MathDB

Problem 4

Part of 2017 Vietnamese Southern Summer School contest

Problems(3)

A nice way to color the square board

Source: Southern Summer School, gr. 10

7/9/2017
In a square board of size 1001 x 1001, we color some mm cells in such a way that: i. Of any two cells that share an edge, at least one is colored. ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored. Determine the smallest possible value of mm.
combinatoricsminimizesquare grid
With Geogebra, geometry is easy

Source: Southern Summer School, gr. 11

7/9/2017
Let ABCABC be a triangle. A point PP varies inside BCBC. Let Q,RQ, R be the points on AC,ABAC, AB in that order, such that PQ∥AB,PR∥ACPQ\parallel AB, PR\parallel AC. 1. Prove that, when PP varies, the circumcircle of triangle AQRAQR always passes through a fixed point XX other than AA. 2. Extend AXAX so that it cuts the circumcircle of ABCABC a second time at point KK. Prove that AX=XKAX=XK.
geometryLocusLocus problems
Funny group of students

Source: Southern Summer School, gr. 12

7/9/2017
In a summer school, there are n>4n>4 students. It is known that, among these students, i. If two ones are friends, then they don't have any common friends. ii If two ones are not friends, then they have exactly two common friends. 1. Prove that 8n−78n-7 must be a perfect square. 2. Determine the smallest possible value of nn.
combinatoricsgraph theory