equal sum of areas, square inside a square

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0 replies

jlacosta

Apr 2, 2025

0 replies

3 var inequality

sqing 0

2 minutes ago

Source: Own

Let $a,b,c>0 ,\frac{a}{b} +\frac{b}{c} +\frac{c}{a} \leq 2\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right).$ Prove that
$a+b+c+2\geq abc$ Let $a,b,c>0 , a^3+b^3+c^3\leq 2(ab+bc+ca).$ Prove that
$a+b+c+2\geq abc$

0 replies

sqing

2 minutes ago

0 replies

Do not try to bash on beautiful geometry

ItzsleepyXD 0

2 minutes ago

Source: Own , Mock Thailand Mathematic Olympiad P9

Let $ABC$ be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$

0 replies

ItzsleepyXD

2 minutes ago

0 replies

already well-known, but yet strangely difficult

Valentin Vornicu 37

N 4 minutes ago by cursed_tangent1434

Source: Romanian ROM TST 2004, problem 6

Let $a,b$ be two positive integers, such that $ab\neq 1$ . Find all the integer values that $f(a,b)$ can take, where $f(a,b) = \frac { a^2+ab+b^2} { ab- 1} .$

37 replies

Valentin Vornicu

May 1, 2004

cursed_tangent1434

4 minutes ago

1 line solution to Inequality

ItzsleepyXD 0

5 minutes ago

Source: Own , Mock Thailand Mathematic Olympiad P8

Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$ such that $x_{n+1}=x_1$ and $x_0=x_n$

0 replies

ItzsleepyXD

5 minutes ago

0 replies

No more topics!

equal sum of areas, square inside a square

parmenides51 2

N May 27, 2021 by naenaendr

Source: 7th QEDMO problem 4 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$ .
IMAGE

2 replies

parmenides51

May 9, 2021

naenaendr

May 27, 2021

equal sum of areas, square inside a square

G H J

Reveal topic tags

geometry

areas

Squares

G H BBookmark kLocked kLocked NReply

Source: 7th QEDMO problem 4 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

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parmenides51

30650 posts

parmenides51

#1 h May 9, 2021, 9:42 AM

Y by

https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png

This post has been edited 2 times. Last edited by parmenides51, May 9, 2021, 9:44 AM

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parmenides51

30650 posts

parmenides51

#2 h May 9, 2021, 9:43 AM

Y by

posted for the image link

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naenaendr

639 posts

naenaendr

#3 h May 27, 2021, 5:18 PM

Y by

Does anyone have any insight into this problem? I made an attempt with it but realized I was incorrectly trying to show using Shoelace that the area of $AA'C'CD$ equals the area of $AA'C'CB$ . Oooops

Any ideas?

Z K Y

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