MathDB

2

Part of 2010 IFYM, Sozopol

Problems(5)

Problem 2 of First round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/13/2019
Let A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 be a right octagon with center OO and λ1\lambda_1,λ2\lambda_2, λ3\lambda_3, λ4\lambda_4 be some rational numbers for which: λ1OA1→+λ2OA2→+λ3OA3→+λ4OA4→=o→\lambda_1 \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}. Prove that λ1=λ2=λ3=λ4=0\lambda_1=\lambda_2=\lambda_3=\lambda_4=0.
geometryoctagonVectorsvector
Problem 2 of Second round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/14/2019
Is it possible to color the cells of a table 19 x 19 in yellow, blue, red, and green so that each rectangle aa x bb (a,b≥2a,b\geq 2) in the table has at least 2 cells in different color?
combinatoricstableColoring
Problem 2 of Third round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/14/2019
Let ABCDABCD be a quadrilateral, with an inscribed circle with center II. Through AA are constructed perpendiculars to ABAB and ADAD, which intersect BIBI and DIDI in points MM and NN respectively. Prove that MN⊥ACMN\perp AC.
geometry
Function

Source:

6/5/2016
Known f:N0→N0f:\mathbb{N}_0 \to \mathbb{N}_0 function for ∀x,y∈N0\forall x,y\in \mathbb{N}_0 the following terms are paid (a).f(0,y)=y+1(a). f(0,y)=y+1 (b).f(x+1,0)=f(x,1)(b). f(x+1,0)=f(x,1) (c).f(x+1,y+1)=f(x,f(x+1,y)).(c). f(x+1,y+1)=f(x,f(x+1,y)). Find the value if f(4,1981)f(4,1981)
function
Find the biggest value

Source:

7/18/2011
If a,b,c>0a,b,c>0 and abc=3abc=3,find the biggest value of:
a2b2a7+a3b3c+b7+b2c2b7+b3c3a+c7+c2a2c7+c3a3b+a7\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}
inequalitiesinequalities unsolved