Let $f(x) = x^{3} - x^{2}$. For a given value of $c$, the graph of $f(x)$, together with the graph of the line $c + x$, split the plane up into regions. Suppose that $c$ is such that exactly two of these regions have finite area. Find the value of $c$ that minimizes the sum of the areas of these two regions.
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN \parallel AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ \parallel AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.
There are 101 people participating in a Secret Santa gift exchange. As usual each person is randomly assigned another person for whom (s)he has to get a gift, such that each person gives and receives exactly one gift and no one gives a gift to themself. What is the probability that the first person neither gives gifts to or receives gifts from the second or third person? Express your answer as a decimal rounded to five decimal places.
Let $a_{1}=3$, and for $n\gt 1$, let $a_{n}$ be the largest real number such that
$$
4\left(a_{n-1}^{2}+a_{n}^{2}\right)=10 a_{n-1} a_{n}-9
$$
What is the largest positive integer less than $a_{8}$?
Let $p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0$, where each $a_i$ is either $1$ or $-1$. Let $r$ be a root of $p$. If $|r| \gt \frac{15}{8}$, what is the minimum possible value of $n$?
Suppose that there exist nonzero complex numbers $a$, $b$, $c$, and $d$ such that $k$ is a root of both the equations $a x^{3} + b x^{2} + c x + d = 0$ and $b x^{3} + c x^{2} + d x + a = 0$. Find all possible values of $k$ (including complex values).
Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Segment $PQ$ is tangent to $\omega_{1}$ at $P$ and to $\omega_{2}$ at $Q$, and $A$ is closer to $PQ$ than $B$. Point $X$ is on $\omega_{1}$ such that $PX \parallel QB$, and point $Y$ is on $\omega_{2}$ such that $QY \parallel PB$. Given that $\angle APQ = 30^{\circ}$ and $\angle PQA = 15^{\circ}$, find the ratio $AX / AY$.
What is the sum of the positive solutions to $2x^{2} - x\lfloor x\rfloor = 5$, where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$?
Call a positive integer almost-square if it can be written as $a \cdot b$, where $a$ and $b$ are integers and $a \leq b \leq \frac{4}{3} a$. How many almost-square positive integers are less than or equal to $1000000$? Your score will be equal to $25-65 \frac{|A-C|}{\min (A, C)}$.
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1 \lt h_2 \lt \dots \lt h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
(This problem was suggested by Kiran Kedlaya, Travis Schedler, and David Speyer.)
Call an $2n$-digit base-10 number special if we can split its digits into two sets of size $n$ such that the sum of the numbers in the two sets is the same. Let $p_{n}$ be the probability that a randomly-chosen $2n$-digit number is special. (We allow leading zeros in $2n$-digit numbers).
a. The sequence $p_{n}$ converges to a constant $c$. Find $c$.
b. Let $q_{n} = p_{n} - c$. There exists a unique positive constant $r$ such that $\frac{q_{n}}{r^{n}}$ converges to a constant $d$. Find $r$ and $d$.
Suppose that a polynomial of the form $p(x) = x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?
Let $f(n) = \sum_{k=1}^{n} \frac{1}{k}$. Then there exist constants $\gamma$, $c$, and $d$ such that
$$
f(n) = \ln(n) + \gamma + \frac{c}{n} + \frac{d}{n^{2}} + O\left(\frac{1}{n^{3}}\right)
$$
where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(c, d)$.
How many 8-digit numbers begin with $1$, end with $3$, and have the property that each successive digit is either one more or two more than the previous digit, considering $0$ to be one more than $9$?
Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint 27 of the remaining squares (not currently occupied by rooks) red.
Prove that no matter how the rooks are arranged and which set of 27 squares are painted, it is always possible to move some or all of the rooks so that:
- All the rooks are still on unpainted squares.
- The rooks are still not attacking each other (no two are in the same row or same column).
- At least one formerly empty square now has a rook on it; that is, the rooks are not on the same 8 squares as before.
A 5-dimensional ant starts at one vertex of a 5-dimensional hypercube of side length $1$. A move is when the ant travels from one vertex to another vertex at a distance of $\sqrt{2}$ away. How many ways can the ant make $5$ moves and end up on the same vertex it started at?
Let $a$, $b$, $c$, and $d$ be positive real numbers satisfying $a b c d = 1$. Prove that
$$
\frac{1}{\sqrt{\frac{1}{2} + a + a b + a b c}} + \frac{1}{\sqrt{\frac{1}{2} + b + b c + b c d}} + \frac{1}{\sqrt{\frac{1}{2} + c + c d + c d a}} + \frac{1}{\sqrt{\frac{1}{2} + d + d a + d a b}} \geq \sqrt{2}.
$$