2020 AMC 10A — Official Competition Problems (January 2020)
📅 2020 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
综合
💡 解题思路
Adding $\frac{3}{4}$ to both sides, $x= \frac{5}{12} - \frac{1}{3} + \frac{3}{4} = \frac{5}{12} - \frac{4}{12} + \frac{9}{12}=\boxed{\textbf{(E) }\frac{5}{6}}$ .
2
第 2 题
统计
The numbers 3, 5, 7, a, and b have an average (arithmetic mean) of 15 . What is the average of a and b ?
💡 解题思路
The arithmetic mean of the numbers $3, 5, 7, a,$ and $b$ is equal to $\frac{3+5+7+a+b}{5}=\frac{15+a+b}{5}=15$ . Solving for $a+b$ , we get $a+b=60$ . Dividing by $2$ to find the average of the two nu
3
第 3 题
规律与数列
Assuming a≠3 , b≠4 , and c≠5 , what is the value in simplest form of the following expression? \[\frac{a-3}{5-c} · \frac{b-4}{3-a} · \frac{c-5}{4-b}\]
💡 解题思路
If $x\neq y,$ then $\frac{x-y}{y-x}=-1.$ We use this fact to simplify the original expression: \[\frac{\color{red}\overset{-1}{\cancel{a-3}}}{\color{blue}\underset{1}{\cancel{5-c}}} \cdot \frac{\color
4
第 4 题
行程问题
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid \0.50 per mile, and her only expense is gasoline at \2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?
💡 解题思路
Since the driver travels $60$ miles per hour and each hour she uses $2$ gallons of gasoline, she spends $\$4$ per hour on gas. If she gets $\$0.50$ per mile, then she gets $\$30$ per hour of driving.
5
第 5 题
规律与数列
What is the sum of all real numbers x for which |x^2-12x+34|=2?
💡 解题思路
Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.
6
第 6 题
数论
How many 4 -digit positive integers (that is, integers between 1000 and 9999 , inclusive) having only even digits are divisible by 5?
💡 解题思路
The units digit, for all numbers divisible by 5, must be either $0$ or $5$ . However, since all digits are even, the units digit must be $0$ . The middle two digits can be 0, 2, 4, 6, or 8, but the th
7
第 7 题
几何·面积
The 25 integers from -10 to 14, inclusive, can be arranged to form a 5 -by- 5 square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
💡 解题思路
Without loss of generality, consider the five rows in the square. Each row must have the same sum of numbers, meaning that the sum of all the numbers in the square divided by $5$ is the total value pe
8
第 8 题
综合
What is the value of \[1+2+3-4+5+6+7-8+·s+197+198+199-200?\]
💡 解题思路
Looking at the numbers, you see that every set of $4$ has $3$ positive numbers and 1 negative number. Calculating the sum of the first couple sets gives us $2+10+18...+394$ . Clearly, this pattern is
9
第 9 题
整数运算
A single bench section at a school event can hold either 7 adults or 11 children. When N bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of N?
💡 解题思路
The least common multiple of $7$ and $11$ is $77$ . Therefore, there must be $77$ adults and $77$ children. The total number of benches is $\frac{77}{7}+\frac{77}{11}=11+7=\boxed{\textbf{(B) }18}$ .~t
10
第 10 题
几何·面积
Seven cubes, whose volumes are 1 , 8 , 27 , 64 , 125 , 216 , and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
💡 解题思路
The volume of each cube follows the pattern of $n^3$ , for $n$ is between $1$ and $7$ .
11
第 11 题
综合
💡 解题思路
We can see that $44^2=1936$ which is less than 2020. Therefore, there are $2020-44=1976$ of the $4040$ numbers greater than $2020$ . Also, there are $2020+44=2064$ numbers that are less than or equal
12
第 12 题
几何·面积
Triangle AMC is isosceles with AM = AC . Medians \overline{MV} and \overline{CU} are perpendicular to each other, and MV=CU=12 . What is the area of \triangle AMC? [图]
💡 解题思路
Since quadrilateral $UVCM$ has perpendicular diagonals, its area can be found as half of the product of the length of the diagonals. Also note that $\triangle AUV$ has $\frac 14$ the area of triangle
13
第 13 题
几何·面积
A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1 , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices (0,0), (0,4), (4,4), and (4,0) . What is the probability that the sequence of jumps ends on a vertical side of the square?
💡 解题思路
Drawing out the square, it's easy to see that if the frog goes to the left, it will immediately hit a vertical end of the square. Therefore, the probability of this happening is $\frac{1}{4} \cdot 1 =
14
第 14 题
综合
Real numbers x and y satisfy x + y = 4 and x · y = -2 . What is the value of \[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\]
A positive integer divisor of 12! is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as \frac{m}{n} , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
The prime factorization of $12!$ is $2^{10} \cdot 3^5 \cdot 5^2 \cdot 7 \cdot 11$ . This yields a total of $11 \cdot 6 \cdot 3 \cdot 2 \cdot 2$ divisors of $12!.$ In order to produce a perfect square
16
第 16 题
几何·面积
A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (2020, 0), (2020, 2020), and (0, 2020) . The probability that the point is within d units of a lattice point is \tfrac{1}{2} . (A point (x, y) is a lattice point if x and y are both integers.) What is d to the nearest tenth ?
Define \[P(x) =(x-1^2)(x-2^2)·s(x-100^2).\] How many integers n are there such that P(n)≤ 0 ?
💡 解题思路
We perform casework on $P(n)\leq0:$
18
第 18 题
整数运算
Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set \{0,1,2,3\}. For how many such quadruples is it true that a· d-b· c is odd? (For example, (0,3,1,1) is one such quadruple, because 0· 1-3· 1 = -3 is odd.)
💡 解题思路
In order for $a\cdot d-b\cdot c$ to be odd, consider parity. We must have (even)-(odd) or (odd)-(even). There are $2(2 + 4) = 12$ ways to pick numbers to obtain an even product. There are $2 \cdot 2 =
19
第 19 题
综合
💡 解题思路
Since we start at the top face and end at the bottom face without moving from the lower ring to the upper ring or revisiting a face, our journey must consist of the top face, a series of faces in the
20
第 20 题
几何·面积
Quadrilateral ABCD satisfies \angle ABC = \angle ACD = 90^{\circ}, AC=20, and CD=30. Diagonals \overline{AC} and \overline{BD} intersect at point E, and AE=5. What is the area of quadrilateral ABCD?
There exists a unique strictly increasing sequence of nonnegative integers a_1 < a_2 < … < a_k such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\] What is k?
💡 解题思路
First, substitute $2^{17}$ with $x$ . Then, the given equation becomes $\frac{x^{17}+1}{x+1}=x^{16}-x^{15}+x^{14}...-x^1+x^0$ by sum of powers factorization. Now consider only $x^{16}-x^{15}$ . This e
22
第 22 题
数论
For how many positive integers n \le 1000 is \[\lfloor \dfrac{998}{n} \rfloor+\lfloor \dfrac{999}{n} \rfloor+\lfloor \dfrac{1000}{n} \rfloor\] not divisible by 3 ? (Recall that \lfloor x \rfloor is the greatest integer less than or equal to x .)
💡 解题思路
If $n$ is a factor of $999$ (and $n \neq 1$ ), then let $999 = an$ for some positive integer $a.$ Then we get $\lfloor 998/n \rfloor = a - 1$ , $\lfloor 999/n \rfloor =a$ , $\lfloor 1000/n \rfloor = a
23
第 23 题
几何·面积
Let T be the triangle in the coordinate plane with vertices (0,0), (4,0), and (0,3). Consider the following five isometries (rigid transformations) of the plane: rotations of 90^{\circ}, 180^{\circ}, and 270^{\circ} counterclockwise around the origin, reflection across the x -axis, and reflection across the y -axis. How many of the 125 sequences of three of these transformations (not necessarily distinct) will return T to its original position? (For example, a 180^{\circ} rotation, followed by a reflection across the x -axis, followed by a reflection across the y -axis will return T to its original position, but a 90^{\circ} rotation, followed by a reflection across the x -axis, followed by another reflection across the x -axis will not return T to its original position.)
💡 解题思路
Label each rotation \( A, B, C, D \), and \( E \) respectively.
24
第 24 题
数论
Let n be the least positive integer greater than 1000 for which \[\gcd(63, n+120) =21 and \gcd(n+63, 120)=60.\] What is the sum of the digits of n ?
💡 解题思路
We know that $\gcd(n+57,63)=21$ and $\gcd(n-57, 120)= 60$ by the Euclidean Algorithm. Hence, let $n+57=21\alpha$ and $n-57=60 \gamma$ , where $\gcd(\alpha,3)=1$ and $\gcd(\gamma,2)=1$ . Subtracting th
25
第 25 题
概率
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
💡 解题思路
Consider the probability that rolling two dice gives a sum of $s$ , where $s \leq 7$ . There are $s - 1$ pairs that satisfy this, namely $(1, s - 1), (2, s - 2), \ldots, (s - 1, 1)$ , out of $6^2 = 36