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Diagnostic - AMC12

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Question 1
AMC12 2025 B · Q2
Jerry wrote down the ones digit of each of the first $2025$ positive squares: $1, 4, 9, 6, 5, 6, \dots$. What is the sum of all the numbers Jerry wrote down?
杰瑞写下了前2025个正整数平方数的个位数：1, 4, 9, 6, 5, 6, \dots。杰瑞写下的所有数字之和是多少？
Question 2
AMC12 2025 B · Q5
Positive integers $x$ and $y$ satisfy the equation $57x+ 22y = 400$. What is the least possible value of $x+y$?
正整数 $x$ 和 $y$ 满足方程 $57x+ 22y = 400$。$x+y$ 的最小可能值是多少？
Question 3
AMC12 2025 B · Q7
What is the value of \[\sum_{n = 2}^{255}\frac{\log_{2}\left(1 + \tfrac{1}{n}\right)}{\left(\log_{2}n\right)\left(\log_{2}(n + 1)\right)}?\]
求 \[\sum_{n = 2}^{255}\frac{\log_{2}\left(1 + \tfrac{1}{n}\right)}{\left(\log_{2}n\right)\left(\log_{2}(n + 1)\right)}\] 的值。
Question 4
AMC12 2025 A · 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
Question 5
AMC12 2025 A · 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？
Question 6
AMC12 2025 A · 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？
Question 7
AMC12 2025 A · 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？
Question 8
AMC12 2025 A · 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？
Question 9
AMC12 2025 B · Q21
Two non-congruent triangles have the same area. Each triangle has sides of length $8$ and $9$, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?
有两个不相容的三角形，它们的面积相同。每个三角形都有边长为$8$和$9$，每个三角形的第三条边长为整数。第三条边的长度的和是多少？
Question 10
AMC12 2025 A · 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 $a > 2b$）。