- 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**- You can only submit your answer once.
- 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!

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 |

**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 \(PQ\).