Find the locus of vertices B that angles lie in the interval

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0 replies

jlacosta

Apr 2, 2025

0 replies

Divisiblity for combinatorial numbers

DinDean 0

30 minutes ago

Source: Vietnam TST 2025

Find all positive integers $k$ satisfying:

There are infinitely many positive numbers $n$ such that $kn+1\nmid \binom{(2025+k)n}{2025n}$ .

0 replies

DinDean

30 minutes ago

0 replies

Integer polynomials

GeoKing 1

N an hour ago by eulerleonhardfan

Let f(x) be a monic polynomial of degree n with integer coefficients, and let d1, · · · , dn be pairwise distinct integers. Suppose that for infinitely many prime numbers p there exists an integer kp for which f(kp + d1) ≡ f(kp + d2) ≡ · · · f(kp + dn) ≡ 0 (mod p). Prove that there exists an integer k0 such that f(k0 + d1) = f(k0 + d2) =· · · = f(k0 + dn) = 0

1 reply

GeoKing

Aug 2, 2021

eulerleonhardfan

an hour ago

Interesting Inequality Problem

Omerking 1

N 2 hours ago by KhuongTrang

Let $a,b,c$ be three non-negative real numbers satisfying $a+b+c+abc=4.$
Prove that
$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1} \leq\frac{6}{13-3ab-3bc-3ca}$

1 reply

Omerking

6 hours ago

KhuongTrang

2 hours ago

Functional Equation

Keith50 1

N 2 hours ago by jasperE3

Source: Own

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)$ holds for all reals $x$ and $y$ .

1 reply

Keith50

Jun 24, 2021

jasperE3

2 hours ago

No more topics!

Find the locus of vertices B that angles lie in the interval

Amir Hossein 2

N Sep 22, 2011 by oneplusone

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$ , find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

2 replies

Amir Hossein

Sep 30, 2010

oneplusone

Sep 22, 2011

Find the locus of vertices B that angles lie in the interval

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Amir Hossein

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Amir Hossein

#1 h Sep 30, 2010, 1:31 PM • 2 Y

Y by Adventure10, Mango247

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$ , find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

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anonymouslonely

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anonymouslonely

#2 h Sep 8, 2011, 8:44 AM • 2 Y

Y by Adventure10, Mango247

I only obtained that the angles have to be not greater than $70$ and the union of some segments but I think it has to be something more than that.
if you have a solution or a hint please post it.

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oneplusone

1459 posts

oneplusone

#3 h Sep 22, 2011, 10:31 AM • 1 Y

Y by Adventure10

We extend $AM$ to $N,D$ such that $2MN=AM$ and $ND=AN$ . Then $\angle ABN,\angle NBD,180-\angle ABD$ must lie in $[40,70]$ . So just draw circles everywhere and the locus is the intersection of the 3 moon-shaped things, which is that tiny curved triangle.

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