Let $f(x) = x^{2} + a x + b$ and $g(x) = x^{2} + c x + d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$, and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(2012, -2012)$, what is the value of $a + c$?
A total of $2^n$ coins are distributed among several children. If one of the children has at least half of the coins, the coins are redistributed: coins are transferred from such a child to each of the other children in such a way that each of them gets as many coins as it had. In the case when one child possesses all the coins there is no possibility for redistribution. What is the greatest number of consecutive redistributions? (For example, if $32$ coins are distributed among $6$ children in the following way: $17$, $2$, $9$, $1$, $2$, $1$, then after one redistribution the children will have: $2$, $4$, $18$, $2$, $4$, $2$ coins, respectively; in the example, that number is $2$).
Explain your answer!
Steph and Jeff each start with the number $4$, and Travis is flipping a coin. Every time he flips a heads, Steph replaces her number $x$ with $2x-1$, and Jeff replaces his number $y$ with $y+8$. Every time he flips a tails, Steph replaces her number $x$ with $\frac{x+1}{2}$, and Jeff replaces his number $y$ with $y-3$. After some (positive) number of coin flips, Steph and Jeff miraculously end up with the same number below $2012$. How many times was the coin flipped?
Let $n\ge3$ be an integer and let $a_2,a_3,\ldots,a_n$ be positive real numbers with $a_2a_3\cdots a_n=1$. Prove that
$$(1+a_2)^2(1+a_3)^3\cdots(1+a_n)^n\gt n^n.$$
Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$?
Given that $P$ is a real polynomial of degree at most $2012$ such that $P(n) = 2^{n}$ for $n = 1, 2, \ldots, 2012$, what choice(s) of $P(0)$ produce the minimal possible value of $P(0)^{2} + P(2013)^{2}$?
Consider triangle $ABC$ where $BC = 7$, $CA = 8$, and $AB = 9$. $D$ and $E$ are the midpoints of $BC$ and $CA$, respectively, and $AD$ and $BE$ meet at $G$. The reflection of $G$ across $D$ is $G'$, and $G'E$ meets $CG$ at $P$. Find the length $PG$.
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x \gt y$ and $x$ is to the left of $y$, and replaces the pair $(x, y)$ by either $(y+1, x)$ or $(x-1, x)$. Prove that she can perform only finitely many such iterations.
For each positive integer $n$, there is a circle around the origin with radius $n$. Rainbow Dash starts off somewhere on the plane, but not on a circle. She takes off in some direction in a straight path. She moves $\frac{\sqrt{5}}{5}$ units before crossing a circle, then $\sqrt{5}$ units, then $\frac{3 \sqrt{5}}{5}$ units. What distance will she travel before she crosses another circle?
A fixed circle $k$ is given, together with three collinear points $E$, $F$ and $G$ such that $E$ and $G$ lie outside the circle and $F$ lies inside the circle. Prove that if $ABCD$ is an arbitrary quadrangle, inscribed in the circle $k$ such that the extensions of the sides $AB$, $AD$ and $DC$ pass through $E$, $F$ and $G$ respectively, then its side $BC$ passes through a fixed point, collinear with $E$, $F$ and $G$, which doesn't depend on the quadrangle $ABCD$.
Find all natural numbers $n$ for which there are nonnegative integers $a_1,\ldots,a_n$ such that
$$\frac1{2a_1}+\frac1{2a_2}+\cdots+\frac1{2a_n}=\frac1{3a_1}+\frac1{3a_2}+\cdots+\frac n{3a_n}=1.$$
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{1,2, \ldots, n\}$ such that the sums of the different pairs are different integers not exceeding $n$?
Mark and William are playing a game. Two walls are placed 1 meter apart, with Mark and William each starting an orb at one of the walls. Simultaneously, they release their orbs directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed (William's reverses direction at the same speed). Initially, Mark's orb is moving at $\frac{1}{1000}$ meters/s, and William's orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens?
You have a twig of length $1$. You repeatedly do the following: select two points on the twig independently and uniformly at random, make cuts on these two points, and keep only the largest piece. After $2012$ repetitions, what is the expected length of the remaining piece?
Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it will take them to finish the food?
Let $A$, $B$, $C$, and $D$ be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines $\overline{AB}$, $\overline{AC}$, $\overline{AD}$, $\overline{BC}$, $\overline{BD}$, and $\overline{CD}$ all have positive slope?