Found problems: 2529
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?
Let $a, b$, and $c$ be positive integers satisfying
$a^4 + a^2b^2 + b^4 = 9633$
$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$.
What is the sum of all distinct values of $a + b + c$?
Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality:
$\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$
In $\triangle ABC, BC=a, CA=b, AB=c,$ and $\Gamma$ is its circumcircle.
$(1)$ Determine a necessary and sufficient condition on $a,b$ and $c$ if there exists a unique point $P(P\neq B, P\neq C)$ on the arc $BC$ of $\Gamma$ not passing through point $A$ such that $PA=PB+PC$.
$(2)$ Let $P$ be the unique point stated in $(1)$. If $AP$ bisects $BC$, prove that $\angle BAC<60^{\circ}$.
In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.
Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively.
[i]Aurel Bârsan[/i]
Show that for positive reals $a, b, c, d$ we have
\[ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 \]
Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$.Then prove that \[a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}\]
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
\[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\]
When does equality hold?
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
Let $ a$, $ b$, $ c$ be positive real numbers such that $ a \plus{} b \plus{} c \equal{} 1$. Prove inequality:
\[ \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.\]
Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that
\[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]
Let $-1 \le x, y \le 1$. Prove the inequality $$2\sqrt{(1- x^2)(1 - y^2) } \le 2(1 - x)(1 - y) + 1 $$
Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$ Find the maximum value of the expression $$x^3+2y$$
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$
Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.
Let $n $ be a positive integer. Determine whether there exists a positive real number $\epsilon >0$ (depending on $n$) such that for all positive real numbers $x_1,x_2,\dots ,x_n$, the inequality $$\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},$$ holds.
Suppose that $m$ and $n$ are positive integers such that $m<n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m,n)$ satisfy these conditions?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }2007$
Find all real solutions to the following system of equations. Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]
Solve in the set of real numbers:
\[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1,
\]
\[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3.
\]
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$