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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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[*]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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Geometry Proof
strongstephen   6
N 27 minutes ago by strongstephen
Proof that choosing four distinct points at random has an equal probability of getting a convex quadrilateral vs a concave one.
not cohesive proof alert!

NOTE: By choosing four distinct points, that means no three points lie on the same line on the Gaussian Plane.

Start by picking three of the four points. Next, graph the regions where the fourth point would make the quadrilateral convex or concave. In diagram 1 below, you can see the regions where the fourth point would be convex or concave. Of course, there is the centre region (the shaded triangle), but in an infinite plane, the probability the fourth point ends up in the finite region approaches 0.

Next, I want to prove to you the area of convex/concave, or rather, the probability a point ends up in each area, is the same. Referring to the second diagram, you can flip each concave region over the line perpendicular to the angle bisector of which the region is defined. (Just look at it and you'll get what it means.) Now, each concave region has an almost perfect 1:1 probability correspondence to another convex region. The only difference is the finite region (the triangle, shaded). Again, however, the actual significance (probability) of this approaches 0.

If I call each of the convex region's probability P(a), P(c), and P(e) and the concave ones P(b), P(d), P(f), assuming areas a and b are on opposite sides (same with c and d, e and f) you can get:
P(a) = P(b)
P(c) = P(d)
P(e) = P(f)

and P(a) + P(c) + P(e) = P(convex)
and P(b) + P(d) + P(f) = P(concave)

therefore:
P(convex) = P(concave)
6 replies
strongstephen
Today at 4:54 AM
strongstephen
27 minutes ago
J
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n and n+100 have odd number of divisors (1995 Belarus MO Category D P2)
jasperE3   4
N an hour ago by KTYC
Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors.
4 replies
jasperE3
Apr 6, 2021
KTYC
an hour ago
J
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Closed form expression of 0.123456789101112....
ReticulatedPython   3
N 3 hours ago by ReticulatedPython
Is there a closed form expression for the decimal number $$0.123456789101112131415161718192021...$$which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
3 replies
ReticulatedPython
3 hours ago
ReticulatedPython
3 hours ago
J
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primes and perfect squares
Bummer12345   5
N 3 hours ago by Shan3t
If $p$ and $q$ are primes, then can $2^p + 5^q + pq$ be a perfect square?
5 replies
Bummer12345
Yesterday at 5:08 PM
Shan3t
3 hours ago
J
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Putnam 2016 A1
Kent Merryfield   15
N Today at 10:51 AM by anudeep
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 $2016.$
15 replies
Kent Merryfield
Dec 4, 2016
anudeep
Today at 10:51 AM
J
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Determinant is 1
Entrepreneur   2
N Today at 8:27 AM by Entrepreneur
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 $1.$
2 replies
Entrepreneur
Yesterday at 7:14 PM
Entrepreneur
Today at 8:27 AM
J
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Can cos(√2 t) be expressed as a polynomial in cost?
tom-nowy   1
N Today at 7:17 AM by Aiden-1089
Source: Question arising while viewing https://artofproblemsolving.com/community/c51293h3562250
Can $\cos ( \sqrt{2}\, t )$ be expressed as a polynomial in $\cos t$ with real coefficients?
1 reply
tom-nowy
Today at 7:10 AM
Aiden-1089
Today at 7:17 AM
J
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36x⁴ + 12x² - 36x + 13 > 0
fxandi   2
N Today at 7:14 AM by MeKnowsNothing
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
2 replies
fxandi
Yesterday at 10:02 PM
MeKnowsNothing
Today at 7:14 AM
J
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2024 Putnam A1
KevinYang2.71   20
N Today at 5:50 AM by thelateone
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[ 2a^n+3b^n=4c^n. \]
20 replies
KevinYang2.71
Dec 10, 2024
thelateone
Today at 5:50 AM
J
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2025 OMOUS Problem 4
enter16180   2
N Yesterday at 8:57 PM by Acridian9
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Find all matrices $M \in M_{n}(\mathbb{C})$ such that following equality holds

$$ \operatorname{rank}(M)+\operatorname{rank}\left(M^{2023}-M^{2025}\right)=\operatorname{rank}\left(M-M^{2}\right)+\operatorname{rank}\left(M^{2023}+M^{2024}\right) $$
2 replies
enter16180
Apr 18, 2025
Acridian9
Yesterday at 8:57 PM
J
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Poker hand
Aksudon   1
N Yesterday at 6:32 PM by lucaminiati
Problem: In a standard 52-card deck, how many different five-card poker hands are there of 'two pairs'?

Can someone please explain what is logically wrong with the following solution? (It gives double of the right solution which supposed to be 123552).

13\binom{4}{2}*12\binom{4}{2}*44=247104

Thanks
1 reply
Aksudon
Yesterday at 5:14 PM
lucaminiati
Yesterday at 6:32 PM
J
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Sequence with GCD involved
mathematics2004   3
N Yesterday at 5:54 PM by anudeep
Source: 2021 Simon Marais, A2
Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and
\[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \]for all integers $n \ge 1$.
Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$.
Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.
3 replies
mathematics2004
Nov 2, 2021
anudeep
Yesterday at 5:54 PM
J
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Putnam 2000 B2
ahaanomegas   20
N Yesterday at 5:05 PM by reni_wee
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
20 replies
ahaanomegas
Sep 6, 2011
reni_wee
Yesterday at 5:05 PM
J
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f(x)f'(x)≥cos, f(∞)=undef. if f is bounded
jasperE3   2
N Yesterday at 4:21 PM by Rohit-2006
Source: VJIMC 2013 1.1
Let $f:[0,\infty)\to\mathbb R$ be a differentiable function with $|f(x)|\le M$ and $f(x)f'(x)\ge\cos x$ for $x\in[0,\infty)$, where $M>0$. Prove that $f(x)$ does not have a limit as $x\to\infty$.
2 replies
jasperE3
May 30, 2021
Rohit-2006
Yesterday at 4:21 PM
J
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J
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Sequence
lgx57   8
N Apr 30, 2025 by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Apr 30, 2025
J
Sequence
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lgx57
37 posts
lgx57
#1 h  Apr 27, 2025, 12:56 PM
Y by
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
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lgx57
37 posts
lgx57
#2 h  Apr 27, 2025, 12:57 PM
Y by
I can only find that $a_n \sim \sqrt{2n}$.
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aops-g5-gethsemanea2
3456 posts
aops-g5-gethsemanea2
#3 h  Apr 27, 2025, 2:05 PM
Y by
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?
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lgx57
37 posts
lgx57
#4 h  Apr 27, 2025, 2:19 PM
Y by
aops-g5-gethsemanea2 wrote:
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?

Just find a function $f$ ,s.t. $a_n=f(n)$
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steve4916
16 posts
steve4916
#5 h  Apr 27, 2025, 2:24 PM
Y by
now prove me if im wrong but there is no simple closed form for this
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lgx57
37 posts
lgx57
#6 h  Apr 27, 2025, 2:48 PM
Y by
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?
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johnnie.walker
2 posts
johnnie.walker
#7 h  Apr 29, 2025, 1:42 PM
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jasperE3
11296 posts
jasperE3
#8 h  Apr 29, 2025, 3:38 PM
Y by
lgx57 wrote:
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?

why would there be a closed form
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Vivaandax
82 posts
Vivaandax
#9 h  Apr 30, 2025, 5:08 AM
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You can bound the value of a_n quite well (consider IMO Shortlist 1975 Problem 14), but there is not an explicit formula to calculate a_n.
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