Geo
Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 3$. A point $P$ on segment $AB$ is such that DPC is a right angle and $AP < BP$. What is the length of $AP$?
1
For which of the following pairs (a, b) is $a^2 + b = b^2 + a?$ (-2022, 2021), (-2022, 2023), (-2021, 2020), (2021, -2022)
(-2022, 2023)
How many positive integers $N$ less than or equal to $1000$ are there such that 75% of $N$'s divisors are multiples of $3$?
37
There are $10$ integers written on a blackboard with mean $n$ for some positive integer $n$. When a positive integer $k$ is added to the list, the mean of the $11$ numbers is $2n-1$. Which of the following numbers could be $k$? (2027, 2028, 2029, 2030)
2029
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$, $BC = 6$, $DA = 5$, and $AB < CD$. The angle bisectors of angles $DAB$ and $ABC$ intersect at a point $X$ on segment $CD$. If $AB = AX$, what is the perimeter of trapezoid $ABCD$?
28
A cubic polynomial $P(x) = ax^3 + bx + c$ with $a$ not equal $0$ has three roots $r, s,$ and $t$. What polynomial with leading coefficient $a$ has roots $r+s$, $s+t$, $t+r$?
ax^3+bx-c
David has eight weights labeled 1 to 8. Weight number $n$ weights $2^n$. He randomly selects at least two of the weights and puts them on a scale. How many different scale readings can he get?
247
The town of hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. What is the largest number that can't be the total number of living things in Hamlet?
47
Line segment $AB$ is a diameter of a circle with $AB = 2$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the incenter of triangle $ABC$ traces out a closed curve. What is the area of the region bounded by this curve?
pi - 2
Two right circular cones have vertices facing down. The first cone has radius 1 and height 7. The second cone has radius 4 and height 4. The first cone is initially full of water. The water is poured from the first cone to the second until the height of the water is the same in both cones. What is the height of water in the cones?
(7cbrt20)/10
Katy is going bowling, but she sucks at bowling. The bowling lane is $5$ meters long, and is surrounded by gutters on both sides. For every meter forward Katy's ball travels, given that it was previously in the center of the lane, there is a 1/2 chance of it going off-center and a 1/2 chance of it remaining in the center. Given that the ball was previously off-center, there is a 3/8 chance of it going into the gutter, 1/8 chance of it moving back to the center, and a 1/2 chance of it remaining off-center. If the ball is not in the gutter by the time it reaches the pins, Katy scores some points. If the ball is initally launched from the center of the lane, the probability that Katy scores a nonzero amount of points is m/n. Find m+n.
695
Let $k>1$ be the smallest positive integer such that the sum of the first $2020$ positive integers is a divisor of the sum of the first $2020k$ positive integers. What is the sum of the digits of $k$?
13