Hard inequality

by JK1603JK, May 1, 2025, 3:55 AM

Let $a,b,c\in R: abc\neq 0$ and $a+b+c=0$ then prove $$|\frac{a-b}{c}|+|\frac{b-c}{a}|+|\frac{c-a}{b}|\ge 6$$
inequalities
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BMO 2024 SL A5

by MuradSafarli, Apr 27, 2025, 12:44 PM

Let \(\mathbb{R}^+ = (0, \infty)\) be the set of positive real numbers.
Find all non-negative real numbers \(c \geq 0\) such that there exists a function \(f : \mathbb{R}^+ \to \mathbb{R}^+\) with the property:
\[ f(y^2f(x) + y + c) = xf(x+y^2) \]for all \(x, y \in \mathbb{R}^+\).
algebra
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BMO 2024 SL A3

by MuradSafarli, Apr 27, 2025, 12:42 PM

A3.
Find all triples \((a, b, c)\) of positive real numbers that satisfy the system:
\[ \begin{aligned} 11bc - 36b - 15c &= abc \\ 12ca - 10c - 28a &= abc \\ 13ab - 21a - 6b &= abc. \end{aligned} \]
algebra
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Arbitrary point on BC and its relation with orthocenter

by falantrng, Apr 27, 2025, 11:47 AM

In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
This post has been edited 1 time. Last edited by falantrng, Apr 27, 2025, 4:38 PM
geometry
BMO
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hard problem

by Cobedangiu, Apr 21, 2025, 1:51 PM

Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
inequalities
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6 variable inequality

by ChuongTk17, Nov 29, 2024, 7:41 AM

Given real numbers a,b,c,d,e,f in the interval [-1;1] and positive x,y,z,t such that $$2xya+2xzb+2xtc+2yzd+2yte+2ztf=x^2+y^2+z^2+t^2$$. Prove that: $$a+b+c+d+e+f \leq 2$$
This post has been edited 3 times. Last edited by ChuongTk17, Nov 29, 2024, 10:55 AM
Reason: Changed title
inequalities
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this is where procrastination this week has led me

by linny.t, Feb 15, 2024, 1:58 AM

i speedran 2 projects in 4 hrs that i woke up at 4 am to finish on monday
im planning on doing the same for my ss notes
i dont have the energy to do them rn
and
i have to read 117 pages in a book that i cant comprehend in 2 days
i love procrastination
im starting to get in the habit of posting a little more often so hopefully i can keep this blog alive
i have to "go to sleep now"
gn guys!!!
procrastination
0 work completed

Something nice

by KhuongTrang, Nov 1, 2023, 12:56 PM

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM
inequalities
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Yet another domino problem

by juckter, Apr 9, 2019, 11:12 AM

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
combinatorics
EGMO 2019
Hi
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Bounding is hard

by whatshisbucket, Jun 28, 2018, 7:13 AM

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$holds for all integers $n>N.$

Proposed by Carl Schildkraut
This post has been edited 1 time. Last edited by whatshisbucket, Jun 29, 2018, 1:06 AM
algebra
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One on reals

by Rushil, Oct 10, 2005, 5:03 AM

If $a,b,c$ are positive real numbers such that $abc= 1$, Prove that \[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]
inequalities
logarithms
ineqstd
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