MathDB

4

Part of 2013 IFYM, Sozopol

Problems(4)

Problem 4 of Second round

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/22/2020
Let k<<nk<<n denote that k<nk<n and k∣nk\mid n. Let f:{1,2,...,2013}→{1,2,...,M}f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\} be such that, if n≤2013n\leq 2013 and k<<nk<<n, then f(k)<<f(n)f(k)<<f(n). What’s the least possible value of MM?
number theoryfunction
Sum of reciprocals maximum

Source: Festival of Young Mathematicians, Bulgaria 2013

9/17/2013
Let a,b,ca,b,c be real numbers for which a+b+c+d=19a+b+c+d=19 and a2+b2+c2+d2=91a^2+b^2+c^2+d^2=91. Find the maximal value of 1a+1b+1c+1d\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}.
inequalities proposedinequalities
m^6 = n^(n+1)+n-1

Source: Turkey TST 2013 - Day 2 - P1

4/2/2013
Find all pairs of integers (m,n)(m,n) such that m6=nn+1+n−1m^6 = n^{n+1} + n -1.
algebrapolynomialnumber theorygreatest common divisormodular arithmeticnumber theory proposednice problem
Problem 4 of Finals

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

1/25/2020
Let aia_i, i=1,2,...,ni=1,2,...,n be non-negative real numbers and ∑i=1nai=1\sum_{i=1}^na_i =1. Find max⁡S=∑i∣jaiaj\max S=\sum_{i\mid j}a_i a_j .
number theoryalgebraSum