MathDB

2

Part of 2014 Bosnia Herzegovina Team Selection Test

Problems(2)

Regular polygon

Source: Bosnia and Herzegovina TST 2014 day 2 problem 2

5/11/2014
It is given regular nn-sided polygon, n≥6n \geq 6. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
floor functionLaTeXcombinatorics unsolvedcombinatorics
Difficult inequality

Source: Bosnia and Herzegovina TST 2014 day 1 problem 2

5/10/2014
Let aa ,bb and cc be distinct real numbers. a)a) Determine value of 1+aba−b⋅1+bcb−c+1+bcb−c⋅1+cac−a+1+cac−a⋅1+aba−b \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b}
b)b) Determine value of 1−aba−b⋅1−bcb−c+1−bcb−c⋅1−cac−a+1−cac−a⋅1−aba−b \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b}
c)c) Prove the following ineqaulity 1+a2b2(a−b)2+1+b2c2(b−c)2+1+c2a2(c−a)2≥32 \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2}
When does eqaulity holds?
inequalitiesinequalities unsolved