $x_n > 4$ for all $n \ge 0$ , by induction. Observe that \[x_{n+1} - 4 = \frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \frac{(x_n - 4)(x_n+5)}{x_n+6}.\] so (since $x_n$ is clearly po
25
第 25 题
规律与数列
How many sequences of 0 s and 1 s of length 19 are there that begin with a 0 , end with a 0 , contain no two consecutive 0 s, and contain no three consecutive 1 s? This problem is just a simplified version of 2001 AIME I, Problem 14 , and a slightly harder version of CLMC 2025 C4 .
💡 解题思路
Let $f(n)$ be the number of valid sequences of length $n$ (satisfying the conditions given in the problem).