orthogonality

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0 replies

jlacosta

Mar 2, 2025

0 replies

Inspired by Mihaela Berindeanu

sqing 1

N 4 minutes ago by lbh_qys

Source: Own

Let $a,b,c>0 .$ Prove that
$(4a+b+c)(a+b)(b+c)(c+a)\geq (ab+bc+ca)(2a+b+c)^2$

1 reply

sqing

Yesterday at 2:08 PM

lbh_qys

4 minutes ago

Max amount of equal numbers among (a_i^2 + a_j^2)/(a_i + a_j)

mshtand1 1

N 6 minutes ago by Rushery_10

Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 9.8

Given $2025$ pairwise distinct positive integer numbers $a_1, a_2, \ldots, a_{2025}$ , find the maximum possible number of equal numbers among the fractions of the form
$\frac{a_i^2 + a_j^2}{a_i + a_j}$
Proposed by Mykhailo Shtandenko

1 reply

mshtand1

Mar 14, 2025

Rushery_10

6 minutes ago

Combinatorics from Iranian TST 2017

bgn 20

N 16 minutes ago by ezpotd

Source: Iranian TST 2017, first exam, day1, problem 2

In the country of Sugarland, there are $13$ students in the IMO team selection camp.
$6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation.
Is it possible that all $13$ students have a chance of being a team member?

Proposed by Morteza Saghafian

20 replies

bgn

Apr 5, 2017

ezpotd

16 minutes ago

C-B=60 <degrees>

Sasha 26

N 34 minutes ago by shendrew7

Source: Moldova TST 2005, IMO Shortlist 2004 geometry problem 3

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$ . The line $AO$ meets the side $BC$ at $D$ . The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$ , respectively. Extend the sides $BA$ and $CA$ beyond $A$ , and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$ . Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$ .

Proposed by Hojoo Lee, Korea

26 replies

Sasha

Apr 10, 2005

shendrew7

34 minutes ago

No more topics!

orthogonality

karimeow 3

N Yesterday at 2:35 PM by cj13609517288

Given a cyclic quadrilateral ABCD inscribed in the circle (O). Let E and F be the intersections of AD with BC and AC with BD, respectively. Prove that the circle with diameter EF is orthogonal to (O).

3 replies

karimeow

Yesterday at 6:57 AM

cj13609517288

Yesterday at 2:35 PM

orthogonality

G H J

Reveal topic tags

geometry

G H BBookmark kLocked kLocked NReply

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karimeow

3 posts

karimeow

#1 h Yesterday at 6:57 AM

Y by

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Curious_Droid

33 posts

Curious_Droid

#2 h Yesterday at 8:44 AM

Y by

Consider inversion about $(ABCD)$ . Note that $E, F$ get sent to the Miquel points of $ABCD$ , $ACBD$ , respectively. By Brokard, these Miquel points lie on the circle with diameter $EF$ . Hence we see that this inversion fixes $(EF)$ , so $(EF)$ is orthogonal to $(ABCD)$

Z K Y

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karimeow

3 posts

karimeow

#3 h Yesterday at 9:44 AM

Y by

do you have another solution which don’t use inversion?

Z K Y

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