many equal squares inside a rectangle (Argentina 2017 OMA L1 p2)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Eazy equation clap

giangtruong13 0

2 hours ago

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

2 hours ago

0 replies

inequality

revol_ufiaw 3

N 3 hours ago 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

4 hours ago

MS_asdfgzxcvb

3 hours ago

How to judge a number is prime or not?

mingzhehu 0

3 hours ago

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

mingzhehu

3 hours ago

0 replies

What is an isogonal conjugate and why is it useful?

EaZ_Shadow 6

N 3 hours ago 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

3 hours ago

No more topics!

many equal squares inside a rectangle (Argentina 2017 OMA L1 p2)

parmenides51 2

N Jan 26, 2025 by BR1F1SZ

The following figure is made up of small congruent squares enclosed in a rectangle. The horizontal side of the rectangle measures $73$ and the vertical side measures $94$ . Calculate the side lenght of each small square.
IMAGE

2 replies

parmenides51

May 15, 2020

BR1F1SZ

Jan 26, 2025

many equal squares inside a rectangle (Argentina 2017 OMA L1 p2)

G H J

Reveal topic tags

geometry

Squares

rectangle

G H BBookmark kLocked kLocked NReply

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parmenides51

30629 posts

parmenides51

#1 h May 15, 2020, 11:42 PM

Y by

http://4.bp.blogspot.com/-jvXkuWo0ojs/Xr8ovWHZ2fI/AAAAAAAAL8w/jVZDjuM57YgNFKFGgVik8POdwV6uAVu9QCK4BGAYYCw/s400/2017oma.png

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natmath

8219 posts

natmath

#2 h May 16, 2020, 8:24 AM

Y by

kindof bashy

Define the angle on the rightside of the bottom most square as $\theta$ and the sidelength of each square as $s$
It can be seen that $s(7\sin\theta+2\cos\theta)=94$ and $s(4\sin\theta+5\cos\theta)=73$
Dividing both equations and simplifying gives us $511\sin\theta+146\cos\theta=376\sin\theta+470\cos\theta$
$135\sin\theta=324\cos\theta$
$5\sin\theta=12\cos\theta$
From there, it is apparent that $\sin\theta=12/13$ and $\cos\theta=5/13$
$s(84+10)=94*13$
$s=\boxed{13}$

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BR1F1SZ

544 posts

BR1F1SZ

#3 h Jan 26, 2025, 1:51 AM

Y by

https://cdn.artofproblemsolving.com/images/3/c/b/3cbdd35d1d6d7eb4674d4d9c6b20938e5a2fa665.png

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