Found problems: 5
2013 Online Math Open Problems, 24
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \][i]Proposed by Evan Chen[/i]
2014 Online Math Open Problems, 18
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
[i]Proposed by Evan Chen[/i]
2017 Online Math Open Problems, 30
We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA\cdot BC=PB\cdot CA=PC\cdot AB$ and $QA\cdot BC=QB\cdot CA=QC\cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal P$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ=2YZ$. Find the greatest integer that is less than $100$ times the area of $\mathcal S$.
[i]Proposed by Michael Ren[/i]
2016 Online Math Open Problems, 28
Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$.
[i]Proposed by Vincent Huang[/i]
2016 Online Math Open Problems, 26
Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$.
[i]Proposed by Vincent Huang[/i]