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AMC12 2025 A

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AMC12 · 2025 (A)

Q1
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1{:}30$, traveling due north at a steady $8$ miles per hour. Betsy leaves on her bicycle from the same point at $2{:}30$, traveling due east at a steady $12$ miles per hour. At what time will they be exactly the same distance from their common starting point?
安迪和贝齐都住在数学城。安迪在1:30骑自行车离开数学城，向正北方向以稳定的8英里/小时速度行驶。贝齐在2:30从同一地点骑自行车出发，向正东方向以稳定的12英里/小时速度行驶。他们何时距离共同起点恰好相等？
Q2
A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews are now in the box?
一个盒子含有10磅坚果混合物，其中50%是花生，20%是腰果，30%是杏仁。加入第二种坚果混合物（20%花生，40%腰果，40%杏仁），结果新混合物中花生比例为40%。现在盒子里有多少磅腰果？
Q3
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is $15$. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from $12$ to $14$. If Ash plays with the teachers, the average age on that team will decrease from $55$ to $52$. How old is Ash?
一支学生队将与一支教师队进行知识竞赛。学生和教师总数为15人。阿什是其中一名学生的表亲，想加入竞赛。如果阿什加入学生队，该队的平均年龄将从12岁增加到14岁。如果阿什加入教师队，该队的平均年龄将从55岁减少到52岁。阿什多大年龄？
Q4
Agnes writes the following four statements on a blank piece of paper. $\bullet$ At least one of these statements is true. $\bullet$ At least two of these statements are true. $\bullet$ At least two of these statements are false. $\bullet$ At least one of these statements is false. Each statement is either true or false. How many false statements did Agnes write on the paper?
阿格尼斯在一张白纸上写下以下四个陈述。 $\bullet$ 这些陈述中至少有一个是真。 $\bullet$ 这些陈述中至少有两个是真。 $\bullet$ 这些陈述中至少有两个是假。 $\bullet$ 这些陈述中至少有一个是假。 每个陈述要么真要么假。阿格尼斯写了多少个假陈述？
Q5
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k,$ where $0 < k < 1.$ The spaces between squares are alternately shaded, as shown in the figure (which is not necessarily drawn to scale). The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k?$
下图中，外部正方形包含无限多个正方形，每个正方形有相同的中心且边与外部正方形平行。相邻正方形的边长比为 $k$，其中 $0 < k < 1$。正方形之间的空间交替着色，如图所示（图未按比例绘制）。 着色部分的面积是原正方形面积的64%。$k$ 等于多少？
stem
Q6
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
六把椅子围成一圈摆放。两名学生和两名老师随机选择四把椅子坐下。两名学生坐在相邻的两把椅子上且两名老师也坐在相邻的两把椅子的概率是多少？
Q7
In a certain alien world, the maximum running speed $v$ of an organism is dependent on its number of toes $n$ and number of eyes $m$. The relationship can be expressed as $v = kn^am^b$ centimeters per hour, where k, a, b are integer constants. In a population where all organisms have 5 toes, $\log v = 4+2\log m$; and in a population where all organisms have 25 eyes, $\log v = 4 + 4 \log n$, where all logs are in base 10. What is $k+a+b$?
在某个外星世界，有机体的最大奔跑速度$v$取决于其脚趾数$n$和眼睛数$m$。关系可表示为$v = kn^am^b$厘米/小时，其中$k, a, b$为整数常数。在所有有机体都有5个脚趾的种群中，$\log v = 4+2\log m$；在所有有机体都有25只眼睛的种群中，$\log v = 4 + 4 \log n$，所有对数均为10为底。求$k+a+b$？
Q8
Pentagon $ABCDE$ is inscribed in a circle, and $\angle BEC = \angle CED = 30^\circ$. Let line $AC$ and line $BD$ intersect at point $F$, and suppose that $AB = 9$ and $AD = 24$. What is $BF$?
五边形$ABCDE$内接于圆中，且$\angle BEC = \angle CED = 30^\circ$。直线$AC$与直线$BD$相交于点$F$，已知$AB = 9$，$AD = 24$。求$BF$？
stem
Q9
Let $w$ be the complex number $2+i$, where $i=\sqrt{-1}$. What real number $r$ has the property that $r$, $w$, and $w^2$ are three collinear points in the complex plane?
设复数$w=2+i$，其中$i=\sqrt{-1}$。求实数$r$，使得$r$、$w$和$w^2$在复平面中三点共线。
Q10
In the figure shown below, major arc $\widehat{AD}$ and minor arc $\widehat{BC}$ have the same center, $O$. Also, $A$ lies between $O$ and $B$, and $D$ lies between $O$ and $C$. Major arc $\widehat{AD}$, minor arc $\widehat{BC}$, and each of the two segments $\overline{AB}$ and $\overline{CD}$ have length $2\pi$. What is the distance from $O$ to $A$?
如图所示，大弧$\widehat{AD}$和小弧$\widehat{BC}$有相同的圆心$O$。此外，$A$位于$O$和$B$之间，$D$位于$O$和$C$之间。大弧$\widehat{AD}$、小弧$\widehat{BC}$以及线段$\overline{AB}$和$\overline{CD}$的长度均为$2\pi$。求$O$到$A$的距离。
stem
Q11
The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are $A(2,31), B(8,27),$ and $C(18,27)$?
三角形的垂心是三条（可能延长）高线的并发交点。顶点为 $A(2,31)$、$B(8,27)$ 和 $C(18,27)$ 的三角形的垂心的坐标之和是多少？
Q12
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of $4,4,$ and $5$ is \[\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}.\] What is the harmonic mean of all the real roots of the $4050$th degree polynomial \[\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)...(2025x^2-4x-3)?\]
一个数列的调和平均数是该数列倒数的算术平均数的倒数。例如，$4,4$ 和 $5$ 的调和平均数为 \[\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}.\] 下列 $4050$ 次多项式 \[\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\dots(2025x^2-4x-3)?\] 的所有实根的调和平均数是多少？
Q13
Let $C = \{1, 2, 3, \dots, 13\}$. Let $N$ be the greatest integer such that there exists a subset of $C$ with $N$ elements that does not contain five consecutive integers. Suppose $N$ integers are chosen at random from $C$ without replacement. What is the probability that the chosen elements do not include five consecutive integers?
令 $C = \{1, 2, 3, \dots, 13\}$。令 $N$ 为最大整数，使得存在 $C$ 的一个 $N$ 元子集不包含五个连续整数。從 $C$ 中不放回地随机选择 $N$ 个整数。所选元素不包含五个连续整数的概率是多少？
Q14
Points $F$, $G$, and $H$ are collinear with $G$ between $F$ and $H$. The ellipse with foci at $G$ and $H$ is internally tangent to the ellipse with foci at $F$ and $G$, as shown below. The two ellipses have the same eccentricity $e$, and the ratio of their areas is $2025$. (Recall that the eccentricity of an ellipse is $e = \tfrac{c}{a}$, where $c$ is the distance from the center to a focus, and $2a$ is the length of the major axis.) What is $e$?
点 $F$、$G$ 和 $H$ 共线，且 $G$ 在 $F$ 和 $H$ 之间。以 $G$ 和 $H$ 为焦点的椭圆内切于以 $F$ 和 $G$ 为焦点的椭圆，如下图所示。 两个椭圆具有相同的离心率 $e$，且面积比为 $2025$。（回想椭圆的离心率为 $e = \tfrac{c}{a}$，其中 $c$ 是中心到焦点的距离，$2a$ 是长轴长度。）$e$ 是多少？
stem
Q15
A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$ is not an element of the set. For example, $\{1,4,6\}$ and the empty set are sum-free, but $\{1,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1,2,3,...,20\}$?
若一个数集称为和自由集，即当 $x$ 和 $y$（不一定不同）为该集合的元素时，$x+y$ 不在该集合中。例如，$\{1,4,6\}$ 和空集是和自由集，但 $\{1,4,5\}$ 不是。$\{1,2,3,...,20\}$ 的和自由子集的最大可能元素个数是多少？
Q16
Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector of $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$. What is $BP$?
三角形 $\triangle ABC$ 的边长 $AB = 80$，$BC = 45$，$AC = 75$。$\angle B$ 的角平分线与侧边 $\overline{AB}$ 的高线交于点 $P$。$BP$ 等于多少？
Q17
The polynomial $(z + i)(z + 2i)(z + 3i) + 10$ has three roots in the complex plane, where $i = \sqrt{-1}$. What is the area of the triangle formed by these three roots?
多项式 $(z + i)(z + 2i)(z + 3i) + 10$ 在复平面中有三个根，其中 $i = \sqrt{-1}$。这三个根形成的三角形的面积是多少？
Q18
How many ordered triples $(x, y, z)$ of different positive integers less than or equal to $8$ satisfy $xy > z$, $xz > y$, and $yz > x$?
有多少个不同的正整数有序三元组 $(x, y, z)$（每个不超过 $8$）满足 $xy > z$，$xz > y$，$yz > x$？
Q19
Let $a$, $b$, and $c$ be the roots of the polynomial $x^3 + kx + 1$. What is the sum\[a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + c^2a^3?\]
设 $a$，$b$，$c$ 是多项式 $x^3 + kx + 1$ 的根。求 \[a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + c^2a^3\] 的值。
Q20
The base of the pentahedron shown below is a $13 \times 8$ rectangle, and its lateral faces are two isosceles triangles with base of length $8$ and congruent sides of length $13$, and two isosceles trapezoids with bases of length $7$ and $13$ and nonparallel sides of length $13$. What is the volume of the pentahedron?
如下图所示的五面体的底面为 $13 \times 8$ 矩形，其侧面为两个底边长 $8$、等腰边长 $13$ 的等腰三角形，以及两个底边长分别为 $7$ 和 $13$、非平行边长 $13$ 的等腰梯形。 该五面体的体积是多少？
stem
Q21
There is a unique ordered triple $(a,k,m)$ of nonnegative integers such that \[\frac{4^a + 4^{a+k}+4^{a+2k}+\cdots + 4^{a+mk}}{2^a + 2^{a+k} + 2^{a+2k}+ \cdots + 2^{a+mk}} = 964.\] What is $a+k+m$?
存在唯一的非负整数有序三元组 $(a,k,m)$ 使得 \[\frac{4^a + 4^{a+k}+4^{a+2k}+\cdots + 4^{a+mk}}{2^a + 2^{a+k} + 2^{a+2k}+ \cdots + 2^{a+mk}} = 964.\] 求 $a+k+m$？
Q22
Three real numbers are chosen independently and uniformly at random between $0$ and $1$. What is the probability that the greatest of these three numbers is greater than $2$ times each of the other two numbers? (In other words, if the chosen numbers are $a \geq b \geq c$, then $a > 2b$.)
独立均匀随机地在 $[0,1]$ 中选择三个实数。求这三个数中最大的那个大于另外两个的 $2$ 倍的概率？（换言之，若所选数字为 $a \geq b \geq c$，则 $a > 2b$）。
Q23
Call a positive integer fair if no digit is used more than once, it has no $0$s, and no digit is adjacent to two greater digits. For example, $196, 23$ and $12463$ are fair, but $1546, 320,$ and $34321$ are not. How many fair positive integers are there?
称正整数为公平数，若无数字重复使用、无 $0$，且无数字邻接两个更大的数字。例如，$196, 23$ 和 $12463$ 是公平数，但 $1546, 320,$ 和 $34321$ 不是。公平正整数有多少个？
Q24
A circle of radius $r$ is surrounded by $12$ circles of radius $1,$ externally tangent to the central circle and sequentially tangent to each other, as shown. Then $r$ can be written as $\sqrt a + \sqrt b + c,$ where $a, b, c$ are integers. What is $a+b+c?$
半径为 $r$ 的圆被 $12$ 个半径为 $1$ 的圆包围，这些圆与中心圆外切，并依次相切，如图所示。然后 $r$ 可以写成 $\sqrt a + \sqrt b + c$，其中 $a, b, c$ 是整数。求 $a+b+c$？
stem
Q25
Polynomials $P(x)$ and $Q(x)$ each have degree $3$ and leading coefficient $1$, and their roots are all elements of $\{1,2,3,4,5\}$. The function $f(x) = \tfrac{P(x)}{Q(x)}$ has the property that there exist real numbers $a < b < c < d$ such that the set of all real numbers $x$ such that $f(x) \leq 0$ consists of the closed interval $[a,b]$ together with the open interval $(c,d)$. How many ordered pairs of polynomials $(P, Q)$ are possible?
多项式 $P(x)$ 和 $Q(x)$ 均为次数 $3$，首项系数 $1$，根均为集合 $\{1,2,3,4,5\}$ 的元素。函数 $f(x) = \tfrac{P(x)}{Q(x)}$ 有性质：存在实数 $a < b < c < d$，使得 $f(x) \leq 0$ 的所有实数 $x$ 的集合为闭区间 $[a,b]$ 与开区间 $(c,d)$ 的并集。可能的多项式有序对 $(P, Q)$ 有多少个？
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