Problems of the Week

Contribute a problem

2018-03-05 Advanced


You have 2016 sticks of the same length in a box. You pick a stick at random, break it into two equal halves, and put them back in for a total of 2017 sticks. You repeat this process of random picking and breaking indefinitely.

What is the maximum value of xx such that, at any time during this process, you are guaranteed to have at least xx sticks of the same length?

We have a partition of 2018. If the maximum value of the product of the numbers in the partition can be expressed as a×bc, a \times b^c, where aa and bb are primes and cc is an integer, then what is a+b+c?a+b+c?

The first diagram shows 3 circles inscribed in a semicircle. The orange line shows their centers connected together with the endpoints of the diameter of the semicircle.

If we inscribe infinitely many circles, not just 3, the resulting orange curve is part of what kind of curve?

This problem can be viewed as the 3D analog of Marion's theorem.

Imagine that each edge of a tetrahedron is trisected. Then, through each of these 12 points and its two opposite vertices, a plane is constructed for a total of 12 planes.

Now, let VV denote the volume of the tetrahedron, and VMV_M the volume of the 3D figure carved out by the 12 planes inside the tetrahedron. If VM=ABV,V_M=\frac{A}{B}V, where AA and BB are coprime positive integers, find A+B.A+B.

The 3D figure in question is shown below:

Let Sn=11n+1+122n+1+12+133n+S_n=\dfrac{1}{1^n}+\dfrac{1+\frac{1}{2}}{2^n}+\dfrac{1+\frac{1}{2}+\frac{1}{3}}{3^n}+\cdots.

Then, for positive even numbers mm, there is a beautiful relationship between SmS_{m} and the Riemann zeta function ζ()\zeta(\cdot): S2=2ζ(3)S4=3ζ(5)ζ(2)ζ(3)S6=4ζ(7)ζ(2)ζ(5)ζ(3)ζ(4)S8=5ζ(9)ζ(2)ζ(7)ζ(3)ζ(6)ζ(4)ζ(5)Sm=m+22ζ(m+1)k=2m2ζ(k)ζ(mk). \begin{array} { l r c } S_2 &=& 2\zeta(3) \\ S_4 &=& {3\zeta(5)} \quad {-\zeta(2)\zeta(3)} \\ S_6 &=& {4\zeta(7)}\quad {-\zeta(2)\zeta(5)} \quad {-\zeta(3)\zeta(4)} \\ S_8 &=& {5\zeta(9)}\quad {-\zeta(2)\zeta(7)} \quad {-\zeta(3)\zeta(6)} \quad {-\zeta(4)\zeta(5)} \\ & \vdots & \\ S_{m} &=& \displaystyle \frac{m+2}2 \zeta(m+1) - \sum_{k=2}^{\frac m2} \zeta(k) \zeta(m-k). \end{array} However, there is also a relationship between positive odd numbers mm and the Riemann zeta function. Find this relationship and submit your answer as π4S3. \dfrac{\pi^4}{S_3}.

Bonus: Prove the pattern shown above.


Problem Loading...

Note Loading...

Set Loading...