Diagnostic - AMC10

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?

A Pascal-like triangle has $10$ as the top row and $10$ followed by $1$ as the second row. In each subsequent row the first number is $10$, the last number is $1$, and, as in the standard Pascal Triangle, each other in the row is the sum of the two numbers directly above it. The first four rows are shown below. \[\large{10}\] \[\large{10}\qquad\large{1}\] \[\large{10}\qquad\large{11}\qquad\large{1}\] \[\large{10}\qquad\large{21}\qquad\large{12}\qquad\large{1}\] What is the sum of the digits of the sum of the numbers in the 11th row?

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?

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?

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 $x+y+z+p+q$?

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?

A circle has been divided into 6 sectors of different sizes. Then 2 of the sectors are painted red, 2 painted green, and 2 painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below. How many different colorings are possible?

Consider a decreasing sequence of n positive integers \[x_1 > x_2 > \cdots > x_n\] that satisfies the following conditions: What is the greatest possible value of n?

A frog hops along the number line according to the following rules: What is the probability that the frog reaches $4?$

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 $a+b+c+d+e$?