# Problem of the Week

## Rules

• Problem of the Week will be posted once a week at 3:00 PM EDT on Thursday
• Most PotWs will have something to do with the class that week, though not necessarily
• No computational aids are to be used when solving the PotW (including calculators, Wolfram Alpha, computer programs, etc.)
• If you are caught cheating you will be disqualified from all future PotWs and ineligible to receive YCMA swag at the end of the year
• No discussion or collaboration on active PotWs
• Though we don't require full solutions, we will ask for a short sketch/key idea behind your solution to ensure integrity. Though we will be lenient, your answer may be disqualified if there is insufficient justification
• Responses are scored based on accuracy, as well as how soon after 3:00 PM EDT they were submitted. Responses will be accepted until the next class (Wednesday, 7:30 EDT)
• There will be a running leaderboard, and first place may be eligible to earn a prize, so good luck!

## Current Problem of the Week

Coming Soon!

Total Score Average Time (hours)
1. Eric Wang 7 3.976
2. Kevin Liu 4 1.1158
3. Eva Lin 3 80.3111
4. Arjun Rastogi 2 3.3917
5. Allen Wang 2 163.5500
6. Abrar Fiaz 1 0.9244
7. Aryaan Jena 1 1.4503
8. Heran Yang 1 168.5353

## Archive

Sample: Let $$x$$,$$y$$,$$z$$ be real numbers such that $x^2+y^2+z^2-\frac{1000}{9}=xy+xz+yz\,\,\text{ and }\,\,x^2+y=y^2+z=z^2+x$ If the maximum possible value of $$x+y$$ can be written as $$\frac{a+\sqrt{b}}{c}$$, for naturals $$a$$,$$b$$,$$c$$ where $$c$$ is minimal, compute $$a+b+c$$

PotW 5/28: Consider the sum $S=\sum_{k=0}^{2020}\frac{4^k}{\binom{2k}{k}}$If it can be written in simplest form as $$\frac{a}{b}$$, compute the last $$3$$ digits of $$3a-b$$

PotW 6/4: Define $f(n)=\sum\limits_{i=0}^\infty\frac{\gcd(n,i)\cdot i}{x^i}.$ If $$f(n)=\frac{P_n(x)}{Q_n(x)}$$ for integer polynomials $$P_n,Q_n$$ which don't share any common factors other than $$\pm 1$$, find the minimal positive integer $$n$$ such that $$2019|P_n(1)$$.

PotW 6/11: Consider sequence $$\{a_i\}$$ with first term $$a_0=\frac{13}{2}$$ which satisfies $$a_n=\lfloor a_{n-1}\rfloor +\frac{1}{a_{n-1}}$$. Compute the numerator of $$a_{2020}\pmod{1013}$$.

PotW 6/25: The sum $\sum\limits_{k=0}^{2018} \left(\cos \left(\frac{\pi k}{2019} \right)\right)^{2020}$ can be expressed in the form $$\frac{m}{n}$$ for relatively prime positive integers $$m$$ and $$n$$. Determine the remainder when $$m$$ is divided by $$1009$$.

PotW 7/2: Let $$ABC$$ be a triangle such that $$AB$$, $$BC$$, and $$CA$$ are in an arithmetic progression in that order. Let $$M$$ be the midpoint of arc $$BC$$ in the circumcircle of $$ABC$$, and let $$I$$ be the incenter of $$ABC$$. If $$AM = 10$$ and $$AC = 8$$, then $$CI^2$$ can be written in the form $$\frac{m}{n}$$ for relatively prime positive integers $$m,n$$. Find $$m+n$$.

PotW 7/30: Circles $$\Omega_1$$ and $$\Omega_2$$ are externally tangent and have radii $$10$$ and $$15$$, respectively. A line intersects $$\Omega_1$$ at $$R$$ and $$S$$ and intersects $$\Omega_2$$ at $$T$$ and $$U$$, in that order. If $$RS = TU = 2ST$$, then $$ST^2$$ can be expressed in the form $$\frac{m}{n}$$ for relatively prime positive integers $$m,n$$. Find the value of $$m+n$$.

PotW 8/13: Tetrahedron $$ABCD$$ satisfies $$AB=BC=CA$$ and $$DA=DB=DC$$. Let $$E,F,G,H$$ be the feet of the altitudes from $$A,B,C,D$$ to their respective faces. If $$E,F,G,H$$ all lie in the interiors of their respective faces, let the maximum value of $$\frac{[EFGH]}{[ABCD]}$$ be $$\frac{m}{n}$$ for relatively prime positive integers $$m,n$$. Compute $$m+n$$.

PotW 8/20: The incircle of triangle $$ABC$$ touches $$BC,CA,AB$$ at $$D,E,F$$ respectively. Let $$I$$ be its incenter, and $$K$$ be the intersection of the $$B$$-midline of $$BDF$$ and the $$C$$-midline of $$CDE$$. If the inradius is $$10$$ and $$AB+AC=3BC=294$$, compute $$KI$$.
Note: the $$X$$-midline of triangle $$XYZ$$ is defined to be the line passing through the midpoints of $$XY$$ and $$XZ$$.

PotW 9/2: Let $$f(n)=n^2+5n+3$$. Denote $$P$$ as the remainder when $$\prod\limits_{n=1}^{2016} f(n)$$ is divided by $$2017$$ and let $$Q$$ be the number of distinct residues $$f(n)$$ achieves mod $$2017$$. Find the last $$3$$ digits of $$PQ$$.