Concurrency with 10 lines

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0 replies

jlacosta

May 1, 2025

0 replies

Interesting inequalities

sqing 2

N 14 minutes ago by sqing

Source: Own

Let $a,b,c\geq 0 ,a+b+c\leq 3.$ Prove that
$a^2+b^2+c^2+2ab+2bc + abc \leq \frac{244}{27}$ $a^2+b^2+c^2+\frac{1}{2}ab +2ca+2bc + abc \leq \frac{73}{8}$ $a^2+b^2+c^2+ab+2ca+2bc + \frac{1}{2}abc \leq \frac{487}{54}$ $a^2+b^2+c^2+a+b+ab+2ca+2bc+2abc\leq 12$

2 replies

1 viewing

sqing

Yesterday at 12:52 PM

sqing

14 minutes ago

thank you !

Nakumi 0

17 minutes ago

Given two non-constant polynomials $P(x),Q(x)$ such that for every real number $c$ , $P(c)$ is a perfect square if and only if $Q(c)$ is a perfect square. Prove that $P(x)Q(x)$ is the square of a polynomial with real coefficients.

0 replies

Nakumi

17 minutes ago

0 replies

Same divisor

sam-n 16

N 21 minutes ago by AbdulWaheed

Source: IMO Shortlist 1997, Q14, China TST 2005

Let $b, m, n$ be positive integers such that $b > 1$ and $m \neq n.$ Prove that if $b^m - 1$ and $b^n - 1$ have the same prime divisors, then $b + 1$ is a power of 2.

16 replies

sam-n

Mar 6, 2004

AbdulWaheed

21 minutes ago

sum of gcd over sets is more then sum of gcd over union

Miquel-point 3

N 36 minutes ago by Jupiterballs

Source: KoMaL A. 882

Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
$\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).$ Proposed by Dávid Matolcsi, Berkeley

3 replies

Miquel-point

Jun 11, 2024

Jupiterballs

36 minutes ago

No more topics!

Concurrency with 10 lines

oVlad 1

N Apr 21, 2025 by kokcio

Source: Romania EGMO TST 2017 Day 1 P1

Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.

1 reply

oVlad

Apr 21, 2025

kokcio

Apr 21, 2025

Concurrency with 10 lines

G H J

Reveal topic tags

geometry

concurrency

G H BBookmark kLocked kLocked NReply

Source: Romania EGMO TST 2017 Day 1 P1

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oVlad

1746 posts

oVlad

#1 h Apr 21, 2025, 1:31 PM

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kokcio

70 posts

kokcio

#3 h Apr 21, 2025, 5:11 PM

Y by

Let $M$ be center of mass of this $5$ points. Let $X$ be centroid of some triangle and $Y$ be midpoint of chord with two other points. Then if line $OM$ intersects line drawn through $X$ at point $P$ , then $\frac{OM}{MP}=\frac{MY}{MX}=\frac{3}{2}$ , so position of $P$ is uniquely determined by the position of points $O,M$ . (if $M=O$ , then $P=O$ ).
Generalization of this problem is in plane geometry by V. Prasolov (problem 14.13): on a circle, $n$ points are given. Through the center of mass of $n-2$ points a straight line is drawn perpendicularly to the chord that connects the two remaining points. Prove that all such straight lines intersect at one point.

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