MathDB

1

Part of 2014 All-Russian Olympiad

Problems(5)

99 numbers around a circle

Source: AllRussian-2014, Grade 9, day1, P1

5/3/2014
On a circle there are 9999 natural numbers. If a,ba,b are any two neighbouring numbers on the circle, then a−ba-b is equal to 11 or 22 or ab=2 \frac{a}{b}=2 . Prove that there exists a natural number on the circle that is divisible by 33.
S. Berlov
modular arithmeticnumber theory proposednumber theory
Grade 9, great divisor

Source: AllRussian-2014, Grade 9, day2, P1

5/3/2014
Define m(n)m(n) to be the greatest proper natural divisor of n∈Nn\in \mathbb{N}. Find all n∈Nn \in \mathbb{N} such that n+m(n)n+m(n) is a power of 1010.
N. Agakhanov
number theory proposednumber theory
18 consecutive integers with 2 prime divisors

Source: All Russian 2014 Grade 10 Day 1 P1

5/3/2014
Let aa be good if the number of prime divisors of aa is equal to 22. Do there exist 1818 consecutive good natural numbers?
number theory proposednumber theory
a?

Source: All Russian2014 Grade 11 Day 1 P1

4/30/2014
Does there exist positive a∈Ra\in\mathbb{R}, such that ∣cos⁡x∣+∣cos⁡ax∣>sin⁡x+sin⁡ax|\cos x|+|\cos ax| >\sin x +\sin ax for all x∈Rx\in\mathbb{R}?
N. Agakhanov
trigonometryalgebra proposedalgebra
easy

Source: All Russian 2014 Grade 11 Day 2 P1

4/29/2014
Call a natural number nn good if for any natural divisor aa of nn, we have that a+1a+1 is also divisor of n+1n+1. Find all good natural numbers.
S. Berlov
number theoryprime numbersnumber theory proposed