MathDB

2

Part of 2012 Turkey Team Selection Test

Problems(3)

Sum of perfect squares

Source: Turkish TST 2012 Problem 5

3/26/2012
A positive integer nn is called good if for all positive integers aa which can be written as a=n2∑i=1nxi2a=n^2 \sum_{i=1}^n {x_i}^2 where x1,x2,…,xnx_1, x_2, \ldots ,x_n are integers, it is possible to express aa as a=∑i=1nyi2a=\sum_{i=1}^n {y_i}^2 where y1,y2,…,yny_1, y_2, \ldots, y_n are integers with none of them is divisible by n.n. Find all good numbers.
number theory proposednumber theory
An isosceles trapezoid

Source: Turkish TST 2012 Problem 2

3/26/2012
In an acute triangle ABC,ABC, let DD be a point on the side BC.BC. Let M1,M2,M3,M4,M5M_1, M_2, M_3, M_4, M_5 be the midpoints of the line segments AD,AB,AC,BD,CD,AD, AB, AC, BD, CD, respectively and O1,O2,O3,O4O_1, O_2, O_3, O_4 be the circumcenters of triangles ABD,ACD,M1M2M4,M1M3M5,ABD, ACD, M_1M_2M_4, M_1M_3M_5, respectively. If SS and TT are midpoints of the line segments AO1AO_1 and AO2,AO_2, respectively, prove that SO3O4TSO_3O_4T is an isosceles trapezoid.
geometrytrapezoidcircumcirclegeometric transformationreflectionrhombusangle bisector
Geometric inequality with centroid

Source: Turkish TST 2012 Problem 8

3/26/2012
In a plane, the six different points A,B,C,A′,B′,C′A, B, C, A', B', C' are given such that triangles ABCABC and A′B′C′A'B'C' are congruent, i.e. AB=A′B′,BC=B′C′,CA=C′A′.AB=A'B', BC=B'C', CA=C'A'. Let GG be the centroid of ABCABC and A1A_1 be an intersection point of the circle with diameter AA′AA' and the circle with center A′A' and passing through G.G. Define B1B_1 and C1C_1 similarly. Prove that AA12+BB12+CC12≤AB2+BC2+CA2 AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2
inequalitiesPythagorean Theoremgeometrygeometry proposed