MathDB

4

Part of 2017 Korea Winter Program Practice Test

Problems(3)

Equilateral triangle with vertices near lattices points

Source: 2017 Korean Winter Program Practice Test 1 Day 1 #4

1/18/2017
For a point PP on the plane, denote by ∥P∥\lVert P \rVert the distance to its nearest lattice point. Prove that there exists a real number L>0L > 0 satisfying the following condition:
For every ℓ>L\ell > L, there exists an equilateral triangle ABCABC with side-length ℓ\ell and ∥A∥,∥B∥,∥C∥<10−2017\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}.
number theorygeometry
Bisecting a set into 2-adic squares and non-squares

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #4

1/21/2017
For a nonzero integer kk, denote by ν2(k)\nu_2(k) the maximal nonnegative integer tt such that 2t∣k2^t \mid k. Given are n(≥2)n (\ge 2) pairwise distinct integers a1,a2,…,ana_1, a_2, \ldots, a_n. Show that there exists an integer xx, distinct from a1,…,ana_1, \ldots, a_n, such that among ν2(x−a1),…,ν2(x−an)\nu_2(x - a_1), \ldots, \nu_2(x - a_n) there are at least n/4n/4 odd numbers and at least n/4n/4 even numbers.
combinatoricsnumber theory
Inequality of four variables

Source: 2017 Korea Winter Program Practice Test 2 #4

8/14/2019
Let a,b,c,da,b,c,d be the area of four faces of a tetrahedron, satisfying a+b+c+d=1a+b+c+d=1. Show that an+bn+cnn+bn+cn+dnn+cn+dn+ann+dn+an+bnn≤1+2n\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2} holds for all positive integers nn.
algebrainequalities