ratio chasing in rectangle, equal areas (2006UQ/QAMT PS Competition 9-10 p3)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Eazy equation clap

giangtruong13 0

Today at 4:03 PM

Find all $x,y,z$ satisfy that: $\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$

0 replies

giangtruong13

Today at 4:03 PM

0 replies

inequality

revol_ufiaw 3

N Today at 2:55 PM by MS_asdfgzxcvb

Prove that that for any real $x \ge 0$ and natural number $n$ ,
$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$

3 replies

revol_ufiaw

Today at 2:05 PM

MS_asdfgzxcvb

Today at 2:55 PM

How to judge a number is prime or not?

mingzhehu 0

Today at 2:45 PM

A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime

0 replies

1 viewing

mingzhehu

Today at 2:45 PM

0 replies

What is an isogonal conjugate and why is it useful?

EaZ_Shadow 6

N Today at 2:40 PM by maxamc

What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.

6 replies

EaZ_Shadow

Dec 28, 2024

maxamc

Today at 2:40 PM

No more topics!

ratio chasing in rectangle, equal areas (2006UQ/QAMT PS Competition 9-10 p3)

parmenides51 3

N Jan 19, 2021 by OlympusHero

In the rectangle $ABCD$ a triangle $AEF$ is drawn, where $E$ lies on $BC$ and $F$ lies on $CD$ , and the triangles $ABE, ECF$ and $ADF$ all have the same area. What is the ratio $CF:FD$ ?

3 replies

parmenides51

Jan 19, 2021

OlympusHero

Jan 19, 2021

ratio chasing in rectangle, equal areas (2006UQ/QAMT PS Competition 9-10 p3)

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

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parmenides51

30629 posts

parmenides51

#1 h Jan 19, 2021, 4:20 PM • 1 Y

Y by son7

In the rectangle $ABCD$ a triangle $AEF$ is drawn, where $E$ lies on $BC$ and $F$ lies on $CD$ , and the triangles $ABE, ECF$ and $ADF$ all have the same area. What is the ratio $CF:FD$ ?

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franzliszt

23531 posts

franzliszt

#2 h Jan 19, 2021, 4:34 PM • 1 Y

Y by son7

I'm not barybashing this one.

WLOG, $ABCD$ is a unit square. Let $BE=x=DF$ and $CF=CE=1-x$ and we have $x=(1-x)^2 \Rightarrow x^2-3x+1=0$ . So $x=\frac{3-\sqrt5}2$ . Answer is $\frac{1-\frac{3-\sqrt5}2}{\frac{3-\sqrt5}2}$ which is not too nice.

Is this the correct answer?

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parmenides51

30629 posts

parmenides51

#3 h Jan 19, 2021, 4:38 PM

Y by

answer is nice after calculations, indeed correct

$\frac{\sqrt5 +1}{2}$