MathDB

5

Part of 2010 IFYM, Sozopol

Problems(5)

Perfect square

Source:

7/4/2011
Let n is a natural number,for which 1+12n2\sqrt{1+12n^2} is a whole number.Prove that 2+21+12n22+2\sqrt{1+12n^2} is perfect square.
number theorygreatest common divisorDiophantine equationnumber theory unsolved
Problem 5 of Second round

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

12/14/2019
Let n>1n>1 be a natural number. Find the real values of the parameter aa, for which the equation 1+xn+1−xn=a\sqrt[n]{1+x}+\sqrt[n]{1-x}=a has a single real root.
algebra
Problem 5 of Third round - Getting a monochromatic coloring of an n-gon

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

12/14/2019
Each vertex of a right nn-gon (n≥3)(n\geq 3) is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which nn can we get from an arbitrary coloring of the nn-gon a monochromatic one (in one color)?
combinatoricspolygonColoring
Problem 5 of Fourth round - m-gons in an n-gon with a certain property

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

12/15/2019
Let A1A2...AnA_1 A_2...A_n be a convex nn-gon. What’s the number of mm-gons with vertices from A1,A2,...,AnA_1,A_2,...,A_n such that between each two adjacent vertices of the mm-gon there are at least kk vertices from the nn-gon?
combinatoricspolygonPolygons
Problem 5 of Finals

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

12/16/2019
We are given ΔABC\Delta ABC, for which the excircle to side BCBC is tangent to the continuations of ABAB and ACAC in points EE and FF respectively. Let DD be the reflection of AA in line EFEF. If it is known that ∠BAC=2∠BDC\angle BAC=2\angle BDC, then determine ∠BAC\angle BAC.
geometryexcircle