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hard problem....

Cobedangiu 1

N an hour ago by arqady

let $a,b,c$ be the lengths of the sides of the triangle. Prove that:
$(a+b+c)(\dfrac{3a-b}{a^2+ab}+\dfrac{3b-c}{b^2+bc}+\dfrac{3c-a}{c^2+ac})\le 9$

1 reply

Cobedangiu

2 hours ago

arqady

an hour ago

Prove the inequality with the condition (a+1)(b+1)(c+1)=8

hlminh 1

N an hour ago by quacksaysduck

Let $a,b,c>0$ such that $(a+1)(b+1)(c+1)=8.$ Prove that $abc(a+b+c)\leq 3.$

1 reply

hlminh

3 hours ago

quacksaysduck

an hour ago

integer functional equation

ABCDE 147

N an hour ago by Adywastaken

Source: 2015 IMO Shortlist A2

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that $f(x-f(y))=f(f(x))-f(y)-1$ holds for all $x,y\in\mathbb{Z}$ .

147 replies

ABCDE

Jul 7, 2016

Adywastaken

an hour ago

number theory FE

pomodor_ap 0

an hour ago

Source: Own, PDC002-P7

Let $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ be a function such that
$f(m) + mn + n^2 \mid f(m)^2 + m^2 f(n) + f(n)^2$ for all $m, n \in \mathbb{Z}^+$ . Find all such functions $f$ .

0 replies

pomodor_ap

an hour ago

0 replies

real+ FE

pomodor_ap 0

an hour ago

Source: Own, PDC001-P7

Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$ for all $x, y \in \mathbb{R}^+$ . Determine all such functions $f$ .

0 replies

pomodor_ap

an hour ago

0 replies

Is this FE solvable?

ItzsleepyXD 2

N 2 hours ago by ItzsleepyXD

Source: Original

Let $c_1,c_2 \in \mathbb{R^+}$ . Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $f(x+c_1f(y))=f(x)+c_2f(y)$

2 replies

ItzsleepyXD

Today at 3:02 AM

ItzsleepyXD

2 hours ago

AM-GM FE ineq

navi_09220114 2

N 2 hours ago by navi_09220114

Source: Own. Malaysian IMO TST 2025 P3

Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$ holds for all reals $x$ , $y$ , $z$ that satisfy $x+y+z\ge 0$ .

Proposed by Ivan Chan Kai Chin

2 replies

navi_09220114

Mar 22, 2025

navi_09220114

2 hours ago

Strange Geometry

Itoz 2

N 2 hours ago by hectorraul

Source: Own

Given a fixed circle $\omega$ with its center $O$ . There are two fixed points $B, C$ and one moving point $A$ on $\omega$ . The midpoint of the line segment $BC$ is $M$ . $R$ is a fixed point on $\omega$ . Line $AO$ intersects $\odot(AMR)$ at $P(\ne A)$ , and line $BP$ intersects $\odot(BOC)$ at $Q(\ne B)$ .

Find all the fixed points $R$ such that $\omega$ is always tangent to $\odot (OPQ)$ when $A$ varies.
Hint

2 replies

Itoz

Yesterday at 2:00 PM

hectorraul

2 hours ago

From Recreatii Matematice 1/2025

mihaig 0

2 hours ago

Source: Own

Given a non-degenerate $\Delta ABC,$
find $x,y,z\geq0$ such that
$x+y+z+\sqrt{\sum_{\text{cyc}}{x^2}-2\sum_{\text{cyc}}{yz\cos A}}=\sum_{\text{cyc}}{\sqrt{y^2-2yz\cos A+z^2}}.$

0 replies

mihaig

2 hours ago

0 replies

Medium geometry with AH diameter circle

v_Enhance 93

N 2 hours ago by waterbottle432

Source: USA TSTST 2016 Problem 2, by Evan Chen

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ . Denote by $M$ , $N$ the midpoints of $\overline{AH}$ , $\overline{BC}$ . Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$ , and meets line $AN$ at a point $Q \neq A$ . The tangent to $\gamma$ at $G$ meets line $OM$ at $P$ . Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$ .

Proposed by Evan Chen

93 replies

v_Enhance

Jun 28, 2016

waterbottle432

2 hours ago

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