AMC10 2027 - Self-practice Set 2

An envelope contains eight coins: two each of denominations $1$, $5$, $10$, and $20$ cents. Two coins are drawn uniformly without replacement. What is the probability that their total value is at least $20$ cents?

In quadrilateral $ABCD,$ $AB = 5,$ $BC = 17,$ $CD = 5,$ $DA = 9,$ and $BD$ is an integer. What is $BD?$

Bob arranges all 2016 coins into a triangle with $N$ rows; row $k$ contains $k$ coins. Find the sum of the digits of $N$.

Alex divides $1270$ by a positive integer and obtains a quotient of $74$. What is the sum of the divisor and the remainder?

Two fair eight-sided dice numbered $1$ through $8$ are rolled independently. What is the probability that the product of their values exceeds their sum?

Segment $BD$ and $AE$ intersect at $C,$ as shown, $AB=BC=CD=CE,$ and $\angle A=\dfrac52\angle B.$ What is the degree measure of $\angle D?$

Real numbers $x,y$ satisfy $x^2+y^2=10x-6y-34$. Find $x+y$.

Laura adds two three-digit positive integers whose six digits are all distinct. Their sum $S$ also has three digits. What is the smallest possible sum of the digits of $S$?

Chloe chooses a real number uniformly from $[0,2017]$. Independently, Laurent chooses one uniformly from $[0,4034]$. What is the probability that Laurent's number exceeds Chloe's?

In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50},$ $E$ is between $B$ and $H,$ and $BE = 1.$ What is the area of the inner square $EFGH?$

Let $0\lt x\lt 1$, $-1\lt y\lt 0$, and $1\lt z\lt 2$. Which expression must be positive?

Let $M=2^4\cdot3^6\cdot5^{10}\cdot7^9$. How many positive divisors of $M$ are odd perfect squares?

A fair die has labels $1,2,2,3,3,4$, and a second fair die has labels $1,3,4,5,6,8$. They are rolled independently. What is the probability that the sum is $5$, $7$, or $9$?

The area of $\triangle EBD$ is one third of the area of the $3$-$4$-$5$ triangle $ABC.$ Segment $DE$ is perpendicular to segment $AB.$ What is $BD?$

Real numbers $x,y$ satisfy $|x|+x+y=8$ and $x+|y|-y=10$. Find $x+y$.

What is the value of
$$M=1-2\cdot2+3\cdot3-4\cdot4+5\cdot5-\cdots-2020\cdot2020+2021\cdot2021?$$

Three fair six-sided dice are rolled. What is the probability that the values on two of the dice sum to the value on the remaining die?

Doug constructs a square window with eight congruent panes as shown. Each pane has height-to-width ratio $5:2$, and all borders around and between the panes are $2\text{ cm}$ wide. What is the window's side length in centimeters?

Find the sum of all real solutions of $|x+2|=2|x-2|$.

On an island, all amounts of money are written in base $b$. A car costs $(440)_b$. Alex pays $(1000)_b$ and receives $(340)_b$ in change. What is the decimal value of $b$?