Problem of the Week


Current Problem of the Week

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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 \(PQ\).