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]

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]

Suppose we have a pack of $2n$ cards, in the order $1, 2, . . . , 2n$. A perfect shuffle of these cards changes the order to $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$ ; i.e., the cards originally in the first $n$ positions have been moved to the places $2, 4, . . . , 2n$, while the remaining $n$ cards, in their original order, fill the odd positions $1, 3, . . . , 2n - 1.$ Suppose we start with the cards in the above order $1, 2, . . . , 2n$ and then successively apply perfect shuffles. What conditions on the number $n$ are necessary for the cards eventually to return to their original order? Justify your answer. [hide="Remark"] Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.[/hide]

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.

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!" Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!" Cat says, "That's cool! My favorite number is not among those four numbers, though." Claire says, "Now I know your favorite number!" What is Cat's favorite number? [i]Proposed by Andrew Wu and Andrew Milas[/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 $p_1 - p_2 + p_3 - p_4 + p_5$?

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?

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

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 $p(z)$ 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 $1.$

How many ways are there to fill in a $2\times 2$ square grid with the numbers $1,2,3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$? [i]Proposed by Andrew Wu[/i]

Carissa is crossing a very, very, very wide street, and did not properly check both ways before doing so. (Don't be like Carissa!) She initially begins walking at $2$ feet per second. Suddenly, she hears a car approaching, and begins running, eventually making it safely to the other side, half a minute after she began crossing. Given that Carissa always runs $n$ times as fast as she walks and that she spent $n$ times as much time running as she did walking, and given that the street is $260$ feet wide, find Carissa's running speed, in feet per second. [i]Proposed by Andrew Wu[/i]

Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States). What is $a^2+b^2+c^2$? 041. The Gentleman's Alliance Cross 042. Glutamine (an amino acid) 051. Grant Nelson and Norris Windross 052. A compact region at the center of a galaxy 061. The value of \verb+'wat'-1+. (See \url{https://www.destroyallsoftware.com/talks/wat}.) 062. Threonine (an amino acid) 071. Nintendo Gamecube 072. Methane and other gases are compressed 081. A prank or trick 082. Three carbons 091. Australia's second largest local government area 092. Angoon Seaplane Base 101. A compressed archive file format 102. Momordica cochinchinensis 111. Gentaro Takahashi 112. Nat Geo 121. Ante Christum Natum 122. The supreme Siberian god of death 131. Gnu C Compiler 132. My TeX Shortcut for $\angle$.

Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.

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 $p+q$.

What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)

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

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.

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]

Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9.

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]

Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$. [i]Proposed by Andrew Wu[/i]

Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$. [i]Proposed by Andrew Wu[/i] (Note: wording changed from original to specify that $a \neq 20, 22$.)

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?

How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.) [i]Proposed by Miles Yamner and Andrew Wu[/i] (Note: wording changed from original to clarify)