MathDB

2

Part of 2017 Iran Team Selection Test

Problems(3)

Combinatorics from Iranian TST 2017

Source: Iranian TST 2017, first exam, day1, problem 2

4/5/2017
In the country of Sugarland, there are 1313 students in the IMO team selection camp. 66 team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these 66 tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all 1313 students have a chance of being a team member?
Proposed by Morteza Saghafian
combinatoricsIranIranian TST
2017 Iran TST2 P2

Source: 2017 Iran TST second exam day1 p2

4/23/2017
Find the largest number nn that for which there exists nn positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.
Proposed by Morteza Saghafian
IranIranian TSTnumber theorycombinatorics
Geometry from Iran TST 2017

Source: 2017 Iran TST third exam day1 p2

4/26/2017
Let PP be a point in the interior of quadrilateral ABCDABCD such that: ∠BPC=2∠BAC  ,  ∠PCA=∠PAD  ,  ∠PDA=∠PAC\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC Prove that: ∠PBD=∣∠BCA−∠PCA∣\angle PBD= \left | \angle BCA - \angle PCA \right |
Proposed by Ali Zamani
geometryTST