Maths Olympiad Prep

Library / IMO Team Selection Contest /6 of 21

IMO Team Selection Contest

Algebra Difficulty 7.8 National Olympiad, round 2 Prove it Estonia

Let x1,…,xnx_1, \dots, x_n be non-negative real numbers, not all of which are zeros.
(i) Prove that
1≤(x1+x22+x33+⋯+xnn)⋅(x1+2x2+3x3+⋯+nxn)(x1+x2+x3+⋯+xn)2≤(n+1)24n. 1 \le \frac{\left(x_1 + \frac{x_2}{2} + \frac{x_3}{3} + \dots + \frac{x_n}{n}\right) \cdot \left(x_1 + 2x_2 + 3x_3 + \dots + nx_n\right)}{\left(x_1 + x_2 + x_3 + \dots + x_n\right)^2} \le \frac{(n+1)^2}{4n}.
(ii) Show that, for each n≥1n \ge 1, both inequalities can hold as equalities.

Solution

Applying AM-GM gives
(∑k=1nxkk)(∑k=1nkxk)=1n⋅(∑k=1nnxkk)(∑k=1nkxk)≤≤1n⋅14(∑k=1nnxkk+∑k=1nkxk)2==14n(∑k=1nxk(nk+k))2≤(n+1)24n(∑k=1nxk)2. \begin{aligned} \left(\sum_{k=1}^{n} \frac{x_k}{k}\right) \left(\sum_{k=1}^{n} kx_k\right) &= \frac{1}{n} \cdot \left(\sum_{k=1}^{n} \frac{nx_k}{k}\right) \left(\sum_{k=1}^{n} kx_k\right) \le \\ &\le \frac{1}{n} \cdot \frac{1}{4} \left(\sum_{k=1}^{n} \frac{nx_k}{k} + \sum_{k=1}^{n} kx_k\right)^2 = \\ &= \frac{1}{4n} \left(\sum_{k=1}^{n} x_k \left(\frac{n}{k} + k\right)\right)^2 \le \frac{(n+1)^2}{4n} \left(\sum_{k=1}^{n} x_k\right)^2. \end{aligned}
(The last inequality is proved by nk+k≤n+1\frac{n}{k} + k \le n + 1, as it is equivalent to (n−k)(k−1)≥0(n-k)(k-1) \ge 0.)
This gives us the necessary upper bound; this bound is achieved for instance if x1=xn=1x_1 = x_n = 1 and x2=⋯=xn−1=0x_2 = \dots = x_{n-1} = 0.

For the lower bound, estimate the numerator by Cauchy-Schwarz inequality:
(∑k=1nxkk)(∑k=1nkxk)≥(∑k=1nxkk⋅kxk)2=(∑k=1nxk)2; \left(\sum_{k=1}^{n} \frac{x_k}{k}\right) \left(\sum_{k=1}^{n} kx_k\right) \ge \left(\sum_{k=1}^{n} \sqrt{\frac{x_k}{k}} \cdot \sqrt{kx_k}\right)^2 = \left(\sum_{k=1}^{n} x_k\right)^2;
the equality holds here if exactly one of xix_is is non-zero.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.