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Aops Community 2023 Turkey Mo (2Nd Round) : National Olympiad Second Round 2023

The document outlines the problems presented in the 2023 Turkey Mathematical Olympiad, specifically detailing the challenges from the second round. It includes a variety of mathematical proofs and problem-solving scenarios involving integers, geometry, and sequences. Each problem requires a unique approach, showcasing advanced mathematical concepts and reasoning skills.

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249 views1 page

Aops Community 2023 Turkey Mo (2Nd Round) : National Olympiad Second Round 2023

The document outlines the problems presented in the 2023 Turkey Mathematical Olympiad, specifically detailing the challenges from the second round. It includes a variety of mathematical proofs and problem-solving scenarios involving integers, geometry, and sequences. Each problem requires a unique approach, showcasing advanced mathematical concepts and reasoning skills.

Uploaded by

310351
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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AoPS Community 2023 Turkey MO (2nd round)

National Olympiad Second Round 2023


www.artofproblemsolving.com/community/c3692324
by Tintarn, AlperenINAN

– Day 1

1 Prove that there exist infinitely many positive integers k such that the equation
n2 + m 2
=k
m4 + n
don’t have any positive integer solution.

2 Let ABC be a triangle and P be an interior point. Let ωA be the circle that is tangent to the
circumcircle of BP C at P internally and tangent to the circumcircle of ABC at A1 internally
and let ΓA be the circle that is tangent to the circumcircle of BP C at P externally and tangent
to the circumcircle of ABC at A2 internally. Define B1 , B2 , C1 , C2 analogously. Let O be the
circumcentre of ABC. Prove that the lines A1 A2 , B1 B2 , C1 C2 and OP are concurrent.

3 Let a 9-digit number be balanced if it has all numerals 1 to 9. Let S be the sequence of the nu-
merals which is constructed by writing all balanced numbers in increasing order consecutively.
Find the least possible value of k such that any two subsequences of S which has consecutive
k numerals are different from each other.

– Day 2

4 Initially given 31 tuplets


(1, 0, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, 0, . . . , 1)
were written on the blackboard. At every move we choose two written 31 tuplets as (a1 , a2 , a3 , . . . , a31 )
and (b1 , b2 , b3 , . . . , b31 ), then write the 31 tuplet (a1 + b1 , a2 + b2 , a3 + b3 , . . . , a31 + b31 ) to the black-
board too. Find the least possible value of the moves such that one can write the 31 tuplets
(0, 1, 1, . . . , 1), (1, 0, 1, . . . , 1), . . . , (1, 1, 1, . . . , 0)
to the blackboard by using those moves.

5 Is it possible that a set consisting of 23 real numbers has a property that the number of the
nonempty subsets whose product of the elements is rational number is exactly 2422?

6 On a triangle ABC, points D, E, F are given on the segments BC, AC, AB respectively such
AB 2
that DE ∥ AB, DF ∥ AC and BD DC = AC 2 holds. Let the circumcircle of AEF meet AD at R and
the line that is tangent to the circumcircle of ABC at A at S again. Let the line EF intersect BC
at L and SR at T . Prove that SR bisects AB if and only if BS bisects T L.

© 2023 AoPS Incorporated 1


Art of Problem Solving is an ACS WASC Accredited School.

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