Three Nagel points collinear

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0 replies

jlacosta

Apr 2, 2025

0 replies

f(x+f(y))=f(x+y)+y

John_Mgr 5

N 2 minutes ago by jasperE3

Determine with proof all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real $x,y:$
$f(x+f(y))=f(x+y)+y$

5 replies

John_Mgr

Mar 28, 2025

jasperE3

2 minutes ago

2 var inquality

sqing 3

N 16 minutes ago by pieMax2713

Source: Own

Let $a,b>0 .$ Prove that
$\dfrac{1}{a^2+b^2}+\dfrac{1}{4ab} \ge \dfrac{\dfrac{3}{2} +\sqrt 2}{(a+b)^2}$ $\dfrac{1}{a^3+b^3}+\dfrac{1}{4ab(a+b)}\ge \dfrac{\dfrac{7}{4} +\sqrt 3}{(a+b)^3}$

3 replies

sqing

Yesterday at 9:28 AM

pieMax2713

16 minutes ago

Inspired by old results

sqing 0

27 minutes ago

Source: Own

Let $a,b,c>0$ and $a+b+c\ge \sqrt { abc}.$ Prove that
$a^2+ 2b^2+3c^2 \ge \frac{6}{11}abc$ $a^2+ b^2+c^2+1 \ge \frac{244}{729}abc$

0 replies

sqing

27 minutes ago

0 replies

Geometry warmup: internally tangent circles

v_Enhance 23

N 38 minutes ago by eg4334

Source: HMMT Invitational Contest 2016, Problem 2

Let $ABC$ be an acute triangle with circumcenter $O$ , orthocenter $H$ , and circumcircle $\Omega$ . Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$ . Assume the points $M$ , $N$ , $O$ , $H$ are distinct and lie on a circle $\omega$ . Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other.

Dhroova Aiylam and Evan Chen

23 replies

v_Enhance

Apr 22, 2016

eg4334

38 minutes ago

No more topics!

Three Nagel points collinear

jayme 3

N Apr 2, 2025 by buratinogigle

Dear Marthlinkers,

1. ABCD a square
2. M a point on the segment CD sothat MA < MB
3. Nm, Na, Nb the Nagel’s points of the triangles MAD, ADM, BCM.

Prove : Nm, Na and Nb are collinear.

Sincerely
Jean-Louis

3 replies