Point moving towards vertices and changing plans again and again

Let $a,b,c,d\geq0$ satisfying
$\left(a+b+c+d-1\right)^2+7\leq\frac83\cdot\left(ab+bc+ca+ad+bd+cd\right).$ Prove
$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}\leq2.$

0 replies

mihaig

an hour ago

0 replies

A geometry problem

Lttgeometry 0

an hour ago

Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$ . The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$ , and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$ . Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$ .
Note: Circle $(AP)$ is the circle with diameter $AP$ , Circle $(AQ)$ is the circle with diameter $AQ$ .

0 replies

Lttgeometry

an hour ago

0 replies

A sharp one with 3 var

mihaig 9

N an hour ago by mihaig

Source: Own

Let $a,b,c\geq0$ satisfying
$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$ Prove
$ab+bc+ca+abc\geq4.$

9 replies

mihaig

May 13, 2025

mihaig

an hour ago

Van der Corput Inequality

EthanWYX2009 0

2 hours ago

Source: en.wikipedia.org/wiki/Van_der_Corput_inequality

Let $V$ be a real or complex inner product space. Suppose that $v,u_{1},\dots ,u_{n}\in V$ and that ${\displaystyle \|v\|=1}.$ Then $\displaystyle \left(\sum _{i=1}^{n}|\langle v,u_{i}\rangle |\right)^{2}\leq \sum _{i,j=1}^{n}|\langle u_{i},u_{j}\rangle |.$

0 replies

EthanWYX2009

2 hours ago

0 replies

Hard Functional Equation in the Complex Numbers

yaybanana 6

N 2 hours ago by jasperE3

Source: Own

Find all functions $f:\mathbb {C}\rightarrow \mathbb {C}$ , s.t :

$f(xf(y)) + f(x^2+y) = f(x+y)x + f(f(y))$

for all $x,y \in \mathbb{C}$

6 replies

yaybanana

Apr 9, 2025

jasperE3

2 hours ago

FE with conditions on $x,y$

Adywastaken 3

N 2 hours ago by jasperE3

Source: OAO

Find all functions $f:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ such that $\forall y>x>0$ ,
$f(x^2+f(y))=f(xf(x))+y$

3 replies

Adywastaken

Friday at 6:18 PM

jasperE3

2 hours ago

Point satisfies triple property

62861 36

N 3 hours ago by cursed_tangent1434

Source: USA Winter Team Selection Test #2 for IMO 2018, Problem 2

Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$ . Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$ . Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$ , respectively. Prove that there exists a point $E$ , lying outside quadrilateral $ABCD$ , such that
[list]
[*] ray $EH$ bisects both angles $\angle BES$ , $\angle TED$ , and
[*] $\angle BEN = \angle MED$ .
[/list]

Proposed by Evan Chen

36 replies

62861

Jan 22, 2018

cursed_tangent1434

3 hours ago

Inspired by 2025 Xinjiang

sqing 1

N 3 hours ago by sqing

Source: Own

Let $a,b,c >0 .$ Prove that
$\left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}+\frac {b}{ c}\right) \left(1+\frac {a}{b}+\frac {b}{ c}+\frac {c}{ a}\right) \geq 12+8\sqrt 2$

1 reply

1 viewing

sqing

Yesterday at 5:32 PM

sqing

3 hours ago

Find the minimum

sqing 27

N 3 hours ago by sqing

Source: Zhangyanzong

Let $a,b$ be positive real numbers such that $a^2b^2+\frac{4a}{a+b}=4.$ Find the minimum value of $a+2b.$

27 replies

sqing

Sep 4, 2018

sqing

3 hours ago

Sharygin 2025 CR P15

Gengar_in_Galar 7

N 3 hours ago by Giant_PT

Source: Sharygin 2025

A point $C$ lies on the bisector of an acute angle with vertex $S$ . Let $P$ , $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$ . Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent.
Proposed by: A.Zaslavsky

7 replies

Gengar_in_Galar

Mar 10, 2025

Giant_PT

3 hours ago

Point moving towards vertices and changing plans again and again

Miquel-point 0

Apr 6, 2025

Source: Romanian IMO TST 1981, Day 3 P6

In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$ . Halfway there, it stops and starts moving in a straight line line towards $B$ . Halfway there, it stops and starts moving in a straight line towards $C$ , and halfway there it stops and starts moving in a straight line towards $A$ , and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$ .