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Let a_1, a_2, \ldots, a_n be positive real numbers. Prove the inequality
\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2

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Let \alpha(n) be the number of digits equal to one in the binary representation of a positive integer n. Prove that:

(a) the inequality \alpha(n) \left(n^2 \right) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1) holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence \left(n_i \right)^{\infty}_1 such that \frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}
goes to zero as i goes to \infty.


Alternative problem: Prove that there exists a sequence a sequence \left(n_i \right)^{\infty}_1 such that \frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}

(d) \infty;
(e) an arbitrary real number \gamma \in (0,1);
(f) an arbitrary real number \gamma \geq 0;

as i goes to \infty.
Let n \geq 2, n \in \mathbb{N} and let p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R} satisfying \frac{1}{2} \leq p \leq 1, 0 \leq a_i, 0 \leq b_i \leq p, i = 1, \ldots, n, and \sum^n_{i=1} a_i = \sum^n_{i=1} b_i. Prove the inequality: \sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.
Suppose that n \geq 2 and x_1, x_2, \ldots, x_n are real numbers between 0 and 1 (inclusive). Prove that for some index i between 1 and n - 1 the
inequality

x_i (1 - x_{i+1}) \geq \frac{1}{4} x_1 (1 - x_{n})
Let w, x, y, z are non-negative reals such that wx + xy + yz + zw = 1. Show that

\frac {w^3}{x + y + z} + \frac {x^3}{w + y + z} + \frac {y^3}{w + x + z} + \frac {z^3}{w + x + y}\geq \frac {1}{3}.
Prove the inequality

a.) \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,

where k\geq 1 is a natural number and a_{1}, a_{2}, ..., a_{k} are arbitrary real numbers.

b.) Using the inequality (1), show that if the real numbers a_{1}, a_{2}, ..., a_{n} satisfy the inequality

a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },

then all of these numbers a_{1}, a_{2}, \ldots, a_{n} are non-negative.
Prove the inequality
\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1
for any x, \alpha with 0 \leq x \leq \frac{\pi }{2} and \frac{\pi}{6} < \alpha < \frac{\pi}{3}.