$$100a + 10b + c = a + b^2 + c^3$$  I've tried restricting the range of the values but I couldn't get  a unique solution. I am looking for a formula if there is one. I wonder which is the best way to solve this problem.
122014
$x^3 + y^2 +z=\overline{zyx}$, where $\overline{zyx}$ denotes the sequence of the digits.  $x^3 +y^2 +z =100z+10y+x$,where $x,y,z&gt;0$  The maximum value of $x$, $y$, $z$ individually can only be $9$.  $\text{Maximum value}= 9^3 + 9^2 + 9 = 819$  So, $100z+10y+z&lt;819$  Then I'm lost here a bit. What do I do next?
77761
Find the greatest value of $a^2\cdot b^3$ where $a,b$ are positive real numbers satisfying $a+b=10$. Determine the values of $a,b$ for which the greatest value is attained.  At first I tried to find the value by putting $a=10-b$, I found a expression, but I got not way to do anything.   If we apply A.M-G.M. inequality, I think we found the minimum value or something else. It is a pre-Olympiad level problem. So I need some answers or some good hints.
66149
Let $t$ be a real number such that $t^2=at+b$ for some positive integers $a$ and $b$. Then for any choice of positive integers $a$ and $b$, $t^3$ is never equal to: (A). $4t+3$ (B). $8t+5$ (C). $10t+3$ (D). $6t+5$  We have $t^3 = at^2+bt$ and so for (A) we have $t^3 = 4t+3 = 4(at^2+bt)+3$. I don't see how to continue since this is a cubic equation. Is there an easier way of finding a contradiction?
92348
$a, b, c$ are integers and $a,b,c\geq 4$ If $4abc=(a+3)(b+3)(c+3)$ and $a, b, c$ are all not equal, then what is the value of $a+b+c$.   I tried solving by taking $a\neq b=c$ and couldn't find any integral value which could satisfy the condition.   How to proceed in the problem??
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.
29565
Find the values of $a,b,c,d\in\mathbb{N}$ such that      $$ 34!=295232799cd9604140847618609643ab0000000 $$   My Attempt:  The factorial of $34$ contains a $3$, so the RHS must be divisible by  $3$. Similarly, it must be divisible by $7$, $11$, $13$, $19$ etc.  But I do not understand how can I calculate $a,b,c,d$ in this equation.
122205
I think I have proved that that a cannot be greater than b (ie a &amp; b must be equal) then using this fact I got a quadratic in terms of a and c, trying out some values I got that (3,3,4) is a solution, but how do I prove this is the only solution or if this is not the only solution then what are the other solutions and how do I find them?
39384
Let $a, b$ and $c$ be positive real numbers. Prove that   $$a^2 b + b^2 c + c^2 a + ab^2 + bc^2 + ca^2 \ge 6abc$$   I have tried factoring with no success for the better part of an hour or so.  How can I go about this?
5974
Let $a,b,$ and $c$ be nonnegative real numbers such that $a^2+b^2+c^2+abc = 4$. Prove that $$0 \leq ab + ac + bc - abc \leq 2.$$   I tried using rearrangement to get $a^2+b^2+c^2+abc = 4 \geq ab+bc+ac+abc$. Then I just need to show that $0\leq ab + ac + bc - abc$ and $4-2abc \leq 2$. I am not sure if this method will work, though.
56464
$ |A |= |B | = |C | = 1 $ ,where A B and C are complex nos  $$ \frac{A^2}{BC}+ \  \frac{B^2}{ \ {CA}} \ +\  \frac{C^2}{ \ {AB}} + 1 = 0$$  Find the possible values of $ |A + B + C |$  Tried substituting cos(theta) + i sin(theta)
9060
I'm having some difficulties continuing this problem. I get that $b^2\mid a$ and $c\mid a$ but I am not sure where to go from there.
2457
Let $a, b, c\geq 0$ s.t.$$(a^2-a+1)(b^2-b+1)(c^2-c+1)=1$$  Show that $ a^2b^2+a^2c^2+b^2c^2\leq 3$.  My idea:  I denote $a+b+c=x$, $ab+bc+ac=y$ and $abc=z$.  Then I have $$x^2+y^2+z^2-xy-xz-yz+2z-y-x=0$$ I have to show that $$y^2-2xz\leq 3$$ I tried to prove it with the sign of trinom, but it doesn't work.
43018
Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.  Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$    Here's my idea:  $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - 2(ab + bc + ca) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) - ((a+b+c)^2 - (a^2 + b^2 + c^2) \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) - (a+b+c)^2 \ge 0$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  And I'm stuck here.  I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge (a+b+c)^2$ or  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a+b+c)$, because $a+b+c = 3$  In the first case using Cauchy-Schwarz Inequality I prove that:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $3(a^2 + b^2 + c^2) \ge (a+b+c)^2$  Now I need to prove that:   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3(a^2 + b^2 + c^2)$   $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a^2 + b^2 + c^2)$   $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge a^2 + b^2 + c^2$   I need I don't know how to continue.  In the second case I tried proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$ and  $a^2 + b^2 + c^2 \ge a+b+c$  Using Cauchy-Schwarz Inequality I proved:  $(a^2 + b^2 + c^2)(1+1+1) \ge (a+b+c)^2$  $(a^2 + b^2 + c^2)(a+b+c) \ge (a+b+c)^2$  $a^2 + b^2 + c^2 \ge a+b+c$  But I can't find a way to prove that $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(a+b+c)$  So please help me with this problem.  P.S   My initial idea, which is proving:  $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) + (a^2 + b^2 + c^2) \ge 3^2 = 9$  maybe isn't the right way to prove this inequality.
70909
I'm stuck at this algebra problem, it seems to me that's what's provided doesn't even at all.  Provided:  $$a+b+c=0$$  Find the value of:  $$\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$$  Like I'm not sure where to start, and the provided clue doesn't even make sense.  There's no way I can think of to factor this big polynomial into a form like $a\times p+b\times q+c\times r=s$ where $p,q,r,s\in\mathbb{Z}$.  Thanks!
13169
If values $a$, $b$, and $c$ are known, is there an efficient way to find $x$ in the equation: $a^{x} \bmod b = c$?  E.g. finding $x$ in $128^{x}\bmod 209 = 39$.
115575
If $a$, $b$, $c$, $d$ belong to $\mathbb{R}$, and   $$(a+b)(c+d)=143 \qquad (a+c)(b+d)=150 \qquad (a+d)(b+c)=169$$   Find the smallest possible value of    $$a^2 + b^2 + c^2+ d^2$$   I thought of adding $7$ to the first equation and make it equal the second, solving, and finding a new equation.  Do it $3$ times, keep substituting, but this is a very very very long approach if it's even an approach.
105383
$a,b,c,d$ are complex numbers satisfying \begin{cases}   a+b+c+d=3 \\   a^2+ b^2+ c^2+ d^2=5 \\   a^3+ b^3+ c^3+ d^3=3 \\   a^4+ b^4+ c^4+ d^4=9 \end{cases}  Find the value of the following: $$a^{2015} + b^{2015} + c^{2015} + d^{2015}$$.