MathDB

4

Part of 2017 Irish Math Olympiad

Problems(2)

subtriangles of equilateral

Source: Irish MO 2017 paper 1 problem 4

12/12/2022
An equilateral triangle of integer side length n≥1n \geq 1 is subdivided into small triangles of unit side length, as illustrated in the figure below for n=5n = 5. In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines. https://cdn.artofproblemsolving.com/attachments/e/9/17e83ad4872fcf9e97f0479104b9569bf75ad0.jpg It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let f(n)f(n) denote the number of distinct colorings satisfying this condition. Determine, with proof, f(n)f(n) for every n≥1n \geq 1
combinatoricsColoring
1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10}

Source: Irish MO 2017 paper 2 problem 4

12/12/2022
Show that for all non-negative numbers a,ba,b, 1+a2017+b2017≥a10b7+a7b2000+a2000b10 1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10} When is equality attained?
algebrainequalities