Lattices and Farey Sequences
Read this topicExplanations 4
- The goal of this essay is to introduce the beautiful Farey sequences, and to prove a result about them using an ingenious geometric interpretation. (This is based on Essay 5 fro…
By Kada Williams, Alex Federspiel, Moor Xu, Toan Pham
- [IMO 2013, problem 6] Let n≥3n\geq 3n≥3 be an integer, and consider a circle with n+1n+1n+1 equally spaced points marked on it. Consider all labellings of these points with the …
By Kada Williams
- Prove that FnF_nFn has 1+∑m=1nφ(m)1+\sum_{m=1}^n \varphi(m)1+∑m=1nφ(m) terms, where φ\varphiφ is the totient function.
By Kada Williams
- Check how we can partition a 1×m1\times m1×m rectangle into 2m2m2m triangles. Also, prove that given TTT, such a partition is always possible (where TTT is triangle CPQCPQCPQ):
By Kada Williams
Problems 2
- [USA TSTST 2013/2] A finite sequence of integers a1,a2,…,ana_1, a_2, \dots, a_na1,a2,…,an is called regular if there exists a real number xxx satisfying ⌊kx⌋=akfor 1≤k≤n.\lef…
- [Vietnamese MO 2013] Write down some numbers a1,a2,…,ana_1,a_2,\ldots, a_na1,a2,…,an from left to right on a line. Step 1, we write a1+a2a_1+a_2a1+a2 between a1,a2a_1,a_2a1…