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Inequalities from SXTB
sqing   21
N 2 hours ago by sqing
T2811. Let $ a,b,c,d> 0 $ and $ a+b+c+d=1. $ Prove that$\frac{a}{a+1}+\frac{2b}{b+2}+\frac{3c}{c+3}+\frac{6d}{d+6} \leq \frac{12}{13}$
21 replies
sqing
Dec 5, 2024
sqing
2 hours ago
J
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Inequalities
sqing   1
N 2 hours ago by sqing
Let $ 0\leq a \leq \frac{1}{2}$. Prove that
$$ \sqrt{\frac{2}{3}a+a^3}+\sqrt{\frac{2}{3}a+(1-2a)^3}+\sqrt{\frac{2}{3}(1-2a)+a^3} \geq \sqrt{\frac{7}{3}}$$$$ \sqrt{\frac{3}{4}a+a^3}+\sqrt{\frac{3}{4}a+(1-2a)^3}+\sqrt{\frac{3}{4}(1-2a)+a^3} \geq \frac{1}{2}\sqrt{\frac{31}{3}}$$$$\sqrt{\frac{11}{8}a+a^3}+\sqrt{\frac{11}{8}a+(1-2a)^3}+\sqrt{\frac{11}{8}(1-2a)+a^3} \geq \frac{\sqrt{2}+ \sqrt{11}+\sqrt{13}}{4}$$$$ \sqrt{\frac{31}{20}a+a^3}+\sqrt{\frac{31}{20}a+(1-2a)^3}+\sqrt{\frac{31}{20}(1-2a)+a^3} \geq \frac{5+6\sqrt{5}+\sqrt{155}}{10\sqrt{2}}$$
1 reply
sqing
Yesterday at 2:45 PM
sqing
2 hours ago
J
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Simultaneous System of Equations
djmathman   6
N 3 hours ago by Math-lover1
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*} abcd &= 2007,\\ a &= \sqrt{55 + \sqrt{k+a}},\\ b &= \sqrt{55 - \sqrt{k+b}},\\ c &= \sqrt{55 + \sqrt{k-c}},\\ d &= \sqrt{55 - \sqrt{k-d}}. \end{align*}
6 replies
djmathman
Sep 17, 2018
Math-lover1
3 hours ago
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[Sipnayan 2021 SHS SF-E2] Sum of 256th powers
aops-g5-gethsemanea2   3
N 3 hours ago by Math-lover1
If $r_1,r_2,\dots,r_{256}$ are the 256 roots (not necessarily distinct) of the equation $x^{256}-2021x^2-3=0$, evaluate $\sum^{256}_{i=1}r_i^{256}$.
3 replies
aops-g5-gethsemanea2
Monday at 1:54 PM
Math-lover1
3 hours ago
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14th PMO Qualifying Stage #10
yes45   2
N 3 hours ago by Math-lover1
If $\frac{\log x}{\log y} = 500$, what is the value of $\frac{\log {\frac{y}{x}}}{\log y}$?

Answer

Solution
2 replies
yes45
Jun 2, 2025
Math-lover1
3 hours ago
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Daily Problem Writing Practice
KSH31415   13
N 5 hours ago by Vivaandax
I'm trying to get better at writing problems so I decided to challenge myself to write one problem for every day this month (June 2025). I will post them in this thread as well as edit this post with all of them in hide tags. If I can, I'll include a difficulty level in the form of AIME placement. If anybody wants to solve them and give feedback on the problem and/or my difficulty rating, please do!

June 1 (AIME P4) - Solved
June 2 (AIME P5) - Solved
June 3 (AIME P12) - Solved

June 1 (AIME P4)
A bag contains $6$ red balls and $6$ blue balls. A draw consists of randomly selecting $2$ of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
13 replies
KSH31415
Monday at 3:13 PM
Vivaandax
5 hours ago
J
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[PMO17 Qualifying II.4] A Consecutive Telescoping Product Sum
Shinfu   2
N 5 hours ago by martianrunner
Simplify

$$\dfrac{1+2+3}{1+2+3+4}\times\dfrac{1+2+3+4+5}{1+2+3+4+5+6}\times\dots\times\dfrac{1+2+3+\dots+19}{1+2+3+\dots+20}$$
$\text{(a) }\frac{1}{5}\qquad\text{(b) }\frac{3}{20}\qquad\text{(c) }\dfrac{1}{7}\qquad\text{(d) }\frac{1}{35}$

Answer Confirmation
2 replies
Shinfu
Yesterday at 8:03 PM
martianrunner
5 hours ago
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13th Philippine Mathematical Olympiad Area Stage #8
qrxz17   1
N Yesterday at 7:57 PM by Mathelets
Problem: Find all complex numbers \(x\) satisfying \(x^3+x^2+x+1=0\).

Answer: \(-1, i, -i\)

Solution: Notice that \(x^3+x^2+x+1\) is a geometric series that can be expressed as
\begin{align*} \frac{x^4-1}{x-1}. \end{align*}
We have

\begin{align*} \frac{x^4-1}{x-1} &= 0 \\ x^4-1&=0 \\ x^4 &= 1. \end{align*}
The solutions to this are the fourth roots of unity. So,
\begin{align*} x^4 = 1 &= \cos (2\pi)+i\sin(2\pi) \\ &= (\cos \theta+i \sin\theta)^4 \\ &= \cos 4\theta+i \sin4\theta. \end{align*}
We then have \(\theta = \frac{2\pi k}{4}\) for \(k=0,1,2,3\). The complex numbers \(x\) that satisfy the equation are \(\boxed{-1, i, -i}\), with \(1\) excluded as a solution. This is because the equation \(\frac{x^4 - 1}{x - 1} = 0,\) is valid only when \(x \ne 1\).
1 reply
qrxz17
Yesterday at 4:15 PM
Mathelets
Yesterday at 7:57 PM
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a problem about cups
pzzd   3
N Yesterday at 7:33 PM by vincentwant
Hello! I’m new to the AoPS forums, so apologies in advance if I’ve posted in the wrong place :-)

Here’s an interesting counting problem I’ve come up with that’s been bothering me for a while:

Say we have a tower of cups such that the top row has one cup, the row directly under that has two cups, the row under that has 3 cups, and so on such that every cup is supported by exactly two cups in the row under it. Each cup in the tower is labelled with a number or letter so that every cup is distinct. If a cup tower has $n$ rows, how many different ways can we pick up all the cups, starting at the top? (Assume that you can’t pick up a cup that’s under another cup, the tower would fall over.)
3 replies
pzzd
Yesterday at 7:11 PM
vincentwant
Yesterday at 7:33 PM
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Min max của pt bậc 2
chunchun.math.2010   2
N Yesterday at 5:46 PM by HS12309
Cho x,y là hai số thực thoa mãn x^2+ y^2+6x-8y+21 =0.Tìm min,max P=x^2+y^2
2 replies
chunchun.math.2010
Yesterday at 3:41 PM
HS12309
Yesterday at 5:46 PM
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a