MathDB

3

Part of 2004 Federal Math Competition of S&M

Problems(3)

Inequalities

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
If a,b,ca,b,c are positive numbers such that abc=1abc = 1, prove the inequality
1b+1a+12+1c+1b+12+1a+1c+12≥2\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} \geq \sqrt{2}
inequalities
Geometric Inequalities

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Let M,N,PM, N, P be arbitrary points on the sides BC,CA,ABBC, CA, AB respectively of an acute-angled triangle ABCABC. Prove that at least one of the following inequalities is satisfied: NP≥12BC;PM≥12CA;MN≥12ABNP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB
inequalities
A problem on Sets

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Let A={1,2,3,...,11}A = \{1,2,3, . . . ,11\}. How many subsets BB of AA are there, such that for each n∈{1,2,...,8}n\in \{1,2, . . . ,8\}, if nn and n+2n+2 are in BB then at least one of the numbers n+1 n+1 and n+3n+3 is also in BB?
Setscombinatorics