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.)

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.)

What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.

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

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 $a + b$.

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 $\angle$.

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]

Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$, $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$. If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$, find $a + b$. [i]Proposed by Andrew Wu[/i]

Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$, inclusive, and that it is divisible by $99$, find the maximum possible value of $\overline{ABCDEF}$. [i]Proposed by Andrew Milas[/i]

The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.

Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?

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]

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

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 $p+q$.

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 $2$.

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]

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]

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 $ATHEM$ 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]

Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

The Duke of Squares left to his three sons a square estate, $100\times 100$ square miles, made up of ten thousand $1\times 1$ square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A $1\times 1$ square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?

At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$. [i]Proposed by Evan Chen[/i]

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?

Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an infinite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses first, what is the probability that he will win? Express your answer as a fraction.

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]