MathDB

2

Part of 2017 Dutch IMO TST

Problems(3)

geometry problem

Source: Netherlands TST for IMO 2017, day 2,problem 2

2/1/2018
The incircle of a non-isosceles triangle ABCABC has centre II and is tangent to BCBC and CACA in DD and EE, respectively. Let HH be the orthocentre of ABIABI, let KK be the intersection of AIAI and BHBH and let LL be the intersection of BIBI and AHAH. Show that the circumcircles of DKHDKH and ELHELH intersect on the incircle of ABCABC.
geometry
combinatorial nT in 2n-gon

Source: Netherlands TST for IMO 2017 day 1,problem 2

2/1/2018
Let n≥4n \geq 4 be an integer. Consider a regular 2n−2n-gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this 2n−2n-gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers mm the integers m+1,m+2,...,m+2nm + 1, m + 2, . . . , m + 2n can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
number theoryprime numbers
algebra problem

Source: Netherlands TST for IMO 2017 day 3 problem 2

2/1/2018
let a1,a2,...ana_1,a_2,...a_n a sequence of real numbers such that a1+....+an=0a_1+....+a_n=0. define bi=a1+a2+....aib_i=a_1+a_2+....a_i for all 1≤i≤n1 \leq i \leq n .suppose bi(aj+1−ai+1)≥0b_i(a_{j+1}-a_{i+1}) \geq 0 for all 1≤i≤j≤n−11 \leq i \leq j \leq n-1. Show that max⁡1≤l≤n∣al∣≥max⁡1≤m≤n∣bm∣\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|
Sequencesalgebra