Find the maximum and minimum values of the expression: $$\frac x{x^2+1} + \frac y{y^2+1} + \frac z{z^2+1}$$where $x,y,z$ are real numbers satisfying the condition $x+y+z=1$. And what is the minimum? I thought the maximum value occurs when x, y and z are real positive, since if a solution is negative, there would be lesser than a positive symmetric. So, it's correct to use AM &gt;= GM dividing by x, y and z all fractions?
122351
Find the maximum value of the expression :   $${\frac {x}{1+x^2}} + {\frac {y}{1+y^2}}+{\frac {z}{1+z^2}}$$   where $x,y,z$ are real numbers satisfying the condition that $x+y+z=1$.   Taking $x=y=z=\frac {1}{3}$, I get the expression as $\frac {3x}{1+x^2}$, which is equal to $\frac {1}{1+{\frac{1}{9}}}$ or $\frac {9}{10}$.  How can I actually solve the problem without making unnecessary assumptions ?
48990
What is the result of  $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion.  Thanks a lot.
122286
Let $x$ and $y$ be real numbers such that $x^2 + 9 y^2-4 x+6 y+4=0$. Find the maximum value of $\displaystyle \frac{4x-9y}{2}$.  My solution: the given function represents an ellipse. Rewriting it, we get $\displaystyle (x-2)^2 + 9(y+\frac{1}{3})^2=1$. To find the maximum of $\displaystyle \frac{4x-9y}{2}$, $x$ should be at its maximum and $y$ at its minimum. Solving the equations, I get that $x = 3$ and $\displaystyle y = -\frac{2}{3}$, but the answer i get is wrong. What am I doing wrong?
40253
My question is this:      $y^2 = \frac{x^5 - 1}{x-1}$ &amp; $x,y \in \mathbb{Z}$     Source: BdMO 2016 Regionals Set 1 Question 3.   I think this is the exact same question. Actually I didn't get the solution in the link. Please don't mark as duplicate.  I tried solving this same problem by writing code; there seems to be $6$ solution pairs to this equation. How do I get the $6$ solutions mathematically? I indeed need a full solution. Please help.
8075
How to find the maximum and minimum value of $\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right|$ (where $|z_1|=|z_2|=|z_3|=1$ are complex numbers.) ?  My try:  $$\begin{align}\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right| &amp;\leq |z_1-z_2|^2 + |z_2-z_3|^2 + |z_3-z_1|^2 \\ &amp;\leq (|z_1|+|z_2|)^2 + (|z_2|+|z_3|)^2 + (|z_3|+|z_1|)^2 \\ &amp;\leq 2^2+2^2+2^2 \leq 12\end{align}$$  However the answer given is $8$. Where am I going wrong and how to do it correctly?
913
Suppose $a_1, a_2,\dots , a_n$ are $n$ positive real numbers with $a_1a_2 \dots a_n = 1$.   Then what is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$ ?    I think $(1 + a_1)(1 + a_2). . .(1 + a_n)$ takes its minimum value when $a_1=a_2=\dots=a_n=1$ and thus the minimum value is $2^n$.  I don't know how to prove it. Please help.
78966
What is the global minimum of the expression   \begin{align} |x-1| &amp;+ |x-2|+|x-5|+|x-6|+|x-8|+|x-9|+|x- 10| \\&amp;+  |x-11|+|x-12|+|x-17|+|x-24|+|x-31|+ |x-32|? \end{align}   I've solved questions of this sort before but there were only 3 terms. I solved those by expanding all the terms in the modulus and drawing a graph. This question came in a paper which requires the student to solve it within 5 minutes. What's a better method?
35089
Is $a,b,c$ are three different positive integers such that :  $$ ab+bc+ca\geq 107 $$  Then what is the minimum value of $a^3+b^3+c^3-3abc$.  I expanded this expression to $(a+b+c)((a+b+c)^2-3(ab+bc+ca))$ and tried to find the minimum value of $(a+b+c)$ by AM-GM inequalities but the problem is that the minimum value occurs at scenario where $a=b=c$, which is ommitted by assumptions of this question. I'd appreciate some help.
43191
Let $x,y,z\in\Bbb R$, and at least one is be a postive integer, and such that  $$x+\frac{1}{y},\;y+\frac{1}{z},\;z+\frac{1}{x}\in\Bbb Z$$ Find the value of $x,y,z$.  My try:  $$\left(x+\dfrac{1}{y}\right)\left(y+\dfrac{1}{z}\right)\left(z+\dfrac{1}{x}\right)=\dfrac{(xy+1)(yz+1)(xz+1)}{xyz}\in Z$$  Then I can't. Thank you.  I konw $(x,y,z)=(1,1,1),(-1,-1,1),(2,0.5,1)$ is such condition,But How prove it?
84151
Q.How do we find out the sum of absolute maximum and absolute minimum values of $f(x,y)=(x+y)^2-(x+y)+1$ on a unit square $\{(x,y):0&lt;x&lt;1,0&lt;y&lt;1\}$?  Using the method of $rt-s^2$ its value is zero. So now how do I search for the Max and main value?
118396
How do I find the maximum value for $x\cos {λ}+y\sin {λ}$ for fixed $x,y$ and varying $λ$ ? I would appreciate if only hint is given.
46266
Let $f\in C[a,b]$. Assume that $\min_{[a,b]}f=m&gt;0$ and $M=\max_{[a,b]}f$. Which one is true?  a. $$\frac{1}{M}\int_a^bf(x)dx+m\int_a^b\frac{1}{f(x)}dx\geq 2\sqrt{\frac{m}{M}}(b-a)$$  b. $$\int_a^bf(x)dx\int\frac{1}{f(x)}dx\geq (b-a)^2$$  c. $$\int f(x)dx\int_a^b\frac{1}{f(x)}dx\leq (b-a)^2$$  For the first one  $$\frac{1}{M}\int _a^bf(x)+m\int _a^b\frac{1}{f(x)}dx\geq {\frac{m}{M}}(b-a)+{\frac{m}{M}}(b-a)$$  and the last two I am not getting. How to find these?
26906
let $x,y,z&gt;0$ such that $x^2+y^2+z^2=1$. Find the minimum of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$  Is the answer $3\sqrt{3}$ by any chance?
112365
Let $a,b,c&gt;0$ be real numbers such that $a+b+c=3$, find the maximum and minimum of:   $$\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}$$   I've tried using AM-GM-HM, but I can't find the maximum or the minimum.  Using intuition, substituting $a=b=c=1$ should give us one of the optimums, but I cannot prove that it is the optimum.
46751
Then the expression will be  $$|z|+\frac{2}{|z|}\ge 2$$  Solving the quadratic inequality   $$|z|\ge1+i$$ and $$|z|\le1-i$$  I don’t know how to solve further. Please help  Thanks!   this is the corrected version of an identical question asked by me. I am sorry for the inconvenience caused
76815
Let $a,b,c$ be positive real numbers satisfying $$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16,$$ then $a+b+c=3$.  I thought about the application of the AM-GM-HM inequality, but in vain. I also thought about splitting 16 into factors and comparing but went nowhere. Any ideas. Thanks beforehand.
3358
Assume that $$ 1a_1+2a_2+\cdots+na_n=1, $$ where the $a_j$ are real numbers. As a function of $n$, what is the minimum value of $$1a_1^2+2a_2^2+\cdots+na_n^2?$$