Found problems: 229
Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?
In a $2024 \times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^\circ$ clockwise or counterclockwise if it is currently on a black square, otherwise it must continue walking in the same direction.
A coloring of the grid is called [i]simple[/i] if it is [b]not[/b] possible for the ant to arrive back at its starting location after some time. How many simple colorings of the grid are maximal, in the sense that adding any black square results in a coloring that is not simple?
Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)
Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.)
Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
Suppose you start at $0$, a friend starts at $6$, and another friend starts at $8$ on the number line. Every second, the leftmost person moves left with probability $\tfrac14$, the middle person with probability $\tfrac13$, and the rightmost person with probability $\tfrac12$. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.
An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die.
Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.)
[i]Proposed by Lohith Tummala[/i]
A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable?
[i]Proposed by Giacomo Rizzo[/i]
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
Compute the number of ordered triples $(x,y,z)$ of integers satisfying
\[x^2+y^2+z^2=9.\]
[i]Proposed by Nathan Xiong[/i]
Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$.
[i]Proposed by David Tang[/i]
Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?
Suppose we have a cubic polynomial $p(x)$ such that $p(0)=0,p(1)=1,$ and $p(x)\leq \sqrt x$ for $0\leq x \leq 1.$ Suppose $p(0.5)$ is maximized. What is the sum of $p(0.25)+p(0.75)?$
[i]Proposed by Ishin Shah[/i]
Adam places down cards one at a time from a standard 52 card deck (without replacement) in a pile. Each time he places a card, he gets points equal to the number of cards in a row immediately before his current card that are all the same suit as the current card. For instance, if there are currently two hearts on the top of the pile (and the third card in the pile is not hearts), then placing a heart would be worth 2 points, and placing a card of any other suit would be worth 0 points. What is the expected number of points Adam will have after placing all 52 cards?
[i]Proposed by Adam Bertelli[/i]
Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
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 $100m + n$.
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023 \\
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$.
[i]Proposed by Howard Halim[/i]
Compute the remainder when $10^{2021}$ is divided by $10101$.
[i]Proposed by Nathan Xiong[/i]
Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$
where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?
Suppose we have a uniformly random function from $\{1, 2, 3, \ldots, 25\}$ to itself. Find the expected value of $$\sum_{x=1}^{25} (f(f(x))-x)^2.$$
Find the number of ordered triples of integers $(a,b,c)$, each between $1$ and $64$, such that
\[
a^2 + b^2 \equiv c^2\pmod{64}.
\]
Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$
Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$.
[i]Proposed by Justin Hsieh[/i]
The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]