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 $1.$

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

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]

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.

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]

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

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]

The [b]Collaptz function[/b] is defined as $$C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}$$ We obtain the [b]Collaptz sequence[/b] of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not [b]collaptzse[/b]. [i]Proposed by Andrew Wu and Jason Wang[/i]

7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers: [list] [*] Exactly one of them is a multiple of $2$; [*] Exactly one of them is a multiple of $3$; [*] Exactly one of them is a multiple of $5$; [*] Exactly one of them is a multiple of $7$; [*] Exactly one of them is a multiple of $11$. [/list] What is the maximum possible sum of the integers that Peggy picked? 8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$? 9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.

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 $ABC$ 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]

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?

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]

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?

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.

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]