MathDB

P1

Part of The Golden Digits 2024

Problems(4)

A strange inequality

Source: The Golden Digits Contest, May 2024, P1

5/19/2024
Let n⩾2n\geqslant 2 be an integer. Prove that for any positive real numbers a1,a2,…,ana_1, a_2,\ldots, a_n, 122∑i=1n2iai2⩾∑1⩽i<j⩽naiaj.\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.Proposed by Andrei Vila
algebrainequalities
A game of debt

Source: 2024 Nepal TST, P4 and The Golden Digits Contest, April 2024, P1

4/21/2024
Vlad draws 100 rays in the Euclidean plane. David then draws a line ℓ\ell and pays Vlad one pound for each ray that ℓ\ell intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David?
Proposed by Vlad Spătaru
combinatoricsgeometrygame
Very simple NT

Source: The Golden Digits Contest, March 2024, P1

4/8/2024
Let k⩾2k\geqslant 2 be a positive integer and n>1n>1 be a composite integer. Let d1<⋯<dmd_1<\cdots<d_m be all the positive divisors of n.n{}. Is it possible for di+di+1d_i+d_{i+1} to be a perfect kk-th power, for every 1⩽i<m1\leqslant i<m?
Proposed by Pavel Ciurea
number theoryDivisors
A neat FE on positive reals

Source: The Golden Digits Contest, February 2024, P1

4/8/2024
Determine all functions f:R+→R+f:\mathbb{R}_+\to\mathbb{R}_+ which satisfy f(yf(x))+x=f(xy)+f(f(x)),f\left(\frac{y}{f(x)}\right)+x=f(xy)+f(f(x)),for any positive real numbers xx and yy.
Proposed by Pavel Ciurea
algebrafunctional equation