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50 Questions

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21	Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$
13	Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$
12	Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube
10	Find minimum of $\frac x {x^2+1} + \frac y{y^2+1} + \frac z{z^2+1}$
7	Solve $x^{2k}+(x-2)^{2k}=2k$ with $k\in \mathbb N$ and $x\in\mathbb{R}^+$

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1,066 Reputation

+10	How many even number in a sequence are there?
+65	Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$
+5	Diophantine equations - Perfect square and Perfect cube related
+10	Prove $x^2+y^2+z^2 \ge 14$ with constraints

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5 Answers

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2	Prove $x^2+y^2+z^2 \ge 14$ with constraints
1	Assume $n$ is even. Prove that $323$ divides $20^n+16^n-3^n-1$.
0	Two Problem: find max, min; number theory: find x, y
0	When is $1^5 + 2^5 + \ldots + n^5$ a square?
0	Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?

18 Tags

2 inequality × 31	0 diophantine-equations × 10
2 homework × 10	0 geometry × 4
2 optimization × 4	0 polynomials × 2
1 elementary-number-theory × 9	0 elliptic-curves × 2
0 number-theory × 18	0 algebra-precalculus × 2

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