MathDB

3

Part of 2013 IMC

Problems(2)

IMC 2013/1/3

Source: Imc 2013 Problem 3

8/8/2013
There are 2n\displaystyle{2n} students in a school (n∈N,n⩾2)\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}. Each week n\displaystyle{n} students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
Proposed by Oleksandr Rybak, Kiev, Ukraine.
IMCcollege contests
IMC 2013/2/3

Source: Imc 2013 Problem 8

8/9/2013
Suppose that v1,v2,...,vd\displaystyle{{v_1},{v_2},...,{v_d}} are unit vectors in Rd\displaystyle{{{\Bbb R}^d}}. Prove that there exists a unitary vector u\displaystyle{u} such that ∣u⋅vi∣≤1d\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}} for i=1,2,...,d\displaystyle{i = 1,2,...,d}.
Note. Here ⋅\displaystyle{ \cdot } denotes the usual scalar product on Rd\displaystyle{{{\Bbb R}^d}}.
Proposed by Tomasz Tkocz, University of Warwick.
vectorinductionlinear algebramatrixanalytic geometrygeometry3D geometry