Found problems: 161
The graph of the equation $\tan(x+y) = \tan(x)+2\tan(y),$ with its pointwise holes filled in, partitions the coordinate plane into congruent regions. Compute the perimeter of one of these regions.
Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$
Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$
A square of side length $1$ is dissected into two congruent pentagons. Compute the least upper bound of the perimeter of one of these pentagons.
The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
Let $f: \{1, 2, 3, \ldots, 9\} \to \{1, 2, 3, \ldots, 9\}$ be a permutation chosen uniformly at random from the $9!$ possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$
A deck of $100$ cards is labeled $1,2,\ldots,100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expected value of the label of the remaining card.
The numbers $1, 2, \ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b,$ and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d.$
Let $N$ denote the number of integers $n$ that satisfy exactly one of $a \le n \le b$ and $c \le n \le d.$ Compute the probability that $N$ is even.
Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).
Let $ABCD$ be a rectangle such that $AB = 20$ and $AD = 24.$ Point $P$ lies inside $ABCD$ such that triangles $PAC$ and $PBD$ have areas $20$ and $24,$ respectively. Compute all possible areas of triangle $PAB.$
If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$
Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.
Three points, $A, B,$ and $C,$ are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\angle{ABC}.$
Find the number of trailing $0$s in the base $12$ expression of $99!$ (Note: $99$ is in base $10$).
Kelvin the frog currently sits at $(0,0)$ in the coordinate plane. If Kelvin is at $(x,y),$ either he can walk to any of $(x,y + 1),$ $(x + 1,y),$ or $(x + 1,y + 1),$ or he can jump to any of $(x,y + 2), (x + 2,y),$ or $(x+1,y+1).$ Walking and jumping from $(x,y)$ to $(x+1,y+1)$ are considered distinct actions. Compute the number of ways Kelvin can reach $(6,8).$
Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.
For any positive integer $n,$ let $f(n)$ be the number of ordered triples $(a,b,c)$ of positive integers such that
[list]
[*] $\max(a,b,c)$ divides $n$ and
[*] $\gcd(a,b,c)=1.$
[/list]
Compute $f(1)+f(2)+\cdots+f(100).$
Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$
Suppose that $a$ and $b$ are positive integers such that $\gcd(a^3 - b^3,(a-b)^3)$ is not divisible by any perfect square except $1.$ Given that $1 \le a-b \le 50,$ compute the number of possible values of $a-b$ across all such $a,b.$
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of $63.$
David starts at the point $A$ and goes up and right along the grid lines to point $B$. At each of the points $C$, $D$, and $E$ there is a bully. Find the number of paths David can take which make him encounter exactly one bully.
[asy]
size(150);
draw((0,0)--(4,0)--(4,3)--(0,3)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((1,0)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
dot((0,0)); label("A", (0,0), W);
dot((4,3)); label("B", (4,3), E);
dot((1,1.5)); label("C", (1,1.5), W);
dot((2,0.5)); label("D", (2,0.5), W);
dot((2.5,2)); label("E", (2.5,2), N);
[/asy]
Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$
Let $a$ and $b$ be positive real numbers that satisfy
\begin{align*}
\sqrt{a-ab}+\sqrt{b-ab}=\frac{\sqrt{6}+\sqrt{2}}{4} \,\,\, \text{and}\,\,\,
\sqrt{a-a^2}+\sqrt{b-b^2}=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2.
\end{align*}
Find the ordered pair $(a, b)$ such that $a>b$ and $a+b$ is maximal.
A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.
Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$