Inspired by JK1603JK

by sqing, May 1, 2025, 9:44 AM

Let $ a,b,c $ be reals such that $ abc\neq 0$ and $ a+b+c=0. $ Prove that
$$\left|\frac{a-b}{c}\right|+k\left|\frac{b-c}{a} \right|+k^2\left|\frac{c-a}{b} \right|\ge 3(k+1)$$Where $ k>0.$
$$\left|\frac{a-b}{c}\right|+2\left|\frac{b-c}{a} \right|+4\left|\frac{c-a}{b} \right|\ge 9$$
inequalities proposed
algebra
inequalities
L

4-var inequality

by RainbowNeos, May 1, 2025, 9:31 AM

Given $a,b,c,d>0$, show that
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]
inequalities
inequalities unsolved
inequalities proposed
L

Find all integer pairs (m,n) such that 2^n! + 1 | 2^m! + 19

by Goblik, May 1, 2025, 9:08 AM

Find all positive integer pairs $(m,n)$ such that $2^{n!} + 1 | 2^{m!} + 19$
This post has been edited 1 time. Last edited by Goblik, an hour ago
Reason: Typo
number theory
L

Inspired by JK1603JK and arqady

by sqing, May 1, 2025, 8:23 AM

Let $ a,b,c $ be reals such that $ abc\neq 0$ and $ a+b+c=0. $ Prove that
$$\left|\frac{a-2b}{c}\right|+\left|\frac{b-2c}{a} \right|+\left|\frac{c-2a}{b} \right|\ge \frac{1+3\sqrt{13+16\sqrt{2}}}{2}$$$$\left|\frac{a-3b}{c}\right|+\left|\frac{b-3c}{a}\right|+\left|\frac{c-3a}{b}\right|\ge 1+2\sqrt{13+16\sqrt{2}} $$
This post has been edited 1 time. Last edited by sqing, 2 hours ago
inequalities proposed
algebra
L

An easiest problem ever

by Asilbek777, May 1, 2025, 8:17 AM

Simplify
Attachments:
algebra
L

problem interesting

by Cobedangiu, Apr 30, 2025, 5:06 AM

Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
This post has been edited 1 time. Last edited by Cobedangiu, Yesterday at 5:06 AM
algebra
L

Inequality with 3 variables and a special condition

by Nuran2010, Apr 29, 2025, 5:06 PM

For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
inequalities
inequalities proposed
L

Twin Prime Diophantine

by awesomeming327., Mar 7, 2025, 8:28 PM

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
number theory
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Junior Balkan Mathematical Olympiad 2024- P3

by Lukaluce, Jun 27, 2024, 11:06 AM

Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$$2020^x + 2^y = 2024^z.$$
Proposed by Ognjen Tešić, Serbia
This post has been edited 1 time. Last edited by Lukaluce, Jun 28, 2024, 12:36 PM
number theory
Diophantine equation
exponent
JBMO
JBMO 2024
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AD is Euler line of triangle IKL

by VicKmath7, Dec 30, 2021, 5:24 PM

Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.

Proposed by Le Viet An, Vietnam
This post has been edited 4 times. Last edited by VicKmath7, Oct 16, 2023, 5:55 PM
geometry
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Tags
Cubic
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Feuerbach
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isogonal center
Isogonal conjugate
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Parry reflection point
Poncelet point
cevian triangle
circumconic
Darboux cubic
Hagge circle
inconic
Kantor-Hervey point
Kiepert perspector
Lester s theorem
Liang-Zelich Theorem
mixtilinear incircle
Morley Point
Orthocorrespondent
Ptolemy triangle
QA-Orthopole-circle
radical axis
Steiner inellipse
X1138
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