$ABCD$ is a square with sides of unit length. Points $E$ and $F$ are taken on sides $AB$ and $AD$ respectively so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. What is this maximum area?

A box of strawberries, containing $12$ strawberries total, costs $\$ 2$. A box of blueberries, containing $48$ blueberries total, costs $ \$ 3$. Suppose that for $\$ 12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$. [i]Proposed by Andrew Wu[/i]

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]

Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$

Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?

A student at Harvard named Kevin Was counting his stones by $11$ He messed up $n$ times And instead counted $9$s And wound up at $2007$. How many values of $n$ could make this limerick true?

Suppose that $a_1 = 1,$ $a_2 = 2$, and for any $n \ge 3$, $a_n = a_1 + a_2 + \cdots + a_{n-1}$. Find $\frac{a_{2021}}{a_{2020}}$. [i]Proposed by Andrew Wu[/i]

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 $yz$.

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]

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?

Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$. [i]Proposed by Andrew Wu, Deyuan Li, and Andrew Milas[/i]

Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only $ 5$ tickets remain, so $ 4$ of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which $ 5$ people can see the game?

Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.

Find the 2000th positive integer that is not the difference between any two integer squares.

You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?

Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

1. If $5x+3y-z=4$, $x=y$, and $z=4$, find $x+y+z$. 2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon? 3. Sirena picks five distinct positive primes, $p_1 < p_2 < p_3 < p_4 < p_5$, and finds that they sum to $192$. If the product $p_1p_2p_3p_4p_5$ is as large as possible, what is $p_1 - p_2 + p_3 - p_4 + p_5$?

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 $X$.)

(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$ (b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.

4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$. Find the maximum possible value of $A \cdot B$. 5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments? 6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.

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

For what values of $a$ does the system of equations \[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?

Let $ABC$ be an acute triangle (with $AB \ne AC$) and $M$ be the midpoint of $BC$. The circle of diameter $AM$ cuts $AC$ at $N$ and $BC$ again at $H$. A point $K$ is taken on $AC$ (between $A$ and $N$) such that $CN = NK$. Segments $AH$ and $BK$ intersect at $L$. The circle that passes through $A$,$K$ and $L$ cuts $AB$ at $P$. Prove that $C$,$L$ and $P$ are collinear.

Let $ABC$ be an isosceles triangle with $\angle{ABC} = \angle{ACB} = 80^\circ$. Let $D$ be a point on $AB$ such that $\angle{DCB} = 60^\circ$ and $E$ be a point on $AC$ such that $\angle{ABE} = 30^\circ$. Find $\angle{CDE}$ in degrees.

A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.