MathDB

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Part of 2016 Czech-Polish-Slovak Match

Problems(2)

Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P1,day 1

7/12/2016
Let PP be a non-degenerate polygon with nn sides, where n>4n > 4. Prove that there exist three distinct vertices A,B,CA, B, C of PP with the following property:If ℓ1,ℓ2,ℓ3\ell_1,\ell_2,\ell_3 are the lengths of the three polygonal chains into which A,B,CA, B, C break the perimeter of PP, then there is a triangle with side lengths ℓ1,ℓ2\ell_1,\ell_2 and ℓ3\ell_3. Remark: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)
geometry
Czech-Polish-Slovak Match 2016

Source: Czech-Polish-Slovak Match 2016,P1,day 2

7/12/2016
Find all quadruplets (a,b,c,d)(a, b, c, d) of real numbers satisfying the system (a+b)(a2+b2)=(c+d)(c2+d2)(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2) (a+c)(a2+c2)=(b+d)(b2+d2)(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2) (a+d)(a2+d2)=(b+c)(b2+c2)(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2) (Slovakia)
algebra