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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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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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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Prove this relation in triangle ABC
Entrepreneur   1
N a few seconds ago by MathIQ.
Source: Conjectured by me
In $\Delta ABC,$ prove that$$\color{blue}{2\angle A=3\angle B\implies c^3a^3(c+a)^2=b^2(c^2+a^2-b^2+2ca)(c^2+a^2-b^2)^2.}$$
1 reply
Entrepreneur
Aug 1, 2024
MathIQ.
a few seconds ago
J
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45 degrees everywhere
Rijul saini   12
N 5 minutes ago by guptaamitu1
Source: India IMOTC 2024 Day 2 Problem 2
Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$.

Proposed by Atul Shatavart Nadig and Shantanu Nene
12 replies
Rijul saini
May 31, 2024
guptaamitu1
5 minutes ago
J
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equation 2025
mohamed-adam   1
N 12 minutes ago by MathIQ.
Source: own
Find all positive integers $a,b$ such that $$a^8-2b^5=2025b$$
1 reply
mohamed-adam
5 hours ago
MathIQ.
12 minutes ago
J
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Graph Theory
ABCD1728   1
N 20 minutes ago by ABCD1728
Can anyone provide the PDF version of "Graphs: an introduction" by Radu Bumbacea (XYZ press), thanks!
1 reply
ABCD1728
Yesterday at 5:10 AM
ABCD1728
20 minutes ago
J
No more topics!
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\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.
parmenides51   7
N Apr 30, 2025 by lightsynth123
Source: Singapore Open Math Olympiad 2017 2nd Round p3 SMO
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
7 replies
parmenides51
Mar 26, 2020
lightsynth123
Apr 30, 2025
J
\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.
G H J
Reveal topic tags
Integer
number theory
Sum
Sum of Squares
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G H BBookmark kLocked kLocked NReply
Source: Singapore Open Math Olympiad 2017 2nd Round p3 SMO
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parmenides51
30653 posts
parmenides51
#1 h  Mar 26, 2020, 5:08 PM
Y by
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
This post has been edited 1 time. Last edited by parmenides51, Mar 28, 2020, 3:42 AM
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Alphatrion
816 posts
Alphatrion
#2 h  Mar 26, 2020, 5:13 PM
Y by
parmenides51 wrote:
Find the smallest positive integer $n$ so that $$\sqrt{\frac{(n+1)(2n+1)}{6}}$$is an integer.
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golema11
83 posts
golema11
#3 h  Mar 26, 2020, 5:17 PM
Y by
$n=1$ satisfy
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usernameyourself
4390 posts
usernameyourself
#4 h  Mar 26, 2020, 5:18 PM
Y by
parmenides51 wrote:
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

[see also here for more unsolved Singapore Contests]

Is it not just 1?
Z K Y
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parmenides51
30653 posts
parmenides51
#5 h  Mar 26, 2020, 5:24 PM • 1 Y
Y by sabkx
by the time is has written 2 in the sum it means more than 2, that was the exact wording anyway

the result is
This post has been edited 1 time. Last edited by parmenides51, Mar 26, 2020, 5:45 PM
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i3435
1350 posts
i3435
#6 h  Mar 26, 2020, 5:46 PM
Y by
Sol
This post has been edited 2 times. Last edited by i3435, Mar 26, 2020, 6:05 PM
Z K Y
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Gryphos
1702 posts
Gryphos
#7 h  Mar 26, 2020, 6:59 PM
Y by
The exact same problem has been posted yesterday, see here.
Z K Y
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lightsynth123
19 posts
lightsynth123
#8 h  Apr 30, 2025, 12:57 AM
Y by
note that the answer is not $1$
This post has been edited 1 time. Last edited by lightsynth123, Apr 30, 2025, 12:58 AM
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