MathDB

1

Part of 2010 Czech-Polish-Slovak Match

Problems(2)

System of equations: a√b - c = a; b√c - a = b; c√a - b = c

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Find all triples (a,b,c)(a,b,c) of positive real numbers satisfying the system of equations ab−c&=a,bc−a&=b,ca−b&=c. a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c.
pigeonhole principlealgebrasystem of equationsalgebra unsolved
Rearranging sides of triangles to make other triangles

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Given any collection of 20102010 nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: [*]let b1≤b2≤⋯≤b2011b_1\le b_2\le\cdots\le b_{2011} denote the lengths of the blue sides; [*]let r1≤r2≤⋯≤r2011r_1\le r_2\le\cdots\le r_{2011} denote the lengths of the red sides; and [*]let w1≤w2≤⋯≤w2011w_1\le w_2\le\cdots\le w_{2011} denote the lengths of the white sides. Find the largest integer kk for which there necessarily exists at least kk indices jj such that bjb_j, rjr_j, wjw_j are the side lengths of a nondegenerate triangle.
combinatorics unsolvedcombinatorics