Found problems: 592
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.
(Edit: Proposed by sir Leonard Giugiuc, Romania)
Prove that for arbitrary positive numbers the following inequality holds
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]
For positive real numbers $a_1, a_2, ..., a_n$ with product 1 prove:
$$\left(\frac{a_1}{a_2}\right)^{n-1}+\left(\frac{a_2}{a_3}\right)^{n-1}+...+\left(\frac{a_{n-1}}{a_n}\right)^{n-1}+\left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^{2}+a_2^{2}+...+a_n^{2}$$
Proposed by Andrei Eckstein
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that
$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
a) Find :
$A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$
b) Prove that for any $(a,b,c) \in A$ next inequality hold :
\begin{align*}
\frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8}
\end{align*}
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that
$$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$
Decide whether there exists a positive real number \( a < 1 \) such that, for any positive real numbers \( x \) and \( y \), the inequality
\[
\frac{2xy^2}{x^2 + y^2} \leq (1 - a)x + ay
\]
holds true.
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.
(Edit: Proposed by sir Leonard Giugiuc, Romania)
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
$$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$
Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.
Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that
$$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]
Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .
Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$.
Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that
$x + y + z \ge \sqrt{3xyz}$.
When does equality hold?
Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of
degree $n$ with integer coefficients. Prove that
$$
\max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}.
$$
Proposed by Géza Kós, Eötvös University, Budapest
Prove that the following inequality holds for all positive real numbers $x,y,z$:
\[\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.\]