MathDB

4

Part of 2015 Iran Team Selection Test

Problems(3)

Find the least k

Source: Iran TST 2015, first exam, day 2 problem 1

5/11/2015
nn is a fixed natural number. Find the least kk such that for every set AA of kk natural numbers, there exists a subset of AA with an even number of elements which the sum of it's members is divisible by nn.
combinatoricsnumber theory
x,y,z collinear

Source: Iran TST 2015,second exam,second day,problem 4

6/4/2015
Let △ABC\triangle ABC be an acute triangle. Point ZZ is on AA altitude and points XX and YY are on the BB and CC altitudes out of the triangle respectively, such that: ∠AYB=∠BZC=∠CXA=90\angle AYB=\angle BZC=\angle CXA=90 Prove that XX,YY and ZZ are collinear, if and only if the length of the tangent drawn from AA to the nine point circle of △ABC\triangle ABC is equal with the sum of the lengths of the tangents drawn from BB and CC to the nine point circle of △ABC\triangle ABC.
geometry
Triangles into each other

Source: Iran TST 2015 Test 3 Day 2 P1

8/18/2023
Ali puts 55 points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle T1T_1 can be placed inside triangle T2T_2 if T1T_1 is congruent to a triangle that is located completely inside T2T_2.)
geometry