Diagnostic - AMC10

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?

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?

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?

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 $(1, 25)$?

On Monday, $6$ students went to the tutoring center at the same time, and each one was randomly assigned to one of the $6$ tutors on duty. On Tuesday, the same $6$ students showed up, the same $6$ tutors were on duty, and the students were again randomly assigned to the tutors. What is the probability that exactly $2$ students met with the same tutor both Monday and Tuesday?

Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing 3 blue bands, 3 red bands, and 3 green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?

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?

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?

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 $r$?

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