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Mathematics Olympiad questions with Solutions R100,00
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Exam (elaborations)

Mathematics Olympiad questions with Solutions

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It has all the questions in the various IMO field that test a high level of critical thinking as well as knowledge in geometry, algebra, number theory and combinatorics. Some are almost impossible but can you solve them? I don’t think you can , so prove me wrong!

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  • January 29, 2025
  • 2
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers
All documents for this subject (179)
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szja626
English (eng), day 1




Saturday, 8. July 2023

Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d1 , d2 , . . . , dk
are all the positive divisors of n with 1 = d1 < d2 < · · · < dk = n, then di divides di+1 + di+2 for
every 1 ⩽ i ⩽ k − 2.

Problem 2. Let ABC be an acute-angled triangle with AB < AC. Let Ω be the circumcircle of
ABC. Let S be the midpoint of the arc CB of Ω containing A. The perpendicular from A to BC
meets BS at D and meets Ω again at E ̸= A. The line through D parallel to BC meets line BE
at L. Denote the circumcircle of triangle BDL by ω. Let ω meet Ω again at P ̸= B.
Prove that the line tangent to ω at P meets line BS on the internal angle bisector of ∠BAC.

Problem 3. For each integer k ⩾ 2, determine all infinite sequences of positive integers a1 , a2 , . . .
for which there exists a polynomial P of the form P (x) = xk + ck−1 xk−1 + · · · + c1 x + c0 , where
c0 , c1 , . . . , ck−1 are non-negative integers, such that

P (an ) = an+1 an+2 · · · an+k

for every integer n ⩾ 1.




Language: English Time: 4 hours and 30 minutes.
Each problem is worth 7 points.

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