Found problems: 2529
Let $a$ and $b$ be two positive reals such that the following inequality
\[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.
[i]Proposed by Fajar Yuliawan[/i]
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:
\[a_1^2 + a_1 - 1 = a_2\]
\[ a_2^2 + a_2 - 1 = a_3\]
\[\hspace*{3.3em} \vdots \]
\[a_{n}^2 + a_n - 1 = a_1\]
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$
\[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\]
Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that
\[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
Prove that if $a,b,c$ are positive numbers with $abc=1$, then
\[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]
Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.
Let $a, b, c$ be non-negative reals which satisfy $a+b+c=1$. Prove that $\frac{\sqrt{a}}{b+1}+\frac{\sqrt{b}}{c+1}+\frac{\sqrt{c}}{a+1}>\frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})$
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$
When does the equality occur?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle.
Prove that
\[AX+AY+BC>AB+AC\]
Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that:
\[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \]
When is there equality?
Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$
Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$
Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \]
[i]Robert Szasz[/i]
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r$
(D.Shvetsov)
(a) Prove that for any positive numbers $x_1,x_2,...,x_k$ ($k > 3$),
$$\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2$$
(b) Prove that for every $k$ this inequality cannot be sharpened, i.e. prove that for every given $k$ it is not possible to change the number $2$ in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers $x_1,x_2,...,x_k$.
(A Prokopiev)
The sides of a triangle have lengths $a,b,c$. Prove that:
\begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?