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RMO Mock Math Problems 2024

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B V Karthikeya
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87 views1 page

RMO Mock Math Problems 2024

Uploaded by

B V Karthikeya
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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RMO Mock

Online Math Club


October 24, 2024

1. Let a, b, c be integers with a3 + b3 + c3 divisible by 18. Prove that abc is divisible by 6.

2. BL is the bisector of an isosceles triangle ABC. A point D is chosen on the Base BC


and a point E is chosen on the lateral side AB so that AE = 21 AL = CD. Prove that
LE = LD.

3. Determine whether there exists positive integers a1 < a2 < · · · < ak such that all sums
ai + aj , where 1 ≤ i < j ≤ k, are unique, and among those sums, there are 1000 consec-
utive integers.

4. Determine all positive integers n such that all positive integers less than or equal to
n and relatively prime to n are pairwise coprime.

5. The infinite sequence P0 (x), P1 (x), P2 (x), . . . , Pn (x), . . . is defined as

P0 (x) = x, Pn (x) = Pn−1 (x − 1) · Pn−1 (x + 1), n ≥ 1.

Find the largest k such that P2014 (x) is divisible by xk .

6. 100 people from 50 countries, two from each countries, stay on a circle. Prove that
one may partition them onto 2 groups in such way that neither no two countrymen,
nor three consecutive people on a circle, are in the same group.

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