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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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Two Rings in Space
nellis25   9
N May 2, 2025 by Waddles2010
Given a regular octahedron where all the edge lengths are 2, the incircle of one face is drawn (shown in red), and the circumcircle of an adjacent face is also drawn (shown in yellow). Find the shortest distance between a point on the red circle and a point on the yellow circle.


[center]IMAGE[/center]
9 replies
nellis25
Jul 13, 2020
Waddles2010
May 2, 2025
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Matching Cards
nellis25   13
N Apr 30, 2025 by srg2015
Two shuffled standard 52-card decks are placed side-by-side.

What is the reciprocal of the probability that at least one card is in the same position in both decks, to nine decimal places?

[center]IMAGE[/center]
13 replies
nellis25
Jan 17, 2022
srg2015
Apr 30, 2025
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Infinite Roots
kchadwick   22
N Apr 29, 2025 by JumpingBobcat30
Which is greater: $2,$ or $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \dots}}}}?$


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22 replies
kchadwick
Oct 28, 2021
JumpingBobcat30
Apr 29, 2025
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AoPS is Not Responsible for Your Broken Picture Frames
nellis25   78
N Apr 29, 2025 by JumpingBobcat30
If you hang a picture with a string looped around two nails, as shown below, removing either nail will not cause the picture to fall.

[center]IMAGE[/center]

Can you find a way to hang a picture with a string looped around two nails so that the picture would fall if either one of the nails were removed?

[center]IMAGE[/center]
78 replies
nellis25
Apr 27, 2020
JumpingBobcat30
Apr 29, 2025
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Marker Problem
nellis25   13
N Apr 26, 2025 by ericheathclifffry
What fraction of this rectangle is shaded?

[center]IMAGE[/center]
13 replies
nellis25
Jan 27, 2021
ericheathclifffry
Apr 26, 2025
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Grogg Tries to Skip School
nellis25   23
N Apr 26, 2025 by ericheathclifffry
Grogg said to his mom, "Why should I go to school when I don't have the time? I sleep $8$ hours a day, which adds up to about $122$ days per year. There's no school on weekends, which is $104$ days per year. We have $60$ days of summer vacation. I need $3$ hours a day to eat - that's $45$ days per year. And I need $2$ hours a day to have fun, or $30$ days per year. If I'm sick $4$ days per year, then look at this:

[center]IMAGE[/center]

There's no more days in the year, so I don't have time to go to school!"

Grogg's mom looked at Grogg's calculations and chuckled. "Your calculations aren't wrong," she said, "but that's no excuse not to go to school!"

What's wrong with Grogg's reasoning?
23 replies
nellis25
Apr 5, 2021
ericheathclifffry
Apr 26, 2025
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Looking for a Match
nellis25   5
N Apr 25, 2025 by ericheathclifffry
Lizzie has three pairs of socks in different colors: one green, one blue, and one red. She grabs three socks randomly from her drawer. If they contain a pair of the same color, she wears it; if they are all different colors, she puts all three back. What is the probability she'll find a pair within her first three pulls?

[center]IMAGE[/center]
5 replies
nellis25
Jun 21, 2021
ericheathclifffry
Apr 25, 2025
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Two Squares and a Semicircle
nellis25   11
N Apr 24, 2025 by ericheathclifffry
The diameter of the semicircle is 10. What's the total area of the two shaded squares?

[center]IMAGE[/center]
11 replies
nellis25
Jun 15, 2020
ericheathclifffry
Apr 24, 2025
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The Magic Chessboard
nellis25   11
N Apr 24, 2025 by ericheathclifffry
I have a magic $8 \times 8$ chessboard: I can choose any row or column and swap the colors of every square in that row or column (so that every white square turns blue, and every blue square turns white). Can I turn the coloring in the left picture to that of the right picture?

[center]IMAGE[/center]
11 replies
nellis25
Jun 15, 2020
ericheathclifffry
Apr 24, 2025
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Sum Blob
nellis25   2
N Apr 24, 2025 by ericheathclifffry
Divide the square below into 4 different sections, each of which sums to the same value. For an added challenge, Calamitous Clod is covering three of the numbers that have the same value. What value is Calamitous Clod covering?

[center]IMAGE[/center]
2 replies
nellis25
Aug 17, 2020
ericheathclifffry
Apr 24, 2025
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Age Conundrum
nellis25   15
N Aug 22, 2024 by kabir_sahil
The sum of Dancer's, Dasher's, and Prancer's ages is 60. When Prancer was born, Dasher's age was double that of Dancer's. How old is Dancer today?

[center]IMAGE[/center]
15 replies
nellis25
Nov 30, 2021
kabir_sahil
Aug 22, 2024
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Age Conundrum
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nellis25
279 posts
nellis25
#1 h  Nov 30, 2021, 6:26 PM
Y by
The sum of Dancer's, Dasher's, and Prancer's ages is 60. When Prancer was born, Dasher's age was double that of Dancer's. How old is Dancer today?
https://data.artofproblemsolving.com/images/keeplearning/Week%2087_Website_2_Age%20Conundrum.png
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MathFan335
343 posts
MathFan335
#2 h  Dec 1, 2021, 4:44 AM • 2 Y
Y by dolphinia, FN04
Sol
This post has been edited 2 times. Last edited by MathFan335, Dec 1, 2021, 4:46 AM
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sigma_notation
2 posts
sigma_notation
#3 h  Dec 1, 2021, 10:43 PM
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dolphinia
1101 posts
dolphinia
#4 h  Mar 1, 2022, 12:28 AM
Y by
boy dancer is pretty young considering santa is pretty old.... :read:
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Dragonsrule
4 posts
Dragonsrule
#5 h  Jul 27, 2022, 6:53 PM
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BabaLama
1256 posts
BabaLama
#6 h  Aug 22, 2022, 11:45 PM • 1 Y
Y by Mango247
20. Prancer + Dasher = 60 = 60/Prancer + Dasher = 60/3 = 20
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Julian.S
437 posts
Julian.S
#7 h  Sep 23, 2022, 4:36 AM
Y by
If we say that Dancer's age at Prancer's birth is x, and Prancer is y years old in the present, then Prancer is y years old, Dancer is x+y, and Dasher is 2x+y. This sums up to 3x+3y = 60, so x+y = 20. Dancer's current age is x+y, or simply 20.
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BabaLama
1256 posts
BabaLama
#8 h  Oct 9, 2022, 2:04 AM
Y by
nellis25 wrote:
The sum of Dancer's, Dasher's, and Prancer's ages is 60. When Prancer was born, Dasher's age was double that of Dancer's. How old is Dancer today?
https://data.artofproblemsolving.com/images/keeplearning/Week%2087_Website_2_Age%20Conundrum.png

20 because 2x + x + 0 = 60 which simplifies to 3x = 60 which equals to x = 20.

Sorry for double posting (this solution should be better this time around).
This post has been edited 1 time. Last edited by BabaLama, Oct 9, 2022, 2:05 AM
Reason: y
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Cokevending56
106 posts
Cokevending56
#9 h  Nov 19, 2022, 5:26 PM
Y by
Dancer is $20$ years old. (I was just playing around with some numbers for this and actually got the right answer :whistling: )
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galaxygamer
2 posts
galaxygamer
#10 h  Mar 9, 2023, 10:39 PM
Y by
I got 20 because Prancer + Dasher = 60 and 60/prancer + dasher = 60/3=20
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SoundersTID
409 posts
SoundersTID
#11 h  Apr 3, 2023, 12:19 AM
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Answer to the right $\Longrightarrow$This is the answer
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MathPerson12321
3749 posts
MathPerson12321
#12 h  Jun 15, 2023, 10:15 PM
Y by
BabaLama wrote:
nellis25 wrote:
The sum of Dancer's, Dasher's, and Prancer's ages is 60. When Prancer was born, Dasher's age was double that of Dancer's. How old is Dancer today?
https://data.artofproblemsolving.com/images/keeplearning/Week%2087_Website_2_Age%20Conundrum.png

20 because 2x + x + 0 = 60 which simplifies to 3x = 60 which equals to x = 20.

Sorry for double posting (this solution should be better this time around).
Used same way.
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JumpingJaguars
17 posts
JumpingJaguars
#13 h  Aug 31, 2023, 5:49 PM
Y by
dancer is 20
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CrazyCat30
77 posts
CrazyCat30
#14 h  Nov 23, 2023, 12:01 AM
Y by
My Solution
This post has been edited 3 times. Last edited by CrazyCat30, Nov 23, 2023, 12:02 AM
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srg2015
209 posts
srg2015
#15 h  Aug 12, 2024, 3:30 AM
Y by
JumpingJaguars wrote:
dancer is 20

I think this is correct
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kabir_sahil
137 posts
kabir_sahil
#16 h  Aug 22, 2024, 2:13 AM
Y by
yeah it should be
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