Geometry / Computation

I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE

1571 replies

rrusczyk

Mar 24, 2025

SmartGroot

Mar 26, 2025

k a March Highlights and 2025 AoPS Online Class Information

jlacosta 0

Mar 2, 2025

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0 replies

jlacosta

Mar 2, 2025

0 replies

Any nice way to do this?

NamelyOrange 1

N 19 minutes ago by HockeyMaster85

Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$ ?

1 reply

NamelyOrange

42 minutes ago

HockeyMaster85

19 minutes ago

Polynomial optimization problem

ReticulatedPython 2

N an hour ago by Mathzeus1024

Let $p(x)=-ax^4+x^3$ , where $a$ is a real number. Prove that for all positive $a$ , $p(x) \le \frac{27}{256a^3}.$

2 replies

ReticulatedPython

Monday at 2:51 PM

Mathzeus1024

an hour ago

a+b+c=3 inequality

JK1603JK 1

N an hour ago by KhuongTrang

Let $a,b,c\ge 0: a+b+c=3$ then prove
$\color{black}{\sqrt{a+b+2c^{2}}+\sqrt{b+c+2a^{2}}+\sqrt{c+a+2b^{2}}\le 3\sqrt{\frac{a^2+b^2+c^2}{ab+bc+ca}+3}.}$

1 reply

JK1603JK

2 hours ago

KhuongTrang

an hour ago

Complex + Radical Evaluation

Saucepan_man02 4

N 2 hours ago by Mathzeus1024

Evaluate: (without calculators)
$(\sqrt{6 - 2 \sqrt{5}} + i \sqrt{2 \sqrt{5} + 10})^5 + (\sqrt{6 - 2 \sqrt{5}} - i \sqrt{2 \sqrt{5} + 10})^5$

4 replies

Saucepan_man02

Mar 17, 2025

Mathzeus1024

2 hours ago

No more topics!

k Geometry / Computation

duttaditya18 81

N Aug 12, 2019 by brainiacmaniac31

An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?

81 replies

duttaditya18

Aug 11, 2019

brainiacmaniac31

Aug 12, 2019

Geometry / Computation

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

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duttaditya18

59 posts

duttaditya18

#1 h Aug 11, 2019, 9:44 AM • 2 Y

Y by Adventure10, Mango247

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Smoothe

176 posts

Smoothe

#2 h Aug 11, 2019, 9:44 AM • 2 Y

Y by Adventure10, Mango247

Ans is 72.

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GeoMetrix

924 posts

GeoMetrix

#4 h Aug 11, 2019, 9:49 AM • 1 Y

Y by Adventure10

$72$ obviously

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duttaditya18

59 posts

duttaditya18

#5 h Aug 11, 2019, 10:05 AM • 1 Y

Y by Adventure10

@amaanmathbuddy_2006 are you sure it is not $76$ because he had moved $4$ at the start, then the ant started moving $20^{\circ}$ every turn.

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Smoothe

176 posts

Smoothe

#6 h Aug 11, 2019, 10:11 AM • 2 Y

Y by Adventure10, Mango247

duttaditya18 wrote:

@amaanmathbuddy_2006 are you sure it is not $76$ because he had moved $4$ at the start, then the ant started moving $20^{\circ}$ every turn.

Sure

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VERYunfortunate

3 posts

VERYunfortunate

#7 h Aug 11, 2019, 10:27 AM • 4 Y

Y by Adventure10, Mango247, Mango247, Mango247

Shouldn't a 160 degree turn be something like $\angle$

This post has been edited 4 times. Last edited by VERYunfortunate, Aug 11, 2019, 10:29 AM

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Akash_Arjun

4 posts

Akash_Arjun

#8 h Aug 11, 2019, 11:27 AM • 2 Y

Y by Adventure10, Mango247

Please show a detailed solution to this problem as I am a beginner.

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Rishi123456

51 posts

Rishi123456

#11 h Aug 11, 2019, 12:50 PM • 2 Y

Y by Adventure10, Mango247

Smoothe wrote:

Ans is 72.

Please check.

Easy question answer will be 72

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zuss77

520 posts

zuss77

#12 h Aug 11, 2019, 1:31 PM • 1 Y

Y by Adventure10

duttaditya18 wrote:

What ridiculous question is that? Obviously ant never returns to the anthill.

Attachments:

This post has been edited 1 time. Last edited by zuss77, Aug 11, 2019, 1:57 PM
Reason: diagram

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rockin.numbers

59 posts

rockin.numbers

#13 h Aug 11, 2019, 1:31 PM • 4 Y

Y by Janstew, Quantum_fluctuations, RudraRockstar, Adventure10

I’ve got 36 and it should be completed in 9 rounds. Try to draw out the diagram

This post has been edited 1 time. Last edited by rockin.numbers, Aug 11, 2019, 1:32 PM
Reason: Left out some part.

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Quantum_fluctuations

1282 posts

Quantum_fluctuations

#14 h Aug 11, 2019, 1:35 PM • 2 Y

Y by Janstew, Adventure10

Yes, it is 36.

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mathsworm

765 posts

mathsworm

#15 h Aug 11, 2019, 1:36 PM • 1 Y

Y by Adventure10

Hey please tell the answer
Some of my friends got 24,36,72
I did 72 by the way

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UjwalKumar

58 posts

UjwalKumar

#16 h Aug 11, 2019, 2:04 PM • 4 Y

Y by RudraRockstar, Agsh2005, killadg, Adventure10

According to me, the answer is obv 36

The ant follows the diagonals of a nonagon.

Attachments:

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zuss77

520 posts

zuss77

#17 h Aug 11, 2019, 2:15 PM • 1 Y

Y by Adventure10

Read more carefully.

Actually, I guess it opposite: don't read carefully, if you want to get good score.

This post has been edited 1 time. Last edited by zuss77, Aug 11, 2019, 2:16 PM

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Fredtheidiot

12 posts

Fredtheidiot

#18 h Aug 11, 2019, 3:02 PM • 2 Y

Y by Adventure10, Mango247

emmm,why do I get 156 :roll:

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zuss77

520 posts

zuss77

#75 h Aug 12, 2019, 11:58 AM • 3 Y

Y by Adventure10, Mango247, Mango247

duttaditya18 wrote:

Let's parse it word by word:

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

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GeoMetrix

924 posts

GeoMetrix

#76 h Aug 12, 2019, 11:59 AM • 1 Y

Y by Adventure10

zuss77 wrote:

duttaditya18 wrote:

Let's parse it word by word:

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

whats wrong in here

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gamerrk1004

1519 posts

gamerrk1004

#77 h Aug 12, 2019, 12:08 PM • 1 Y

Y by Adventure10

I think in this .... after 1 turn it is moving towards the left, not right

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GeoMetrix

924 posts

GeoMetrix

#78 h Aug 12, 2019, 12:09 PM • 2 Y

Y by Vasu090, Adventure10

no bro
it didnt say it had to move towards the right
it said just turn towards the right

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gamerrk1004

1519 posts

gamerrk1004

#79 h Aug 12, 2019, 12:10 PM • 1 Y

Y by Adventure10

Ok ... so you mean that the question asked for the right direction, not its right.
If that's the case, then the question must be considered as a bonus ...

This post has been edited 1 time. Last edited by gamerrk1004, Aug 12, 2019, 12:14 PM
Reason: AoPS error 1234

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GeoMetrix

924 posts

GeoMetrix

#80 h Aug 12, 2019, 12:32 PM • 2 Y

Y by Adventure10, Mango247

i think theres a typo in this problem. the problem should have said that the ant turns towards his right because the right may mean east

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Fredtheidiot

12 posts

Fredtheidiot

#81 h Aug 12, 2019, 1:03 PM • 2 Y

Y by Adventure10, Mango247

amaanmathbuddy_2006 wrote:

i think theres a typo in this problem. the problem should have said that the ant turns towards his right because the right may mean east

Absolutely, the "right" we are talking about is for the ant, not for ourselves.
btw, the answer is 36

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GeoMetrix

924 posts

GeoMetrix

#82 h Aug 12, 2019, 1:05 PM • 2 Y

Y by Adventure10, Mango247

but it should have been menioned
the right can be our right too

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gamerrk1004

1519 posts

gamerrk1004

#83 h Aug 12, 2019, 2:16 PM • 2 Y

Y by Adventure10, Mango247

Fredtheidiot wrote:

amaanmathbuddy_2006 wrote:

i think theres a typo in this problem. the problem should have said that the ant turns towards his right because the right may mean east

Absolutely, the "right" we are talking about is for the ant, not for ourselves.
btw, the answer is 36

You are correct but that is "Logic"
As per the language of the question ... I think @amaanmathbuddy_2006 is also right ....
Please anyone else also verify it ...

This post has been edited 1 time. Last edited by gamerrk1004, Aug 12, 2019, 2:16 PM

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Hexagrammum16

1774 posts

Hexagrammum16

#85 h Aug 12, 2019, 4:05 PM • 2 Y

Y by Adventure10, Mango247

amaanmathbuddy_2006 wrote:

but it should have been menioned
the right can be our right too

When I turn right while walking, I don't see whether it's right or left of Donald Trump or not.

A request to mods: there's been a lot of unnecessary debate about the validity and answer of the question. I myself have posted at least 3 times about 36 against 72. So I think this thread needs some cleaning up

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KC-ian

4 posts

KC-ian

#86 h Aug 12, 2019, 4:21 PM • 1 Y

Y by Adventure10

It should be 36 feet
Solution: Given in attachment.

Attachments:

New Microsoft Office Word Document.docx (11kb)

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RudraRockstar

606 posts

RudraRockstar

#87 h Aug 12, 2019, 5:44 PM • 1 Y

Y by Adventure10

Hexagrammum16 wrote:

amaanmathbuddy_2006 wrote:

but it should have been menioned
the right can be our right too

I absolutely agree. It’s more than evident the answer is 36 (Just look at the diagram guys). I don’t get why there is sooo much debate about this...

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Purrpl_Cat

19 posts

Purrpl_Cat

#88 h Aug 12, 2019, 10:09 PM • 2 Y

Y by Adventure10, Mango247

I thought over it a bit more, and yeah it's 36. Even drew out a diagram with the nonagon thing like everyone else, and seems like nobody else has any evidence to support 72

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ItsAmeYushi

1173 posts

ItsAmeYushi

#89 h Aug 12, 2019, 10:12 PM • 3 Y

Y by brainiacmaniac31, Adventure10, Mango247

zuss77 wrote:

duttaditya18 wrote:

Let's parse it word by word:

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

The pattern being continued is that makes a $160^\circ$ turn to the right and walks $4$ more feet not the $4$ feet east part. This is understood implicitly since the east gives us a starting and it is going walk then turn then walk then turn, etc.

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
t.setheading(0)
while True:    
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
t.setheading(0)
while True:    
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

edit: Didn't read through the whole thread so I didn't notice this earlier, but you can see NikoIsLife's code above which is basically the same as this.

This post has been edited 2 times. Last edited by ItsAmeYushi, Aug 12, 2019, 10:18 PM

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brainiacmaniac31

2170 posts

brainiacmaniac31

#90 h Aug 12, 2019, 10:39 PM • 1 Y

Y by Adventure10

ItsAmeYushi wrote:

zuss77 wrote:

duttaditya18 wrote:

Let's parse it word by word:

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
while True:
    t.setheading(0)
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    t.forward(100)   # and walks 4 more feet.
    answer += 4
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle
 
t = turtle.Turtle()
start = t.position()
 
answer = 0
t.setheading(0)
while True:    
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

import turtle

t = turtle.Turtle()
start = t.position()

answer = 0
t.setheading(0)
while True:    
    t.forward(100)   # It walks 4 feet east
    answer += 4
    t.right(160)     # and then makes a 160 turn to the right
    if t.position() == start: 
        break
turtle.done()
print("The answer is",answer)

edit: Didn't read through the whole thread so I didn't notice this earlier, but you can see NikoIsLife's code above which is basically the same as this.

Nice solution!

Z K Y

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