Found problems: 161
Let $P(x)$ be a quartic polynomial with integer coefficients and leading coefficient $1$ such that $P(\sqrt 2+\sqrt 3+\sqrt 6)=0$. Find $P(1)$.
A four-digit integer in base $10$ is [i]friendly[/i] if its digits are four consecutive digits in any order. A four-digit integer is [i]shy[/i] if there exist two adjacent digits in its representation that differ by $1.$ Compute the number of four-digit integers that are both friendly and shy.
Let $A, E, H, L, T,$ and $V$ be chosen independently and at random from the set $\{0, \tfrac{1}{2}, 1\}.$ Compute the probability that $\lfloor T \cdot H \cdot E \rfloor = L \cdot A \cdot V \cdot A.$
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
Let $ABC$ be a triangle with $AB=13, BC=15, AC=14$. Let $P$ be the point such that $AP$ $=$ $CP$ $=$ $\tfrac12 BP$. Find $AP^2$.
Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.
Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.
Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$
A base $9$ number [i]probably places[/i] if it has a $7$ as one of its digits. Find the number of base $9$ numbers less than or equal to $100$ in base $10$ that probably place.
Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$
Equilateral triangles $ABF$ and $BCG$ are constructed outside regular pentagon $ABCDE.$ Compute $\angle{FEG}.$