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AMC10 2025 A

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AMC10 · 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?
Andy 和 Betsy 都住在 Mathville。Andy 在 1:30 骑自行车离开 Mathville，向正北方向以稳定的 8 英里每小时速度行驶。Betsy 在 2:30 从同一地点骑自行车出发，向正东方向以稳定的 12 英里每小时速度行驶。他们何时将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。将另一种坚果混合物（20% 花生，40% 腰果，40% 杏仁）加入盒子后，新混合物中花生比例为 40%。现在盒子中腰果有多少磅？
Q3
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$?
有多少个面积为正的等腰三角形，其边长均为正整数，且最长边长为 2025？
Q4
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琐碎知识竞赛。学生和教师总数为 15 人。Ash 是其中一名学生的表亲，想加入竞赛。如果 Ash 加入学生队，该队的平均年龄将从 12 岁增加到 14 岁。如果 Ash 加入教师队，该队的平均年龄将从 55 岁下降到 52 岁。Ash 多大年龄？
Q5
Consider the sequence of positive integers $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2 \dots$ What is the $2025$th term in the sequence?
考虑正整数序列 $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2 \dots$ 该序列的第 2025 项是多少？
Q6
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle 20°-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
在一个等边三角形中，每个内角被一对射线三等分。每个顶点处中间20°角内部的交集是一个凸六边形的内部。这个六边形的最小内角的度量是多少度？
stem
Q7
Suppose $a$ and $b$ are real numbers. When the polynomial $x^3+x^2+ax+b$ is divided by $x-1$, the remainder is $4$. When the polynomial is divided by $x-2$, the remainder is $6$. What is $b-a$?
设$a$和$b$是实数。当多项式$x^3+x^2+ax+b$被$x-1$除时，余数是$4$。当被$x-2$除时，余数是$6$。$b-a$是多少？
Q8
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了以下四个陈述。 $\bullet$ 这些陈述中至少有一个是真命题。 $\bullet$ 这些陈述中至少有两个是真命题。 $\bullet$ 这些陈述中至少有两个是假命题。 $\bullet$ 这些陈述中至少有一个是假命题。 每个陈述要么真要么假。阿格尼斯写了多少个假陈述？
Q9
Let $f(x) = 100x^3 - 300x^2 + 200x$. For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$?
设$f(x) = 100x^3 - 300x^2 + 200x$。有几个实数$a$使得$y = f(x - a)$的图像经过点$(1, 25)$？
Q10
A semicircle has diameter $\overline{AB}$ and chord $\overline{CD}$ of length $16$ parallel to $\overline{AB}$. A smaller semicircle with diameter on $\overline{AB}$ and tangent to $\overline{CD}$ is cut from the larger semicircle, as shown below. What is the area of the resulting figure, shown shaded?
一个半圆直径为$\overline{AB}$，弦$\overline{CD}$长为$16$且平行于$\overline{AB}$。一个较小的半圆直径在$\overline{AB}$上且与$\overline{CD}$相切，从较大的半圆中切掉，如下图所示。 阴影所示图形的面积是多少？
stem
Q11
The sequence $1,x,y,z$ is arithmetic. The sequence $1,p,q,z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x+y+z+p+q$?
数列 $1,x,y,z$ 是等差数列。数列 $1,p,q,z$ 是等比数列。两个数列都是严格递增的且仅包含整数，且 $z$ 尽可能小。$x+y+z+p+q$ 的值是多少？
Q12
Carlos uses a $4$-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is $0$. How many $4$-digit passcodes satisfy these conditions?
Carlos 使用一个 4 位密码来解锁他的电脑。在他的密码中，正好有一个数字是偶数，正好有一个（可能不同的）数字是质数，且没有数字是 $0$。有多少个 4 位密码满足这些条件？
Q13
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，每个正方形都有相同的中心且边与外部正方形平行。某正方形边长与其内侧下一个正方形边长的比值为 $k$，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 $64\%$。$k$ 是多少？
stem
Q14
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?
六把椅子围成圆桌排列。两名学生和两名老师随机选四把椅子坐下。两名学生坐在相邻两把椅子的概率，且两名老师也坐在相邻两把椅子的概率是多少？
Q15
In the figure below, $ABEF$ is a rectangle, $\overline{AD}\perp\overline{DE}$, $AF=7$, $AB=1$, and $AD=5$. What is the area of $\triangle ABC$?
下图中，$ABEF$ 是矩形， $\overline{AD}\perp\overline{DE}$， $AF=7$， $AB=1$， $AD=5$。 $ riangle ABC$ 的面积是多少？
stem
Q16
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placement of the other coins. What is the expected number of coins in a jar with the most coins?
有三个罐子。每个三个硬币被随机且独立地放入三个罐子之一。罐子中硬币最多的那个罐子中的硬币数的期望值是多少？
Q17
Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$. What is the tens digit of $N$?
设 $N$ 为唯一的正整数，使得 $273436$ 除以 $N$ 余 $16$，$272760$ 除以 $N$ 余 $15$。$N$ 的十位数字是多少？
Q18
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 \[\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)...(2025x^2-4x-3)?\]
一个数集的调和平均数是该数集倒数的算术平均数的倒数。例如，$4,4$ 和 $5$ 的调和平均数是 \[\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}.\] 多项式 $4050$ 次方程的所有实根的调和平均数是多少？该多项式为 \[\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)...(2025x^2-4x-3)?\]
Q19
An array of numbers is constructed beginning with the numbers $-1$, $3$, and $1$ in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with $-1$ and $1,$ respectively. \[\large{-1}\qquad\large{3}\qquad\large{1}\] \[\large{-1}\qquad\large{2}\qquad\large{4}\qquad\large{1}\] \[\large{-1}\qquad\large{1}\qquad\large{6}\qquad\large{5}\qquad\large{1}\] If the process continues, one of the rows will sum to $12{,}288$. In that row, what is the third number from the left?
一个数字阵列从顶行数字 $-1$、$3$ 和 $1$ 开始构造。每相邻一对数字相加产生下一行的数字。每行开始和结束分别为 $-1$ 和 $1$。 \[\large{-1}\qquad\large{3}\qquad\large{1}\] \[\large{-1}\qquad\large{2}\qquad\large{4}\qquad\large{1}\] \[\large{-1}\qquad\large{1}\qquad\large{6}\qquad\large{5}\qquad\large{1}\] 如果过程继续，有一行之和为 $12{,}288$。在那一行中，距左边第三个数是多少？
Q20
A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of g can be written as $\frac{a\sqrt{b}-c}{d}$, where $a,b,c,$ and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$. What is $a+b+c+d$?
一个直径 $20$ 米的筒仓（右圆柱体）矗立在田野中。MacDonald 位于筒仓中心西 $20$ 米、南 $15$ 米处。McGregor 位于筒仓中心东 $20$ 米、南 $g > 0$ 米处。MacDonald 和 McGregor 之间的视线与筒仓相切。$g$ 的值为 $\frac{a\sqrt{b}-c}{d}$，其中 $a,b,c,d$ 为正整数，$b$ 无任何质数的平方因子，$d$ 与 $a$ 和 $c$ 的最大公因数互质。求 $a+b+c+d$？
stem
Q21
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 $\{1,2,3,...,20\}$?
一个数集被称为无和集（sum-free），如果集合中的任意（不一定不同的）元素 $x$ 和 $y$，$x+y$ 都不在该集合中。例如，$\{1,4,6\}$ 和空集是无和集，但 $\{1,4,5\}$ 不是。在集合 $\{1,2,3,...,20\}$ 中，无和子集最多可能有多少个元素？
Q22
A circle of radius $r$ is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below. What is $r$?
一个半径为 $r$ 的圆被三个圆包围，这些圆的半径分别为 1、2 和 3，它们都与内部圆外切，并且彼此外切，如下图所示。 $r$ 是多少？
stem
Q23
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 $AB = 80$，$BC = 45$，$AC = 75$。角 $B$ 的平分线与侧边 $\overline{AB}$ 的高线交于点 $P$。$BP$ 是多少？
Q24
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?
称一个正整数为公平数（fair），如果没有数字重复使用，不含 $0$，且没有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。有多少个公平正整数？
Q25
A point $P$ is chosen at random inside square $ABCD$. The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\frac{a + b \pi - c \sqrt{d}}{e}$, where $a, b, c, d,$ and $e$ are positive integers, $\text{gcd}(a, b, c, e) = 1$, and $d$ is not divisible by the square of a prime. What is $a+b+c+d+e$?
在正方形 $ABCD$ 内随机选择一点 $P$。直线 $\overline{AP}$ 既不是 $\triangle APB$ 的最短边也不是最长边的概率可以写成 $\frac{a + b \pi - c \sqrt{d}}{e}$，其中 $a, b, c, d,$ 和 $e$ 是正整数，$\text{gcd}(a, b, c, e) = 1$，且 $d$ 不可被任一质数的平方整除。求 $a+b+c+d+e$？
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