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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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jlacosta
Apr 2, 2025
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EMC Wrangle Favorites #1
peace09   6
N 2 minutes ago by JamesYMath
Is it possible to dissect an isosceles right triangle into multiple similar triangles such that none of them are congruent? If so, provide an example. If not, prove it is impossible.
6 replies
peace09
Jul 28, 2022
JamesYMath
2 minutes ago
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STATE SOLUTIONS AND STUFF DROPPED!!!
Soupboy0   46
N 6 minutes ago by PikaPika999
https://www.mathcounts.org/resources/past-competitions
46 replies
+1 w
Soupboy0
Friday at 5:44 PM
PikaPika999
6 minutes ago
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The daily problem!
Leeoz   62
N 8 minutes ago by huajun78
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]

Past Problems!
62 replies
Leeoz
Mar 21, 2025
huajun78
8 minutes ago
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Math Problem I cant figure out how to do without bashing
equalsmc2   0
21 minutes ago
Hi,
I cant figure out how to do these 2 problems without bashing. Do you guys have any ideas for an elegant solution? Thank you!
Prob 1.
An RSM sports field has a square shape. Poles with letters M, A, T, H are located at the corners of the square (see the diagram). During warm up, a student starts at any pole, runs to another pole along a side of the square or across the field along diagonal MT (only in the direction from M to T), then runs to another pole along a side of the square or along diagonal MT, and so on. The student cannot repeat a run along the same side/diagonal of the square in the same direction. For instance, she cannot run from M to A twice, but she can run from M to A and at some point from A to M. How many different ways are there to complete the warm up that includes all nine possible runs (see the diagram)? One possible way is M-A-T-H-M-H-T-A-M-T (picture attached)

Prob 2.
In the expression 5@5@5@5@5 you replace each of the four @ symbols with either +, or, or x, or . You can insert one or more pairs of parentheses to control the order of operations. Find the second least whole number that CANNOT be the value of the resulting expression. For example, each of the numbers 25=5+5+5+5+5 and 605+(5+5)×5+5 can be the value of the resulting expression.

Prob 3. (This isnt bashing I don't understand how to do it though)
Suppose BC = 3AB in rectangle ABCD. Points E and F are on side BC such that BE = EF = FC. Compute the sum of the degree measures of the four angles EAB, EAF, EAC, EAD.

P.S. These are from an RSM olympiad. The answers are
0 replies
equalsmc2
21 minutes ago
0 replies
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2017 Kosovo MO 9th grade
XbenX   5
N Apr 26, 2023 by Rijadinho
5.
Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25.
Find the distance from T to D.
5 replies
XbenX
Jun 21, 2017
Rijadinho
Apr 26, 2023
J
2017 Kosovo MO 9th grade
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Reveal topic tags
geometry
rectangle
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XbenX
590 posts
XbenX
#1 h  Jun 21, 2017, 7:34 PM • 1 Y
Y by Adventure10
5.
Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25.
Find the distance from T to D.
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Mathnerd1223334444
962 posts
Mathnerd1223334444
#2 h  Jun 21, 2017, 8:15 PM • 1 Y
Y by Adventure10
British Flag Theorem
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equinox145111
3342 posts
equinox145111
#3 h  Jun 21, 2017, 8:16 PM • 2 Y
Y by Adventure10, Mango247
Oh that sounds like a fun theorem... What is it?
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Mathnerd1223334444
962 posts
Mathnerd1223334444
#4 h  Jun 21, 2017, 8:20 PM • 2 Y
Y by Adventure10, Mango247
It says that $AT^2+CT^2=BT^2+DT^2$, meaning that $225+625=400+DT^2$, so $DT=\sqrt{450}=\boxed{15\sqrt{2}}$.
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Mathster360
1308 posts
Mathster360
#5 h  Jun 22, 2017, 3:26 AM • 1 Y
Y by Adventure10
equinox145111 wrote:
Oh that sounds like a fun theorem... What is it?

https://artofproblemsolving.com/wiki/index.php?title=Category:Theorems

See here to learn more about theorems found in problem solving.
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Rijadinho
35 posts
Rijadinho
#6 h  Apr 26, 2023, 6:43 PM
Y by
$AT^2+CT^2=BT^2+DT^2 \Rightarrow 15^2+25^2=20^2+DT^2$
$225+625=400+DT^2 \Rightarrow DT^2=450 \Rightarrow DT= \sqrt{450}$ $\blacksquare$
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