Found problems: 60
Rectangle $ABCD$ has $AB=20$ and $BC=15$. $2$ circles with diameters $AB$ and $AC$ intersect again at point $E$. What is the length of $DE$?
Let $ABCDEF$ be a regular hexagon, and let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The intersection of lines $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$.
A fair coin is flipped $6$ times. The probability that the coin lands on the same side $3$ flips in a row at some point can be expressed as a common fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.
For integers $z$, let $\#(z)$ denote the number of integer ordered pairs $(x, y)$ that satisfy $x^2 - xy + y^2 = z$. How many integers $z$ between $0$ and $150$ inclusive satisfy $\#(z) \equiv 6$ (mod $12$)?
Alex scans the list of integers between $1$ and $2020$ inclusive using the following algorithm. First, he reads off perfect squares between $1$ and $2020$ in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are $2$, $6$, $12$, $...$, and removes these numbers from the list. He repeats this algorithm until he reads off $2020$, which is the nth number he has read o so far. Compute $n$.
In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$?
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Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.
Triangle $\vartriangle ABC$ has side lengths $AB = 26$, $BC = 34$, and $CA = 24\sqrt2$. A fourth point $D$ makes a right angle $\angle BDC$. What is the smallest possible length of $\overline{AD}$?
Let $S_n$ be the number of subsets of the first $n$ positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer $n$ such that $S_n$ is a multiple of $2020$.