MathDB

2

Part of 2011 Junior Balkan Team Selection Tests - Romania

Problems(3)

Problem 5.16 (Star Theorem)

Source:

5/27/2018
Let A1A2A3A4A5A_1A_2A_3A_4A_5 be a convex pentagon. Suppose rays A2A3A_2A_3 and A5A4A_5A_4 meet at the point X1X_1. Define X2X_2, X3X_3, X4X_4, X5X_5 similarly. Prove that ∏i=15XiAi+2=∏i=15XiAi+3\displaystyle\prod_{i=1}^{5} X_iA_{i+2} = \displaystyle\prod_{i=1}^{5} X_iA_{i+3} where the indices are taken modulo 5.
Chapter 5
1 nxn square divided into n^2 unit squares, real number in each unti square

Source: 2011 Romania JBMO TST 2.2

6/1/2020
We consider an n×nn \times n (n∈N,n≥2n \in N, n \ge 2) square divided into n2n^2 unit squares. Determine all the values of k∈Nk \in N for which we can write a real number in each of the unit squares such that the sum of the n2n^2 numbers is a positive number, while the sum of the numbers from the unit squares of any k×kk \times k square is a negative number.
combinatoricssquare
a^2 + b^2 \in A for every a, b \in A with a \ne b

Source: 2011 Romania JBMO TST 3.2

6/1/2020
Find all the finite sets AA of real positive numbers having at least two elements, with the property that a2+b2∈Aa^2 + b^2 \in A for every a,b∈Aa, b \in A with a≠ba \ne b
Setsalgebra