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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Intersection of circumcircles of MNP and BOC
Djile   39
N a few seconds ago by bjump
Source: Serbian National Olympiad 2013, Problem 3
Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]
39 replies
Djile
Apr 8, 2013
bjump
a few seconds ago
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Good divisors and special numbers.
Nuran2010   2
N a minute ago by BR1F1SZ
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.
2 replies
Nuran2010
Yesterday at 4:52 PM
BR1F1SZ
a minute ago
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Inspired by old results
sqing   3
N 7 minutes ago by Jamalll
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
3 replies
sqing
Yesterday at 12:29 PM
Jamalll
7 minutes ago
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Question on Balkan SL
Fmimch   0
25 minutes ago
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
0 replies
Fmimch
25 minutes ago
0 replies
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Matrix Row and column relation.
Schro   5
N Yesterday at 5:01 PM by Etkan
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
5 replies
Schro
Monday at 2:54 PM
Etkan
Yesterday at 5:01 PM
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D1023 : MVT 2.0
Dattier   0
Yesterday at 1:04 PM
Source: les dattes à Dattier
Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $$\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$$
Is it true that $$\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$$
0 replies
Dattier
Yesterday at 1:04 PM
0 replies
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Time required for draining out water
Kunihiko_Chikaya   1
N Yesterday at 12:22 PM by Mathzeus1024
Source: Kyoto University 2nd-stage entry exam 2006 /Science, Problem
Let $H>0,\ R>0$ and $O$ be the origin, $P\ (R,\ 0\ ,H).$ In space given the vessel formed by the revolution of the segment $OP$ about $z$-axis. Denote the hight from $O$ to the surface of water by $h.$ When the vessel is filled with water, drain the water out such that the displacement per time is $\sqrt{h}$ at the time of $h,$ that is to say, if the total volume of the water drained out till the time $t$ is $V(t),$ then $\frac{dV}{dt}=\sqrt{h}$ holds. Find the time required for draining out water.
1 reply
Kunihiko_Chikaya
Mar 14, 2006
Mathzeus1024
Yesterday at 12:22 PM
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D1021 : Does this series converge?
Dattier   1
N Yesterday at 9:01 AM by Dattier
Source: les dattes à Dattier
Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?
1 reply
Dattier
Apr 26, 2025
Dattier
Yesterday at 9:01 AM
J
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UC Berkeley Integration Bee 2025 Qualifying Exam
Silver08   18
N Yesterday at 5:29 AM by Silver08
Source: UC Berkeley Integration Bee 2025
Good luck and have fun!!!

1. $$\int_{-\sqrt[3]{2}}^{1}(x^3+x^6)(x^2+2x^5)dx$$
2. $$\int \frac{e^x}{1+e^{2x}}+\frac{x}{1+x^2}dx$$
3. $$\int_0^\pi x^4\cos(x)dx$$
4. $$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\cos^2(x)dx$$
5. $$\int_1^2\frac{\sqrt{x^2+2\sqrt{x^2-1}}}{\sqrt{x^2-1}}dx$$
6. $$\int_2^4\frac{(2-x)}{(x-1)(x-5)}dx$$
7. $$\int_0^\infty \frac{dx}{x^2-2x+2}$$
8. $$\int (e^x+\ln(x))\left(e^x+\frac{1}{x}\right)dx$$
9. $$\int_{-\infty}^{\infty}\frac{\ln(x^2+1)}{x^2}dx$$
10. $$\int_0^1 \frac{dx}{\sqrt{1+\sqrt{x}}-\sqrt{1-\sqrt{x}}}$$
11. $$\int_0^1 2^{\ln(x)}dx$$
12. $$\int \frac{d}{dx}\left [ \frac{e^x}{\ln(x)} \right] \cdot \frac{\ln(x)}{e^x} dx$$
13. $$\int_1^\infty \ln(x)-\frac{1}{2}\ln(x^2+1)dx$$
14. $$\int_0^1 2x\sqrt{x-x^2}dx$$
15. $$\int_0^1 \ln^3(x)dx$$
16. $$\int_0^\pi\frac{\sqrt{\csc(x)-\sin(x)}}{\sqrt{\sin(x)}+1}dx$$
17. $$\int_0^6\frac{x}{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}dx$$
18. $$\int_0^\infty \frac{\sin(x)\cos(2x)\cos^2(x)}{x}dx$$
19. $$\int_0^{\infty}\frac{dx}{1+x+x^2+x^3}$$
20. $$\int_{-\infty}^{\infty}\frac{(x^5-x^3)}{(x^2-1)^4+x^4}dx$$
18 replies
Silver08
Apr 27, 2025
Silver08
Yesterday at 5:29 AM
J
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Polynomial Limit
P162008   1
N Yesterday at 4:05 AM by P162008
Let $p = \lim_{y\to\infty} \left(\frac{2}{y^2} \left(\lim_{z\to\infty} \frac{1}{z^4} \left(\lim_{x\to 0} \frac{((y^2 + y + 1)x^k + 1)^{z^2 + z + 1} - ((z^2 + z + 1)x^k + 1)^{y^2 + y + 1}}{x^{2k}}\right)\right)\right)^y$ where $k \in N$ and $q = \lim_{n\to\infty} \left(\frac{\binom{2n}{n}. n!}{n^n}\right)^{1/n}$ where $n \in N$. Find the value of $p.q.$
1 reply
P162008
Apr 25, 2025
P162008
Yesterday at 4:05 AM
J
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Limit of the sum of a symmetric polynomial evaluated at sequence terms
tom-nowy   0
Yesterday at 1:27 AM
Source: 2018 Osaka City University entrance exam
The real sequence $\{ a_n \}_{n=1,2,3,\ldots} $ satisfies $0 \le a_n \le 1 \, (\forall n \in \mathbb{N} )$ and $\lim_{n\to\infty} a_n = 1$. Find
\[\lim_{n\to\infty} n \left( \;1 \; +\; \sum_{k=1}^n \frac{ (-1)^k}{k+1} \sum_{1 \le j_1 \le j_2 \le \cdots \le j_k \le n} a_{j_1} a_{j_2} \cdots a_{j_k} \right). \]
0 replies
tom-nowy
Yesterday at 1:27 AM
0 replies
J
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Probability
Qebehsenuef   0
Monday at 11:45 PM
Source: OBM
A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?
0 replies
Qebehsenuef
Monday at 11:45 PM
0 replies
J
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Triple Sum
P162008   1
N Monday at 8:11 PM by soryn
Source: From a friend
Evaluate $\Omega = \lim_{n \to \infty} \frac{1}{n^5} \sum_{i=1}^{n} \sum_{k=1}^{2n} \sum_{k=1}^{3n} \frac{i(jk + 6n^2 - 3jn - 2kn) - jkn + n^2(3j + 2k) - 6n^3}{\sqrt{i^2 + j^2 + k^2 - i(a - 2n) + j(b - 4n) + k(c - 6n) + c + 14n^2}}$
1 reply
P162008
Apr 26, 2025
soryn
Monday at 8:11 PM
J
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A Ball-Drawing problem
Vivacious_Owl   9
N Monday at 6:15 PM by Vivacious_Owl
Source: Inspired by a certain daily routine of mine
There are N identical black balls in a bag. I randomly take one ball out of the bag. If it is a black ball, I throw it away and put a white ball back into the bag instead. If it is a white ball, I simply throw it away and do not put anything back into the bag. The probability of getting any ball is the same.
Questions:
1. How many times will I need to reach into the bag to empty it?
2. What is the ratio of the expected maximum number of white balls in the bag to N in the limit as N goes to infinity?
9 replies
Vivacious_Owl
Apr 24, 2025
Vivacious_Owl
Monday at 6:15 PM
J
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J
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Squares in an Octagon
kred9   1
N Apr 15, 2025 by crazydog
Source: 2025 Utah Math Olympiad #1
A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)
1 reply
kred9
Apr 5, 2025
crazydog
Apr 15, 2025
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Squares in an Octagon
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: 2025 Utah Math Olympiad #1
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kred9
1021 posts
kred9
#1 h  Apr 5, 2025, 11:50 PM • 1 Y
Y by PikaPika999
A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)
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crazydog
18 posts
crazydog
#2 h  Apr 15, 2025, 5:54 PM
Y by
it is twelve
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N Quick Reply
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