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Polynomial Roots
gauss1181   54
N Friday at 8:21 PM by SomeonecoolLovesMaths
Source: 1984 USAMO #1
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.
54 replies
gauss1181
Aug 11, 2009
SomeonecoolLovesMaths
Friday at 8:21 PM
J
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Inequality on a Quartic
patrickhompe   137
N Friday at 6:29 PM by mudkip42
Source: USAMO 2014, Problem 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
137 replies
patrickhompe
Apr 29, 2014
mudkip42
Friday at 6:29 PM
J
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Polynomials.....again
kevinmathz   73
N Friday at 5:09 PM by mudkip42
Source: 2020 AOIME #11
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
73 replies
kevinmathz
Jun 7, 2020
mudkip42
Friday at 5:09 PM
J
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Cubic Roots
worthawholebean   41
N Friday at 4:46 PM by mudkip42
Source: AIME 2008II Problem 7
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3+1001x+2008=0.\]Find $ (r+s)^3+(s+t)^3+(t+r)^3$.
41 replies
worthawholebean
Apr 3, 2008
mudkip42
Friday at 4:46 PM
J
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USAJMO problem 3: Inequality
BOGTRO   108
N Jun 18, 2025 by Kempu33334
Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.
108 replies
BOGTRO
Apr 24, 2012
Kempu33334
Jun 18, 2025
J
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Exponents
jeffq   13
N Jun 17, 2025 by NeoAzure
Source: AHSME 1991 problem 20
The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is

$ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $
13 replies
jeffq
Oct 30, 2011
NeoAzure
Jun 17, 2025
J
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90! and its last two nonzero digits
Smartguy   56
N Jun 16, 2025 by mudkip42
Source: AMC 12A 2010, Problem 23
The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$?

$ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$
56 replies
Smartguy
Feb 10, 2010
mudkip42
Jun 16, 2025
J
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Trig equation
AwesomeToad   23
N Jun 9, 2025 by mahyar_ais
Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$.
Find $24\cot^2{x}$.
23 replies
AwesomeToad
Mar 18, 2011
mahyar_ais
Jun 9, 2025
J
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looks like roots of unity filter!
math31415926535   38
N Jun 8, 2025 by Kempu33334
Source: 2022 AIME II Problem 13
There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$
38 replies
math31415926535
Feb 17, 2022
Kempu33334
Jun 8, 2025
J
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Degree Six Polynomial's Roots
ksun48   46
N Jun 7, 2025 by Yiyj
Source: 2014 AIME I Problem 14
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
46 replies
ksun48
Mar 14, 2014
Yiyj
Jun 7, 2025
J
a