Found problems: 161
Points $A$ and $B$ lie on circle $\omega$ with center $O.$ Let $X$ be a point inside $\omega.$ Suppose that $XO=2\sqrt{2}, XA=1, XB=3,$ and $\angle{AXB}=90^\circ.$ Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle{AXB}$ and $\triangle{YXZ}$ are similar with the same orientation. Compute $XY.$
Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.
Suppose $a$ and $b$ are positive integers such that $a^b=2^{2023}.$ Compute the smallest possible value of $b^a.$
For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$
Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$
A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to $\mathcal{P}.$
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Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$
Let $\triangle{ABC}$ be an equilateral triangle. Point $D$ is on segment $\overline{BC}$ such that $BD=1$ and $DC=4.$ Points $E$ and $F$ lie on rays $\overrightarrow{AC}$ and $\overrightarrow{AB},$ respectively, such that $D$ is the midpoint of $\overline{EF}.$ Compute $EF.$
Let $a,b,$ and $c$ be real numbers such that
\begin{align*}
a+b+c &= 100 \\
ab+bc+ca &= 20, \text{ and} \\
(a+b)(a+c) &=24.
\end{align*}
Compute all possible values of $bc.$
An equilateral triangle is inscribed in a circle $\omega.$ A chord of $\omega$ is cut by the perimeter of the triangle into three segments of lengths $55, 121,$ and $55,$ in that order. Compute the sum of all possible side lengths of the triangle.
Find all positive integer pairs $(a,b)$ that satisfy the equation$$a^2b+ab^2+73=8ab+9a+9b.$$
The vertices of a cube are labeled with the integers $1$ through $8,$ with each used exactly once. Let $s$ be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of $s$ over all such labelings.
Let $ABC$ be a triangle such that $AB = 3, AC = 4,$ and $\angle{BAC} = 75^\circ.$ Square $BCDE$ is constructed outside triangle $ABC.$ Compute $AD^2 +AE^2.$
Suppose $x$, $y$, $z$ are pairwise distinct real numbers satisfying
\[
x^2+3y =y^2 +3z = z^2+3x.
\]Find $(x+y)(y+z)(z+x)$.
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$
Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$
The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$
The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.
Compute the number of quadruples $(a,b,c,d)$ of positive integers satisfying $$12a+21b+28c+84d=2024.$$
let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$
Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino.
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The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.
Find
\[\sum_{n=1}^{\infty} \frac{\varphi(n)}{(-4)^n-1},\]where $\varphi(n)$ is the number of positive integers $k \le n$ relatively prime to $n$. (Note $\varphi(1)=1$.)
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$