MathDB

2

Part of 2012 Junior Balkan Team Selection Tests - Romania

Problems(4)

if x^n - 2x = y^n - 2y, then x = y.

Source: 2012 Romania JBMO TST 1.2

6/2/2020
Let xx and yy be two rational numbers and nn be an odd positive integer. Prove that, if xn−2x=yn−2yx^n - 2x = y^n - 2y, then x=yx = y.
algebrarationalequation
min no of rectangular tiles needed to cover the remaining surface in square

Source: 2012 Romania JBMO TST 2.2

6/2/2020
From an n×nn \times n square, n≥2,n \ge 2, the unit squares situated on both odd numbered rows and odd numbers columns are removed. Determine the minimum number of rectangular tiles needed to cover the remaining surface.
combinatoricsSquares
isosceles wanted, semicircle and circle tangent to arc

Source: 2012 Romania JBMO TST3 P2

5/29/2020
Consider a semicircle of center OO and diameter [AB][AB], and let CC be an arbitrary point on the segment (OB)(OB). The perpendicular to the line ABAB through CC intersects the semicircle in DD. A circle centered in PP is tangent to the arc BDBD in FF and to the segments [AB][AB] and [CD][CD] in GG and EE, respectively. Prove that the triangle ADGADG is isosceles.
geometryisoscelestangent circlessemicircle
red and blue vertices of a egular 2n-gon

Source: 2012 Romania JBMO TST 4.2, 10th Vojtech Jarnik Mathematical Competition, Ostrava, 2000

6/2/2020
Let us choose arbitrarily nn vertices of a regular 2n2n-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.
combinatorial geometrycombinatoricsColoring