Found problems: 2529
Real numbers $ x_{1},x_{2},..,x_{n}$ are positive. Prove the inequality:
$ \frac{x_{1}}{x_{2}\plus{}x_{3}}\plus{}\frac{x_{2}}{x_{3}\plus{}x_{4}}\plus{}...\plus{} \frac{x_{n\minus{}1}}{x_{n}\plus{}x_{1}}\plus{}\frac{x_{n}}{x_{1}\plus{}x_{2}}>(\sqrt{2}\minus{}1)n$
Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$
Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression :
$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.
Given the equation
\[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\]
has four real, positive roots, prove that
(a) $ pr \minus{} 16s \geq 0$
(b) $ q^2 \minus{} 36s \geq 0$
with equality in each case holding if and only if the four roots are equal.
Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Find the largest real constant $\lambda$ such that
\[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\]
For all positive real numbers $a,b,c.$
$a,b,c$ are positive numbers.$a+b+c=3$
Prove that:
$\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $
Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.
Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$.
Proposed by Minseok Eli Park (wolfpack)
$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$
Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]
If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$
$$(2-a_n )a_{n+1} =1,$$
prove that $\lim_{n\to \infty} a_n =1.$
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that
\[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \]
[i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$.
Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$.
[i]Proposed by Dirk Laurie, South Africa[/i]
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that:
$ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.
Prove that
\[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \]
for all positive real numbers $a$ and $b.$
Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.)
\[ x_i^2 + x_i - 1 = x_{i+1} \]
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$.
[i]Proposed by Evan Chen[/i]
Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality:
$$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$
$x, y, z$ are positive reals such that $x \leq 1$. Prove that
$$xy+y+2z \geq 4 \sqrt{xyz}$$
Let $x,y$ be positive real numbers such that $x+y=1$. Prove that
$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.
Let $a>b>0$. Prove that $\sqrt2 a^3+ \frac{3}{ab-b^2}\ge 10$
When does equality hold?
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.