Triangle determined by three Simson lines

<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.

1 reply

omega2007

an hour ago

GreekIdiot

29 minutes ago

Difference between being pre-qualified and pre-approved for a mortgage

smitjohn 0

30 minutes ago

Source: Home

In the context of a Southern Home Ownership Programs, it’s essential to understand the difference between pre-qualification and pre-approval. Pre-qualification is an informal estimate of how much you might be able to borrow, based on self-reported financial information. It's a good first step, but it doesn’t carry much weight with sellers.

Pre-approval, however, is a formal process where a lender verifies your income, credit score, and debts. Once pre-approved, you’ll receive a letter showing you're a serious buyer—often giving you an edge in competitive markets. Many home ownership programs require pre-approval before offering down payment assistance or other benefits. Getting pre-approved shows you're financially ready and serious about buying. It also helps you set a realistic home budget and avoid falling for homes you can’t afford. Always aim for pre-approval to give your offer strength and move forward with confidence.

0 replies

smitjohn

30 minutes ago

0 replies

Induction

Mathlover_1 2

N 33 minutes ago by GreekIdiot

Hello, can you share links of same interesting induction problems in algebra

2 replies

Mathlover_1

Mar 24, 2025

GreekIdiot

33 minutes ago

n-gon function

ehsan2004 10

N 44 minutes ago by Zany9998

Source: Romanian IMO Team Selection Test TST 1996, problem 1

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that for every regular $n$ -gon $A_1A_2 \ldots A_n$ we have $f(A_1)+f(A_2)+\cdots +f(A_n)=0$ . Prove that $f(x)=0$ for all reals $x$ .

10 replies

ehsan2004

Sep 13, 2005

Zany9998

44 minutes ago

Functional equations

hanzo.ei 13

N an hour ago by GreekIdiot

Source: Greekldiot

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

13 replies

1 viewing

hanzo.ei

Mar 29, 2025

GreekIdiot

an hour ago

Congruency in sum of digits base q

buzzychaoz 3

N an hour ago by sttsmet

Source: China Team Selection Test 2016 Test 3 Day 2 Q4

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$ . It is given that there exist a positive integer $M$ such that
$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$
holds for all integers $n\ge M$ . Prove that the above equation is true for all positive integers $n$ . (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$ ).

3 replies

buzzychaoz

Mar 26, 2016

sttsmet

an hour ago

Unsolved NT, 3rd time posting

GreekIdiot 11

N an hour ago by GreekIdiot

Source: own

Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint

There are 4 triplets that satisfy

11 replies

GreekIdiot

Mar 26, 2025

GreekIdiot

an hour ago

Bashing??

John_Mgr 2

N an hour ago by GreekIdiot

I have learned little about what bashing mean as i am planning to start geo, feels like its less effort required and doesnt need much knowledge about the synthetic solutions?
what do you guys recommend ? also state the major difference of them... especially of bashing pros and cons..

2 replies

John_Mgr

3 hours ago

GreekIdiot

an hour ago

Inspired by JK1603JK

sqing 13

N an hour ago by sqing

Source: Own

Let $a,b,c\geq 0$ and $ab+bc+ca=1.$ Prove that $\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1}$

13 replies

sqing

Today at 3:31 AM

sqing

an hour ago

A simple power

Rushil 19

N an hour ago by Raj_singh1432

Source: Indian RMO 1993 Problem 2

Prove that the ten's digit of any power of 3 is even.

19 replies

Rushil

Oct 16, 2005

Raj_singh1432

an hour ago

Triangle determined by three Simson lines

Strobenz 1

N Oct 19, 2018 by brokendiamond

Let $ABC$ be a triangle and let $P, Q, R$ be three points lying on the circumcircle of $ABC$ . Prove that if the three Simson lines determined by $P, Q$ and $R$ intersect at $X, Y, Z$ , then the triangle $XYZ$ is similar to the triangle $PQR$ .

I've looked into this problem since forever and I just have no idea how to prove it! Any help would really be appreciated!

1 reply