Found problems: 2529
Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that
\[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
Find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(x+f\left(y\right)\right)f\left(y+f\left(x\right)\right)=\left(2x+f\left(y-x\right)\right)\left(2y+f\left(x-y\right)\right) $$ holds for all integers $ x,y $
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$.
In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.
If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;
If $ x,y$ are positive real numbers with sum $ 2a$, prove that :
$ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$
When does equality hold ?
Babis
Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that
$ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.
let x,y are positive and $ \in R$ that : $ x\plus{}2y\equal{}1$.prove that :
\[ \frac{1}{x}\plus{}\frac{2}{y} \geq \frac{25}{1\plus{}48xy^2}\]
If $a$, $b$, and $c$ are positive numbers, determine the least possible value of the following expression:
$\frac{1}{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}-\frac{2}{\frac{a}{c}+\frac{c}{b}+\frac{b}{a}}$.
Which of these has the smallest maxima on positive real numbers?
$\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$
Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that
\[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\]
for all $n \ge 1$.
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$
When the equality holds?
Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $.
Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]
For all positive real numbers $a,b,c,d$ prove the inequality
\[\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4} \ge 2\sqrt{2}(ad+bc)\]
On the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ find the point $T=(x_0,y_0)$ such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.
[b]i)[/b]If $X,Y,Z$ be the angles of a triangle then show that
\[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}+\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}+\tan {\frac{Z}{2}}\tan {\frac{X}{2}}=1\]
[b]ii)[/b] Prove using [b](i)[/b] or otherwise that
\[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}\leq\frac {1}{3\sqrt{3}}\]
Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.
if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \]
A. Khrabrov
Consider the $n \times n$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table.
[asy]
import graph;
size(3.5cm);
for (int x=0; x<=5; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (5, y), linewidth(.5pt));
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(0, 4, "1");
foo(1, 4, "2");
foo(2, 4, "3");
foo(3, 4, "$\dots$");
foo(4, 4, "$n$");
foo(0, 3, "2");
foo(1, 3, "4");
foo(2, 3, "6");
foo(3, 3, "$\dots$");
foo(4, 3, "$2n$");
foo(0, 2, "3");
foo(1, 2, "6");
foo(2, 2, "9");
foo(3, 2, "$\dots$");
foo(4, 2, "$3n$");
foo(0, 1, "$\vdots$");
foo(1, 1, "$\vdots$");
foo(2, 1, "$\vdots$");
foo(3, 1, "$\ddots$");
foo(4, 1, "$\vdots$");
foo(0, 0, "$n$");
foo(1, 0, "$2n$");
foo(2, 0, "$3n$");
foo(3, 0, "$\dots$");
foo(4, 0, "$n^2$");
[/asy]
We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, in the case $n = 5$, here is one such possible path, with all the numbers along the path circled:
[asy]
import graph;
size(3.5cm);
for (int x=0; x<=5; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (5, y), linewidth(.5pt));
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
draw(Circle((0.5,4.5),0.5));
draw(Circle((1.5,4.5),0.5));
draw(Circle((2.5,4.5),0.5));
draw(Circle((2.5,3.5),0.5));
draw(Circle((3.5,3.5),0.5));
draw(Circle((3.5,2.5),0.5));
draw(Circle((3.5,1.5),0.5));
draw(Circle((3.5,0.5),0.5));
draw(Circle((4.5,0.5),0.5));
foo(0, 4, "1");
foo(1, 4, "2");
foo(2, 4, "3");
foo(3, 4, "4");
foo(4, 4, "5");
foo(0, 3, "2");
foo(1, 3, "4");
foo(2, 3, "6");
foo(3, 3, "8");
foo(4, 3, "10");
foo(0, 2, "3");
foo(1, 2, "6");
foo(2, 2, "9");
foo(3, 2, "12");
foo(4, 2, "15");
foo(0, 1, "4");
foo(1, 1, "8");
foo(2, 1, "12");
foo(3, 1, "16");
foo(4, 1, "20");
foo(0, 0, "5");
foo(1, 0, "10");
foo(2, 0, "15");
foo(3, 0, "20");
foo(4, 0, "25");
[/asy]
If we add up the circled numbers in the example above (including the start and end squares), we get $93$. Considering all such possible paths on the $n \times n$ grid:
(a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Express your answer in terms of $n$, and prove that it is correct.
(b) What is the largest sum we can possibly get when we add up the numbers along such a path? Express your answer in terms of $n$, and prove that it is correct.
In the Cartesian plane, let $S_{i,j} = \{(x,y)\mid i \le x \le j\}$. For $i=0,1,\ldots,2012$, color $S_{i,i+1}$ pink if $i$ is even and gray if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon $P$ [i]pinxtreme[/i] if it lies completely in the region $S_{0,2013}$ and has at least one vertex on each of the lines $x=0$ and $x=2013$. Given that the minimum value of $d(P)$ over all non-degenerate convex pinxtreme polygons $P$ in the plane can be expressed in the form $\frac{(1+\sqrt{p})^2}{q^2}$ for positive integers $p,q$, find $p+q$.
[i]Victor Wang.[/i]
Let $0\le a\le b\le c$ be real numbers. Prove that
\[(a+3b)(b+4c)(c+2a)\ge 60abc \]
Determine all triples $(a,b,c)$ of real numbers such that
$$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$
Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that
$\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$
Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove:
$$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$