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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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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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[*]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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Inequalities
sqing   4
N 2 hours ago by sqing
Let $ a,b> 0 ,\frac{a}{2b+1}+\frac{b}{3}+\frac{1}{2a+1} \leq 1.$ Prove that
$$ a^2+b^2 -ab\leq 1$$$$ a^2+b^2 +ab \leq3$$Let $ a,b,c> 0 , \frac{a}{2b+1}+\frac{b}{2c+1}+\frac{c}{2a+1} \leq 1.$ Prove that
$$ a +b +c +abc \leq 4$$
4 replies
sqing
Today at 3:11 AM
sqing
2 hours ago
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Inequalities
sqing   19
N 2 hours ago by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
19 replies
sqing
May 21, 2025
sqing
2 hours ago
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Primes and automorphisms
CatalinBordea   2
N 3 hours ago by a_0a
Source: Romanian District Olympiad 2016, Grade XII, Problem 3
Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $
2 replies
CatalinBordea
Oct 5, 2018
a_0a
3 hours ago
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[19th PMO Area Stage I. 11]
ACalculationError   2
N 3 hours ago by Shan3t
How many real number \(x\) satisfy
\[ (\lvert x^2-12x+20 \rvert^{\log x^2})^{-1+\log x}=\lvert x^2-12x+20 \rvert^{1+\log(1/x)}? \]
Answer Confirmation
2 replies
ACalculationError
3 hours ago
Shan3t
3 hours ago
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helpppppppp me
stupid_boiii   2
N 3 hours ago by pacoga
Given triangle ABC. The tangent at ? to the circumcircle(ABC) intersects line BC at point T. Points D,E satisfy AD=BD, AE=CE, and ∠CBD=∠BCE<90 ∘ . Prove that D,E,T are collinear.
2 replies
stupid_boiii
May 22, 2025
pacoga
3 hours ago
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a tst 2013 test
Math2030   2
N 3 hours ago by wh0nix
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
2 replies
Math2030
Today at 5:26 AM
wh0nix
3 hours ago
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[Sipnayan SHS] Written Round, Average, #4.6
LilKirb   2
N 3 hours ago by LilKirb
Define the function
\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]where
\[a_k = \begin{cases} 3^k, & \text{if } k \text{ is even}, \\ 0, & \text{if } k \text{ is odd}. \end{cases} \]Find the remainder when \( f(10^9 + 2) \) is divided by \( 2^{20} + 1 \)
2 replies
LilKirb
5 hours ago
LilKirb
3 hours ago
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Japanese Olympiad
parkjungmin   7
N 3 hours ago by Gauler
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
7 replies
parkjungmin
May 10, 2025
Gauler
3 hours ago
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A rather difficult question
BeautifulMath0926   5
N 4 hours ago by evankuang
I got a difficult equation for users to solve:
Find all functions f: R to R, so that to all real numbers x and y,
1+f(x)f(y)=f(x+y)+f(xy)+xy(x+y-2) holds.
5 replies
BeautifulMath0926
Apr 13, 2025
evankuang
4 hours ago
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Inequalities
sqing   1
N 4 hours ago by sqing
Let $ a,b>0 $ and $ \frac { a}{b} +\frac {4b}{a}=5. $ Prove that
$$ \frac {a^{2}+2}{a+2b} + \frac {b^{2}+\frac{1}{2}}{a} \geq \frac{1}{6}\sqrt {\frac {385}{2} }$$$$ \frac {a^{2}+2}{a+2b} + \frac {b^{2}+\frac{1}{3}}{a} \geq \frac {5\sqrt {7}}{6} $$$$ \frac {a^{2}+2}{a+2b} + \frac {b^{2}+\frac{1}{5}}{a} \geq \frac {\sqrt {161}}{6} $$$$ \frac {a^{2}+2}{a+2b} + \frac {b^{2}+3}{a} \geq \frac {\sqrt {455}}{6} $$$$ \frac {a^{2}+2}{a+2b} + \frac {b^{2}+5}{a} \geq \frac {\sqrt {665}}{6} $$
1 reply
sqing
4 hours ago
sqing
4 hours ago
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Original Problem Unit 1 - Logarithms
ACalculationError   2
N 5 hours ago by ACalculationError
Let \( a, b, c \) be positive real numbers such that \( abc = 1 \). Compute the minimum value of \(\log(1 + a^2) + \log(1 + b^2) + \log(1 + c^2).\)

Answer Confirmation
2 replies
ACalculationError
5 hours ago
ACalculationError
5 hours ago
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[Sipnayan SHS] Written Round, Difficult, #4.13
LilKirb   0
5 hours ago
The remainder when $3 \times 6 \times \cdots \times 2019 \times 2022$ is divided by $2031$ is $r.$ What is the remainder when $r$ is divided by $1000?$
0 replies
LilKirb
5 hours ago
0 replies
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Different Paths Probability
Qebehsenuef   3
N 6 hours ago by Etkan
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?
3 replies
Qebehsenuef
Apr 28, 2025
Etkan
6 hours ago
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Find the expected END time for the given process
superpi   2
N Today at 1:32 PM by Hello_Kitty
This problem suddenly popped up in my head. But I don't know how to deal with it.

There are N bulbs. All the bulbs' available time follows same exponential distribution with parameter lambda(Or any arbitrary distribution with mean $\mu$). We do following operations
1. First, turn on the all $N$ bulbs
2. For each $k >= 2$ bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
3. Repeat 2 until all the bulbs goes out

What is the expected terminate time for the above process for given $N, k, \lambda$?

Or, is there any more conditions to complete the problem?

2 replies
superpi
Yesterday at 4:33 PM
Hello_Kitty
Today at 1:32 PM
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determinant practice
Semcio   1
N Apr 16, 2025 by Saucepan_man02
let $A=(\varepsilon^{i \cdot j})_{ij}$ where$1\leq i,j \leq p$ and $\varepsilon=e^{\frac{2i\pi}{p}}$ is root of unity
calculate $\det(A)$
1 reply
Semcio
Jul 12, 2020
Saucepan_man02
Apr 16, 2025
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determinant practice
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determinant
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Semcio
121 posts
Semcio
#1 h  Jul 12, 2020, 6:03 PM
Y by
let $A=(\varepsilon^{i \cdot j})_{ij}$ where$1\leq i,j \leq p$ and $\varepsilon=e^{\frac{2i\pi}{p}}$ is root of unity
calculate $\det(A)$
This post has been edited 2 times. Last edited by Semcio, Jul 13, 2020, 11:10 AM
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Saucepan_man02
1358 posts
Saucepan_man02
#2 h  Apr 16, 2025, 2:07 AM
Y by
I got $\det(A) = (-1)^{n+1} \prod_{n \ge j>i \ge 1} (\epsilon^j-\epsilon^i)$
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