Interesting inequality

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0 replies

jlacosta

Apr 2, 2025

0 replies

Arbitrary point on BC and its relation with orthocenter

falantrng 16

N 41 minutes ago by MathLuis

Source: Balkan MO 2025 P2

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

16 replies

falantrng

Yesterday at 11:47 AM

MathLuis

41 minutes ago

all functions satisfying f(x+yf(x))+y = xy + f(x+y)

falantrng 27

N an hour ago by MathLuis

Source: Balkan MO 2025 P3

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$f(x+yf(x))+y = xy + f(x+y).$
Proposed by Giannis Galamatis, Greece

27 replies

falantrng

Yesterday at 11:52 AM

MathLuis

an hour ago

Projections on collections of lines

Assassino9931 1

N an hour ago by awesomeming327.

Source: Balkan MO Shortlist 2024 C6

Let $\mathcal{D}$ be the set of all lines in the plane and $A$ be a set of $17$ points in the plane. For a line $d\in \mathcal{D}$ let $n_d(A)$ be the number of distinct points among the orthogonal projections of the points from $A$ on $d$ . Find the maximum possible number of distinct values of $n_d(A)$ (this quantity is computed for any line $d$ ) as $A$ varies.

1 reply

Assassino9931

3 hours ago

awesomeming327.

an hour ago

weird Condition

B1t 4

N an hour ago by MathLuis

Source: Mongolian TST 2025 P4

In triangle $ABC$ , where $AC < AB$ , the internal angle bisectors of angles $\angle A$ , $\angle B$ , and $\angle C$ meet the sides $BC$ , $AC$ , and $AB$ at points $D$ , $E$ , and $F$ , respectively. Let $I$ be the incenter of triangle $AEF$ , and let $G$ be the foot of the perpendicular from $I$ to line $BC$ . Prove that if the quadrilateral $DGEF$ is cyclic, then the center of its circumcircle lies on segment $AD$ .

4 replies

1 viewing

B1t

Yesterday at 1:37 PM

MathLuis

an hour ago

No more topics!

Interesting inequality

A_E_R 2

N Apr 13, 2025 by A_E_R

Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)

2 replies

A_E_R

Apr 13, 2025

A_E_R

Apr 13, 2025

Interesting inequality

G H J

Reveal topic tags

algebra

Algebra inequality

inequalities

G H BBookmark kLocked kLocked NReply

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A_E_R

5 posts

A_E_R

#1 h Apr 13, 2025, 9:41 AM • 1 Y

Y by cubres

Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)

This post has been edited 2 times. Last edited by A_E_R, Apr 13, 2025, 11:51 AM
Reason: size

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sqing

41864 posts

sqing

#2 h Apr 13, 2025, 12:02 PM • 1 Y

Y by A_E_R

Let $a,b,c,d>0$ and $a(b^2+c^2)\geq 5bcd.$ Prove that $(a^2+b^2+c^2+d^2)(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2})\geq\frac{2401}{100}$
*

A_E_R wrote:

Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)

Z K Y

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A_E_R

5 posts

A_E_R

#3 h Apr 13, 2025, 4:12 PM

Y by

Someone told me that the answer was 49^2/100, but I don't know solution,can you explain it ?

sqing wrote:

Let $a,b,c,d>0$ and $a(b^2+c^2)\geq 5bcd.$ Prove that $(a^2+b^2+c^2+d^2)(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2})\geq\frac{2401}{100}$
*

A_E_R wrote:

Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)

Z K Y

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