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Complex Numbers Question
franklin2013   2
N Today at 1:59 PM by Xx_BABAI_xX
Hello everyone! This is one of my favorite complex numbers questions. Have fun!

$f(z)=z^{720}-z^{120}$. How many complex numbers $z$ are there such that $|z|=1$ and $f(z)$ is an integer.

Hint
2 replies
franklin2013
Apr 20, 2025
Xx_BABAI_xX
Today at 1:59 PM
J
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Inequalities
sqing   17
N Today at 1:26 PM by sqing
Let $ a,b,c> 0 $ and $ ab+bc+ca\leq 3abc . $ Prove that
$$ a+ b^2+c\leq a^2+ b^3+c^2 $$$$ a+ b^{11}+c\leq a^2+ b^{12}+c^2 $$
17 replies
sqing
Yesterday at 1:54 PM
sqing
Today at 1:26 PM
J
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Geometric inequality
ReticulatedPython   1
N Today at 12:43 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=n$, where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$, where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$
1 reply
ReticulatedPython
Yesterday at 5:12 PM
vanstraelen
Today at 12:43 PM
J
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Binomial Sum
P162008   0
Today at 12:34 PM
Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
0 replies
P162008
Today at 12:34 PM
0 replies
J
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Triple Sum
P162008   0
Today at 12:24 PM
Find the value of

$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{k.2^n + 2m + 1}$
0 replies
P162008
Today at 12:24 PM
0 replies
J
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Binomial Sum
P162008   0
Today at 12:03 PM
The numbers $p$ and $q$ are defined in the following manner:

$p = 99^{98} - \frac{99}{1} 98^{98} + \frac{99.98}{1.2} 97^{98} - \frac{99.98.97}{1.2.3} 96^{98} + .... + 99$

$q = 99^{100} - \frac{99}{1} 98^{100} + \frac{99.98}{1.2} 97^{100} - \frac{99.98.97}{1.2.3} 96^{100} + .... + 99$

If $p + q = k(99!)$ then find the value of $\frac{k}{10}.$
0 replies
P162008
Today at 12:03 PM
0 replies
J
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Polynomial Limit
P162008   0
Today at 11:55 AM
If $P_{n}(x) = \prod_{k=0}^{n} \left(x + \frac{1}{2^k}\right) = \sum_{k=0}^{n} a_{k} x^k$ then find the value of $\lim_{n \to \infty} \frac{a_{n - 2}}{a_{n - 4}}.$
0 replies
P162008
Today at 11:55 AM
0 replies
J
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Telescopic Sum
P162008   0
Today at 11:40 AM
Compute the value of $\Omega = \sum_{r=1}^{\infty} \frac{14 - 9r - 90r^2 - 36r^3}{7^r r(r + 1)(r + 2)(4r^2 - 1)}$
0 replies
P162008
Today at 11:40 AM
0 replies
J
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CHINA TST 2017 P6 DAY1
lingaguliguli   0
Today at 9:03 AM
When i search the china TST 2017 problem 6 day I i crossed out this lemme, but don't know to prove it, anyone have suggestion? tks
Given a fixed number n, and a prime p. Let f(x)=(x+a_1)(x+a_2)...(x+a_n) in which a_1,a_2,...a_n are positive intergers. Show that there exist an interger M so that 0<v_p((f(M))< n + v_p(n!)
0 replies
lingaguliguli
Today at 9:03 AM
0 replies
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Combinatoric
spiderman0   1
N Today at 6:44 AM by MathBot101101
Let $ S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$
1 reply
spiderman0
Yesterday at 7:46 AM
MathBot101101
Today at 6:44 AM
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