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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]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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Problem 3 IMO 2005 (Day 1)
Valentin Vornicu   121
N 13 minutes ago by Rayvhs
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
Hojoo Lee, Korea
121 replies
Valentin Vornicu
Jul 13, 2005
Rayvhs
13 minutes ago
J
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geo problem saved from graveyard
CrazyInMath   1
N 28 minutes ago by Curious_Droid
Source: 3rd KYAC Math-A P5
Given triangle $ABC$ and orthocenter $H$. The foot from $H$ to $BC, CA, AB$ is $D, E, F$ respectively. A point $L$ satisfies that $\odot(LBA)$ and $\odot(LCA)$ are both tangent to $BC$. A circle passing through $B, E$ and tangent to $\odot(BHC)$ intesects $BC$ at another point $P$. $X$ is an arbitrary point on $\odot(PDE)$, and $Y$ is the second intesection point of $\odot(BXE)$ and $\odot(CXD)$.
Prove that $H, Y, L, C$ are concyclic.

Proposed by CrazyInMath.
1 reply
CrazyInMath
Feb 8, 2025
Curious_Droid
28 minutes ago
J
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From a well-known prob
m4thbl3nd3r   3
N 29 minutes ago by aaravdodhia
Find all primes $p$ so that $$\frac{7^{p-1}-1}{p}$$can be a perfect square
3 replies
1 viewing
m4thbl3nd3r
Oct 10, 2024
aaravdodhia
29 minutes ago
J
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weird conditions in geo
Davdav1232   1
N 42 minutes ago by NO_SQUARES
Source: Israel TST 7 2025 p1
Let \( \triangle ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) be a point on \( AC \). Let \( L \) be a point inside the triangle such that \( \angle CLD = 90^\circ \) and
\[ CL \cdot BD = BL \cdot CD. \]Prove that the circumcenter of triangle \( \triangle BDL \) lies on line \( AB \).
1 reply
Davdav1232
2 hours ago
NO_SQUARES
42 minutes ago
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Romanian National Olympiad 1997 - Grade 11 - Problem 2
Filipjack   1
N 5 hours ago by loup blanc
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 2
Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
1 reply
Filipjack
Apr 6, 2025
loup blanc
5 hours ago
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Serious qustion
Thayaden   2
N 6 hours ago by ReticulatedPython
Let $F_n$ be then $n$-th fibbiance number. As $n$ gets bigger and bigger, we have,
$$\frac{F_{n+1}}{F_n}\approx\varphi,$$my question is dose,
$$\lim_{n\rightarrow \infty}\frac{F_{n+1}}{F_n}=\varphi.$$My reservations about this is that $\varphi\in\mathbb{R}\setminus\mathbb{Q}$ and $F_n\in\mathbb{Z}^+$ so $\frac{F_{n+1}}{F_n}\in\mathbb{Q}$. So, if the limit holds, does that mean that if $S$ is a set and $P$ is a set, for each $s\in S$ that $s\not\in P$ we can have, for $\text{Range}(f)=S$ we can have,
$$\lim_{x\rightarrow n}f(x)\in P,$$for some $n$?
2 replies
Thayaden
6 hours ago
ReticulatedPython
6 hours ago
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Putnam 2010 B5
Kent Merryfield   25
N Today at 2:59 PM by Rohit-2006
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
25 replies
Kent Merryfield
Dec 6, 2010
Rohit-2006
Today at 2:59 PM
J
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Determinant problem
Entrepreneur   3
N Today at 2:49 PM by Entrepreneur
Source: Hall & Knight
If a determinant is of $n^{\text{th}}$ order, and if the constituents of its first, second, ..., $n^{\text{th}}$ rows are the first $n$ figurate numbers of the first, second, ..., $n^{\text{th}}$ orders respectively, show that it's value is $1.$
3 replies
Entrepreneur
May 5, 2025
Entrepreneur
Today at 2:49 PM
J
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Integration Bee Kaizo
Calcul8er   56
N Today at 2:16 PM by franklin2013
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
56 replies
Calcul8er
Mar 2, 2025
franklin2013
Today at 2:16 PM
J
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AB=BA if A-nilpotent
KevinDB17   2
N Today at 1:01 PM by loup blanc
Let A,B 2 complex n*n matrices such that AB+I=A+B+BA
If A is nilpotent prove that AB=BA
2 replies
KevinDB17
Mar 30, 2025
loup blanc
Today at 1:01 PM
J
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Putnam 2016 A1
Kent Merryfield   16
N Today at 10:49 AM by sangsidhya
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\](the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$
16 replies
Kent Merryfield
Dec 4, 2016
sangsidhya
Today at 10:49 AM
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Putnam 1954 B1
sqrtX   7
N Today at 3:42 AM by justaguy_69
Source: Putnam 1954
Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.
7 replies
sqrtX
Jul 17, 2022
justaguy_69
Today at 3:42 AM
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What is the limit?
Disjeje   2
N Today at 1:30 AM by Alphaamss
Let’s say An=(sin(n))^n
Does An converge if n reaches infinity?
2 replies
Disjeje
Yesterday at 5:45 AM
Alphaamss
Today at 1:30 AM
J
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Summation
Saucepan_man02   5
N Today at 1:17 AM by Saucepan_man02
If $P = \sum_{r=1}^{50} \sum_{k=1}^{r} (-1)^{r-1} \frac{\binom{50}{r}}{k}$, then find the value of $P$.

Ans
5 replies
Saucepan_man02
May 3, 2025
Saucepan_man02
Today at 1:17 AM
J
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Integral-Summation Duality
Mathandski   3
N Apr 29, 2025 by ihategeo_1969
Source: Friend at school gave it to me
Given a continuous function $f$ such that $f(2x) = 3 f(x)$ and $\int_0^1 f(x) \, dx = 1$, evaluate $\int_1^2 f(x) \, dx$.
3 replies
Mathandski
Apr 28, 2025
ihategeo_1969
Apr 29, 2025
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Integral-Summation Duality
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Source: Friend at school gave it to me
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Mathandski
757 posts
Mathandski
#1 h  Apr 28, 2025, 7:58 PM • 1 Y
Y by compoly2010
Given a continuous function $f$ such that $f(2x) = 3 f(x)$ and $\int_0^1 f(x) \, dx = 1$, evaluate $\int_1^2 f(x) \, dx$.
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zhoujef000
316 posts
zhoujef000
#3 h  Apr 28, 2025, 8:30 PM
Y by
am i high or is this trivial
This post has been edited 1 time. Last edited by zhoujef000, Apr 28, 2025, 9:09 PM
Reason: hehe
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Mathandski
757 posts
Mathandski
#4 h  Apr 29, 2025, 12:45 AM • 1 Y
Y by compoly2010
Darn how did I miss the trivial solution ;-;
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ihategeo_1969
235 posts
ihategeo_1969
#5 h  Apr 29, 2025, 10:09 AM
Y by
See that \[\int_1^2 f(x) dx=\int_0^2 f(x) dx -\int_0^1 f(x) dx=2 \int_0^1 f(2x) dx-1=6 \int_0^1 f(x) dx-1=\boxed{5}\]
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