Found problems: 2529
Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying
$$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$
Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0$, knowing that they are real, positive and that:
\[ \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.\]
Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that\[\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.\]
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$
Show that for all real numbers $ a,b,c,d,$ the following inequality holds:
\[ (a\plus{}b\plus{}c\plus{}d)^2 \leq 3 (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} 6ab\]
(Russia 1195) If x, y > 0, prove that
x/(x^4 + y^2) + y/(y^4 + x^2) <= 1/(xy)
Thoughts?
By the way, this was in Kiran Kedlaya's MOP notes and said to be from Russia 1995, but John Scholes' Kalva archive doesn't have this problem under Russia 1995. Odd.
Hint:
[hide]This was in the Power Mean Inequality section in the lecture notes.[/hide]
Show that for any real number $x$:
\[ x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0 . \]
For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression:
$$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$
[i]Proposed by Serhii Torba[/i]
$ a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4 $ Prove that
$a^3b+b^3c+c^3b \le 3$
Prove that if $a > 0$, $b > 0$, $abc=1$, then
$$a+b+c \ge 3$$
On a circle $\omega$ with center O and radius $r$ three different points $A, B$ and $C$ are chosen. Let $\omega_1$ and $\omega_2$ be the circles that pass through $A$ and are tangent to line $BC$ at points $B$ and $C$, respectively.
(a) Show that the product of the areas of $\omega_1$ and $\omega_2$ is independent of the choice of the points $A, B$ and $C$.
(b) Determine the minimum value that the sum of the areas of $\omega_1$ and $\omega_2$ can take and for what configurations of points $A, B$ and $C$ on $\omega$ this minimum value is reached.
Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$.
\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]
Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
Let $a,b,c$ be positive real numbers with sum $6$. Find the maximum value of
\[S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.\]
The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that
\[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\]
Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]
Given $x_1,x_2,\dots,x_n>0,n\geq 5$, show that
\[\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}\]
For $x,y,z > 0$ and $xyz=1$, prove that
\[\frac{x^{9}+y^{9}}{x^{6}+x^{3}y^{3}+y^{6}}+\frac{x^{9}+z^{9}}{x^{6}+x^{3}z^{3}+z^{6}}+\frac{y^{9}+z^{9}}{y^{6}+y^{3}z^{3}+z^{6}}\geq 2\]
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.
Given is a natural number $n \geq 3$. Solve the system of equations:
$\[
\begin{cases}
\tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\
\tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\
& \dots \\
\tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1) \\
\end{cases}
\]$
Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that
\[ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. \]
[i]Mircea Becheanu[/i]
Initially the numbers $i^3-i$ for $i=2,3 \ldots 2n+1$ are written on a blackboard, where $n\geq 2$ is a positive integer. On one move we can delete three numbers $a, b, c$ and write the number $\frac{abc} {ab+bc+ca}$. Prove that when two numbers remain on the blackboard, their sum will be greater than $16$.
Let a,b, and c be positive reals. Prove:
$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$