In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane, and Sammy on the outer. They will race for one complete lap, measured by the inner track. What is the square of the distance between Willy and Sammy's starting positions so that they will both race the same distance? Assume that they are of point size and ride perfectly along their respective lanes

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 $1.$

16. Suppose trapezoid $JANE$ is inscribed in a circle of radius $25$ such that the center of the circle lies inside the trapezoid. If the two bases of $JANE$ have side lengths $14$ and $30$ and the average of the lengths of the two legs is $\sqrt{m}$, what is $m$? 17. What is the radius of the circle tangent to the $x$-axis, the line $y=\sqrt{3}x$, and the circle $(x-10\sqrt{3})^2+(y-10)^2=25$? 18. Find the smallest positive integer $n$ such that $3n^3-9n^2+5n-15$ is divisible by $121$ but not $2$.

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.

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?

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?

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]

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]

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]

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

Ali defines a [i]pronunciation[/i] of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } \vert \text{ THEN } \vert \text{ A}$, $\text{ATH } \vert \text{ E } \vert \text{ NA}$, $\text{ATHENA}$, and $\text{AT } \vert \text{ HEN } \vert \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.) [i]Proposed by Andrew Wu, with significant inspiration from ali cy[/i]

Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.

For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that $Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$. [hide="hint"] Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]

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?

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.

Let $ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $D$ be a point on arc $\widehat{BC}$ of its circumcircle $\Omega$. Suppose that the angle bisectors of $\angle ADB$ and $\angle ADC$ meet $AB$ and $AC$ at $E$ and $F$, respectively, and that $EF$ and $BC$ meet at $G$. Line $GD$ meets $\Omega$ at $T$. If the maximum possible value of $AT^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a,b) = 1$, find $a + b$. [i]Proposed by Andrew Wu[/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]

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 \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

A polygon in the plane (with no self-intersections) is called $\emph{equitable}$ if every line passing through the origin divides the polygon into two (possibly disconnected) regions of equal area. Does there exist an equitable polygon which is not centrally symmetric about the origin? (A polygon is centrally symmetric about the origin if a $180$-degree rotation about the origin sends the polygon to itself.)

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?

Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled? [i]Proposed by Andrew Wu[/i]

You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.

Including the original, how many ways are there to rearrange the letters in PRINCETON so that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive?

Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions: [list] [*] She rolled a $6$ exactly three times; [*] The product of her first three rolls is the same as the product of her last three rolls. [/list] How many distinct sequences of six rolls could Kara have rolled? [i]Proposed by Andrew Wu[/i]

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 $p_1 - p_2 + p_3 - p_4 + p_5$?