MathDB

5

Part of 2021 Iran Team Selection Test

Problems(2)

Best Geo ever made

Source: Iranian TST 2021, second exam day 2, problem 2

5/22/2021
Point XX is chosen inside the non trapezoid quadrilateral ABCDABCD such that ∠AXD+∠BXC=180\angle AXD +\angle BXC=180. Suppose the angle bisector of ∠ABX\angle ABX meets the DD-altitude of triangle ADXADX in KK, and the angle bisector of ∠DCX\angle DCX meets the AA-altitude of triangle ADXADX in LL.We know BK⊥CXBK \perp CX and CL⊥BXCL \perp BX. If the circumcenter of ADXADX is on the line KLKL prove that KL⊥ADKL \perp AD.
Proposed by Alireza Dadgarnia
geometrytrapezoidangle bisectorcircumcircle
Nice numbers which are not really nice

Source: Iranian TST 2021, first exam day 2, problem 5

5/17/2021
Call a triple of numbers Nice if one of them is the average of the other two. Assume that we have 2k+12k+1 distinct real numbers with k2k^2 Nice triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by Morteza Saghafian
Sequencealgebraarithmetic sequence