Rice University and Stanford University write questions and corresponding solutions for a high school math tournament. The Rice group writes 10 questions every hour but make a mistake in calculating their solutions 10% of the time. The Stanford group writes 20 problems every hour and makes solution mistakes 20% of the time. Each school works for 10 hours and then sends all problems to Smartie to be checked. However, Smartie isn’t really so smart, and only 75% of the problems she thinks are wrong are actually incorrect. Smartie thinks 20% of questions from Rice have incorrect solutions, and that 10% of questions from Stanford have incorrect solutions. This problem was definitely written and solved correctly. What is the probability that Smartie thinks its solution is wrong?

Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define \[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\] for $n\ge 0.$ Evaluate \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?

Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.

Consider a finite set of points $T\in\mathbb{R}^n$ contained in the $n$-dimensional unit ball centered at the origin, and let $X$ be the convex hull of $T$. Prove that for all positive integers $k$ and all points $x\in X$, there exist points $t_1, t_2, \dots, t_k\in T$, not necessarily distinct, such that their centroid \[\frac{t_1+t_2+\dots+t_k}{k}\]has Euclidean distance at most $\frac{1}{\sqrt{k}}$ from $x$. (The $n$-dimensional unit ball centered at the origin is the set of points in $\mathbb{R}^n$ with Euclidean distance at most $1$ from the origin. The convex hull of a set of points $T\in\mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)

Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179; /* image dimensions */ /* draw figures */ draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2)); draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2)); draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2)); /* dots and labels */ dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle); label("$O_1$", (-4.252707018231291,-1.545940604327141), N * labelscalefactor); dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle); label("$O_2$", (21.704189347819636,-1.250863978037686), NE * labelscalefactor); dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle); label("$A$", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor); dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle); label("$B$", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] [i]Proposed by Deyuan Li and Andrew Milas[/i]

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

A pack of MIT students are holding an escape room, where students may compete in teams of 4, 5, or 6. There is \$60 dollars worth of prize money in Amazon gift cards for the winning team. If each gift card can contain any whole number of dollars, what is the minimum number of gift cards required so that the prize money can be distributed evenly among any team? [i]Lightning 4.1[/i]

13. The triangle with vertices $(0,0), (a,b)$, and $(a,-b)$ has area $10$. Find the sum of all possible positive integer values of $a$, given that $b$ is a positive integer. 14. Elsa is venturing into the unknown. She stands on $(0,0)$ in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as $\frac{a}{b}$ in simplest terms. Find $a+b$. (A lattice point is a point with integer coordinates.) 15. Let $ABCD$ be a square with side length $4$. Points $X, Y,$ and $Z$, distinct from points $A, B, C,$ and $D$, are selected on sides $AD, AB,$ and $CD$, respectively, such that $XY = 3, XZ = 4$, and $\angle YXZ = 90^{\circ}$. If $AX = \frac{a}{b}$ in simplest terms, then find $a + b$.

10. Prair picks a three-digit palindrome $n$ at random. If the probability that $2n$ is also a palindrome can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. (A palindrome is a number that reads the same forwards as backwards; for example, $161$ and $2992$ are palindromes, but $342$ is not.) 11. If two distinct integers are picked randomly between $1$ and $50$ inclusive, the probability that their sum is divisible by $7$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. 12. Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rockets is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, find $p+q$.

Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D,E$, and $F$ are chosen on $AB, BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1$, find $a + b$. (Here $[DEF]$ denotes the area of triangle $DEF$.) [i]Proposed by Vismay Sharan[/i]

You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties: \begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*} $\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$ Find a general formula for $x\diamond y$, assuming that $y\gex>0$.

The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$. [i]Proposed By Pedram Safaei[/i]

Let $m>1$ be a fixed positive integer. For a nonempty string of base-ten digits $S$, let $c(S)$ be the number of ways to split $S$ into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by $m$. These strings are allowed to have leading zeroes. In terms of $m$, what are the possible values that $c(S)$ can take? For example, if $m=2$, then $c(1234)=2$ as the splits $1234$ and $12|34$ are valid, while the other six splits are invalid.

The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrive once every one-hour block at some uniformly random time (once at a random time between $\text{9am}$ and $\text{10am}$, once at a random time between $\text{10am}$ and $\text{11am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.

Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.