MathDB

3

Part of 2017 Iran Team Selection Test

Problems(3)

Geometry from Iranian TST 2017

Source: Iranian TST 2017, first exam, day1, problem 3

4/5/2017
In triangle ABCABC let IaI_a be the AA-excenter. Let ω\omega be an arbitrary circle that passes through A,IaA,I_a and intersects the extensions of sides AB,ACAB,AC (extended from B,CB,C) at X,YX,Y respectively. Let S,TS,T be points on segments IaB,IaCI_aB,I_aC respectively such that ∠AXIa=∠BTIa\angle AXI_a=\angle BTI_a and ∠AYIa=∠CSIa\angle AYI_a=\angle CSI_a.Lines BT,CSBT,CS intersect at KK. Lines KIa,TSKI_a,TS intersect at ZZ. Prove that X,Y,ZX,Y,Z are collinear.
Proposed by Hooman Fattahi
geometryIranIranian TSTcollinear
2017 Iran TST2 p3

Source: 2017 Iran TST second exam day1 p3

4/23/2017
There are 2727 cards, each has some amount of (11 or 22 or 33) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a match such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a match be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a match?
Proposed by Amin Bahjati
combinatoricsIranian TSTIran
Functional Equation from Iran TST 2017

Source: 2017 Iran TST third exam day1 p3

4/26/2017
Find all functions f:R+×R+→R+f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+ that satisfy the following conditions for all positive real numbers x,y,z:x,y,z: f(f(x,y),z)=x2y2f(x,z)f\left ( f(x,y),z \right )=x^2y^2f(x,z) f(x,1+f(x,y))≥x2+xyf(x,x)f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)
Proposed by Mojtaba Zare, Ali Daei Nabi
algebrafunctional equationfunction