MathDB

4

Part of 2016 Korea Winter Program Practice Test

Problems(4)

Combinatoral Geometry

Source: 2016 Korea Winter Camp 1st Test #8

1/25/2016
There are nn lattice points in a general position. (no three points are collinear) A convex polygon PP covers the said nn points. (the borders are included) Prove that, for large enough nn and a positive real ϵ\epsilon, the perimeter of PP is no less than (2+ϵ)n(\sqrt{2}+\epsilon)n.
geometryKorea
How many solutions?

Source: 2016 Korea Winter Program Test1 Day1 #4

1/27/2016
p(x)p(x) is an irreducible polynomial with integer coefficients, and qq is a fixed prime number. Let ana_n be a number of solutions of the equation p(x)≡0mod  qnp(x)\equiv 0\mod q^n.
Prove that we can find MM such that {an}n≥M\{a_n\}_{n\ge M} is constant.
polynomialnumber theorynumber theory proposedalgebra
Strange Inequality

Source: 2016 Korea Winter Camp 2nd Test #4

1/25/2016
Let x,y,z≥0x,y,z \ge 0 be real numbers such that (x+y−1)2+(y+z−1)2+(z+x−1)2=27(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27.
Find the maximum and minimum of x4+y4+z4x^4+y^4+z^4
inequalities
Maximum length of strictly monotone sequence

Source: 2016 Korea Winter Camp 2nd Test #8

1/25/2016
Let a1,a2,⋯a100a_1, a_2, \cdots a_{100} be a permutation of 1,2,⋯1001,2,\cdots 100. Define l(k)l(k) as the maximum mm such that there exists i1,i2⋯imi_1, i_2 \cdots i_m such that ai1>ai2>⋯>aima_{i_1} > a_{i_2} > \cdots > a_{i_m} or ai1<ai2<⋯<aima_{i_1} < a_{i_2} < \cdots < a_{i_m}, where i1=ki_1=k and i1<i2<⋯<imi_1<i_2< \cdots <i_m
Find the minimum possible value for ∑i=1100l(i)\sum_{i=1}^{100} l(i).
combinatorics