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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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Name of a point on a circle
clarkculus   1
N 3 hours ago by martianrunner
Is there a name for the point $P'$ with respect to a circle $\Gamma$, a diameter $\ell$, and a given point $P$, such that $P'$ is the reflection of the $P$-antipode about $\ell$? Equivalently, $P'$ is the the other intersection of $\Gamma$ and the line through $P$ parallel to $\ell$.
1 reply
clarkculus
3 hours ago
martianrunner
3 hours ago
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A Collection of Good Problems from my end
SomeonecoolLovesMaths   5
N 4 hours ago by Math-lover1
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3
5 replies
SomeonecoolLovesMaths
Yesterday at 8:16 AM
Math-lover1
4 hours ago
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GCD of consecutive terms
nsato   33
N 6 hours ago by reni_wee
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
33 replies
nsato
Mar 14, 2006
reni_wee
6 hours ago
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Largest Divisor
4everwise   19
N 6 hours ago by reni_wee
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
19 replies
4everwise
Dec 22, 2005
reni_wee
6 hours ago
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Can anyone solve this binomial identity
sasu1ke   0
Yesterday at 6:19 PM


\[ \sum_{0 \leq k \leq l} (l - k) \binom{m}{k} \binom{q + k}{n} = \binom{l + q + 1}{m + n + 1}, \]\[ \text{integers } l, m \geq 0,\quad \text{integers } n \geq q \geq 0. \]
0 replies
sasu1ke
Yesterday at 6:19 PM
0 replies
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Rolles theorem
sasu1ke   4
N Yesterday at 6:13 PM by sasu1ke

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[ f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1. \]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

4 replies
sasu1ke
Saturday at 9:00 PM
sasu1ke
Yesterday at 6:13 PM
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[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1
parmenides51   3
N Yesterday at 6:04 PM by iamahana008
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?

(Here $[t]$ denotes the area of the geometrical figure$ t$.)
3 replies
parmenides51
Jan 18, 2021
iamahana008
Yesterday at 6:04 PM
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Putnam 1947 B1
sqrtX   3
N Yesterday at 5:38 PM by Levieee
Source: Putnam 1947
Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$
$$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$Prove that
$$\lim_{x\to \infty} f(x)$$exists and is less than $1+ \frac{\pi}{4}.$
3 replies
sqrtX
Apr 3, 2022
Levieee
Yesterday at 5:38 PM
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Sequences and GCD problem
BBNoDollar   0
Yesterday at 3:23 PM
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
Yesterday at 3:23 PM
0 replies
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Sum of digits is 18
Ecrin_eren   13
N Yesterday at 3:20 PM by NamelyOrange
How many 5 digit numbers are there such that sum of its digits is 18
13 replies
Ecrin_eren
Saturday at 1:10 PM
NamelyOrange
Yesterday at 3:20 PM
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Inequalities
sqing   1
N Yesterday at 3:18 PM by DAVROS
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
1 reply
sqing
Yesterday at 12:46 PM
DAVROS
Yesterday at 3:18 PM
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Double Sum
P162008   2
N Yesterday at 3:01 PM by Etkan
Evaluate $\sum_{a=1}^{\infty} \sum_{b=1}^{a + 2} \frac{1}{ab(a + 2)}.$
2 replies
P162008
Yesterday at 12:19 PM
Etkan
Yesterday at 3:01 PM
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Summation
Saucepan_man02   4
N Yesterday at 2:47 PM by Etkan
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
4 replies
Saucepan_man02
Saturday at 9:40 AM
Etkan
Yesterday at 2:47 PM
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an algebra problem
Asyrafr09   2
N Yesterday at 12:03 PM by pooh123
Determine all real number($x,y,z$) that satisfy
$$x=1+\sqrt{y-z^2}$$$$y=1+\sqrt{z-x^2}$$$$z=1+\sqrt{x-y^2}$$
2 replies
Asyrafr09
Yesterday at 10:05 AM
pooh123
Yesterday at 12:03 PM
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How to solve this problem
xiangovo   1
N Apr 21, 2025 by loup blanc
Source: website
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
1 reply
xiangovo
Mar 19, 2025
loup blanc
Apr 21, 2025
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How to solve this problem
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Reveal topic tags
algebraic geometry
superior algebra
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Source: website
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xiangovo
1 post
xiangovo
#1 h  Mar 19, 2025, 11:54 AM
Y by
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
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loup blanc
3591 posts
loup blanc
#2 h  Apr 21, 2025, 11:09 AM
Y by
With a computer, I find
for $r\leq 5$, the number of such points (up to scaling) over $\mathbb{F}_{5^r}$ is $5^r+1$.
Then I conjectured that the required result is $5^{18}+1$, but this is probably false. -see below-
EDIT 1. The considered projective curve is the Klein quartic over $\mathbb{F}_q$.
Indeed, the number of its points is $q+1$ except when $q=1 \mod 7$.
Unfortunately $5^r=1 \mod 7$ when $r=6,12,18,\cdots$.
EDIT 2. Over $\mathbb{F}_{5^6}$, the number of points is $16376$ and not $5^6+1=15626$.

I think that @ xiangovo is a joker.
This post has been edited 3 times. Last edited by loup blanc, Apr 21, 2025, 5:55 PM
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