A sharp one with 4 var

by mihaig, May 25, 2025, 4:05 AM

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
$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}\leq2.$

inequalities

A geometry problem

by Lttgeometry, May 25, 2025, 4:03 AM

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 $AQ$ .

This post has been edited 2 times. Last edited by Lttgeometry, an hour ago

geometry

Van der Corput Inequality

by EthanWYX2009, May 25, 2025, 3:36 AM

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 $\displaystyle \left(\sum _{i=1}^{n}|\langle v,u_{i}\rangle |\right)^{2}\leq \sum _{i,j=1}^{n}|\langle u_{i},u_{j}\rangle |.$

inequalities

Inspired by 2025 Xinjiang

by sqing, May 24, 2025, 5:32 PM

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

inequalities proposed

algebra

inequalities

FE with conditions on $x,y$

by Adywastaken, May 23, 2025, 6:18 PM

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

This post has been edited 1 time. Last edited by Adywastaken, Yesterday at 6:08 AM

function

algebra

positive

A sharp one with 3 var

by mihaig, May 13, 2025, 7:20 PM

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

inequalities

Hard Functional Equation in the Complex Numbers

by yaybanana, Apr 9, 2025, 3:29 PM

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

Sharygin 2025 CR P15

by Gengar_in_Galar, Mar 10, 2025, 12:10 PM

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

This post has been edited 1 time. Last edited by Gengar_in_Galar, Mar 11, 2025, 9:54 AM

geometry

Sharygin Geometry Olympiad

Trignometry

tangent circles

Find the minimum

by sqing, Sep 4, 2018, 12:24 AM

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

inequalities proposed

Point satisfies triple property

by 62861, Jan 22, 2018, 5:00 PM

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

ray $EH$ bisects both angles $\angle BES$ , $\angle TED$ , and
$\angle BEN = \angle MED$ .

Proposed by Evan Chen

geometry

Submit

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math_exploration

by mihaig, May 25, 2025, 4:05 AM

by Lttgeometry, May 25, 2025, 4:03 AM

by EthanWYX2009, May 25, 2025, 3:36 AM

by sqing, May 24, 2025, 5:32 PM

by Adywastaken, May 23, 2025, 6:18 PM

by mihaig, May 13, 2025, 7:20 PM

by yaybanana, Apr 9, 2025, 3:29 PM

by Gengar_in_Galar, Mar 10, 2025, 12:10 PM

by sqing, Sep 4, 2018, 12:24 AM

by 62861, Jan 22, 2018, 5:00 PM