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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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number theory FE
pomodor_ap   0
12 minutes ago
Source: Own, PDC002-P7
Let $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ be a function such that
$$f(m) + mn + n^2 \mid f(m)^2 + m^2 f(n) + f(n)^2$$for all $m, n \in \mathbb{Z}^+$. Find all such functions $f$.
0 replies
pomodor_ap
12 minutes ago
0 replies
J
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real+ FE
pomodor_ap   0
14 minutes ago
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
0 replies
pomodor_ap
14 minutes ago
0 replies
J
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Is this FE solvable?
ItzsleepyXD   2
N 23 minutes ago by ItzsleepyXD
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
2 replies
ItzsleepyXD
Today at 3:02 AM
ItzsleepyXD
23 minutes ago
J
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AM-GM FE ineq
navi_09220114   2
N 23 minutes ago by navi_09220114
Source: Own. Malaysian IMO TST 2025 P3
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.

Proposed by Ivan Chan Kai Chin
2 replies
navi_09220114
Mar 22, 2025
navi_09220114
23 minutes ago
J
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complex analysis
functiono   1
N 2 hours ago by Mathzeus1024
Source: exam
find the real number $a$ such that

$\oint_{|z-i|=1} \frac{dz}{z^2-z+a} =\pi$
1 reply
functiono
Jan 15, 2024
Mathzeus1024
2 hours ago
J
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Computational Calculus
Munmun5   0
2 hours ago
1. Consider the set of all continuous and infinitely differentiable functions $f$ with domain $[0,2025]$ satisfying $f(0)=0,f'(0)=0,f'(2025)=1$ and $f''$ is strictly increasing on $[0,2025]$ Compute smallest real M such that all functions in this set ,$f(2025)<M$ .
2. Polynomials $A(x)=ax^3+abx^2-4x-c,B(x)=bx^3+bcx^2-6x-a,C(x)=cx^3+cax^2-9x-b$ have local extrema at $b,c,a$ respectively. find $abc$ . Here $a,b,c$ are constants .
3. Let $R$ be the region in the complex plane enclosed by curve $$f(x)=e^{ix}+e^{2ix}+\frac{e^{3ix}}{3}$$for $0\leq x\leq 2\pi$. Compute perimeter of $R$ .
0 replies
Munmun5
2 hours ago
0 replies
J
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Why is this series not the Fourier series of some Riemann integrable function
tohill   0
4 hours ago
$\sum_{n=1}^{\infty}{\frac{\sin nx}{\sqrt{n}}}$ (0<x<2π)
0 replies
tohill
4 hours ago
0 replies
J
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Converging product
mathkiddus   10
N Today at 4:30 AM by HacheB2031
Source: mathkiddus
Evaluate the infinite product, $$\prod_{n=1}^{\infty} \frac{7^n - n}{7^n + n}.$$
10 replies
mathkiddus
Apr 18, 2025
HacheB2031
Today at 4:30 AM
J
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Find the formula
JetFire008   4
N Today at 12:36 AM by HacheB2031
Find a formula in compact form for the general term of the sequence defined recursively by $x_1=1, x_n=x_{n-1}+n-1$ if $n$ is even.
4 replies
JetFire008
Yesterday at 12:23 PM
HacheB2031
Today at 12:36 AM
J
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$f\circ g +g\circ f=0\implies n$ even
al3abijo   4
N Yesterday at 10:37 PM by alexheinis
Let $n$ a positive integer . suppose that there exist two automorphisms $f,g$ of $\mathbb{R}^n$ such that $f\circ g +g\circ f=0$ .
Prove that $n$ is even.
4 replies
al3abijo
Yesterday at 9:05 PM
alexheinis
Yesterday at 10:37 PM
J
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2025 OMOUS Problem 6
enter16180   2
N Yesterday at 9:06 PM by loup blanc
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $A=\left(a_{i j}\right)_{i, j=1}^{n} \in M_{n}(\mathbb{R})$ be a positive semi-definite matrix. Prove that the matrix $B=\left(b_{i j}\right)_{i, j=1}^{n} \text {, where }$ $b_{i j}=\arcsin \left(x^{i+j}\right) \cdot a_{i j}$, is also positive semi-definite for all $x \in(0,1)$.
2 replies
enter16180
Apr 18, 2025
loup blanc
Yesterday at 9:06 PM
J
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Sum of multinomial in sublinear time
programjames1   0
Yesterday at 7:45 PM
Source: Own
A frog begins at the origin, and makes a sequence of hops either two to the right, two up, or one to the right and one up, all with equal probability.

1. What is the probability the frog eventually lands on $(a, b)$?

2. Find an algorithm to compute this in sublinear time.
0 replies
programjames1
Yesterday at 7:45 PM
0 replies
J
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Find the answer
JetFire008   1
N Yesterday at 6:42 PM by Filipjack
Source: Putnam and Beyond
Find all pairs of real numbers $(a,b)$ such that $ a\lfloor bn \rfloor = b\lfloor an \rfloor$ for all positive integers $n$.
1 reply
JetFire008
Yesterday at 12:31 PM
Filipjack
Yesterday at 6:42 PM
J
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Pyramid packing in sphere
smartvong   2
N Yesterday at 4:23 PM by smartvong
Source: own
Let $A_1$ and $B$ be two points that are diametrically opposite to each other on a unit sphere. $n$ right square pyramids are fitted along the line segment $\overline{A_1B}$, such that the apex and altitude of each pyramid $i$, where $1\le i\le n$, are $A_i$ and $\overline{A_iA_{i+1}}$ respectively, and the points $A_1, A_2, \dots, A_n, A_{n+1}, B$ are collinear.

(a) Find the maximum total volume of $n$ pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of $n$.

(b) Find the maximum total volume of $n$ pyramids that can be fitted in the sphere, in terms of $n$.

(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as $n$ tends to infinity.

Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
2 replies
smartvong
Apr 13, 2025
smartvong
Yesterday at 4:23 PM
J
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J
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2022 IMOC Problems
CrazyInMath   16
N Sep 25, 2022 by CrazyInMath
Source: 2022 IMOC
Hello everyone. Here are the problems from 2022 IMOC.
This year the camp was held from August 15 to August 21, but USJL was too busy to post the problems, so I posted them instead. Sorry for the big delay.
This year's shortlist has 6 problems in each category.

My English and LaTeX skill is not so proficient, so there may be errors. Tell me if you find any.

What is IMOC

edit: the FE problems had been fixed.
edit: Fixed a minor typo in G2 and a severe typo in G3.
16 replies
CrazyInMath
Sep 5, 2022
CrazyInMath
Sep 25, 2022
J
2022 IMOC Problems
G H J
Reveal topic tags
IMOC
L
G H BBookmark kLocked kLocked NReply
Source: 2022 IMOC
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CrazyInMath
445 posts
CrazyInMath
#1 h  Sep 5, 2022, 12:49 PM • 3 Y
Y by rama1728, David-Vieta, parmenides51
Hello everyone. Here are the problems from 2022 IMOC.
This year the camp was held from August 15 to August 21, but USJL was too busy to post the problems, so I posted them instead. Sorry for the big delay.
This year's shortlist has 6 problems in each category.

My English and LaTeX skill is not so proficient, so there may be errors. Tell me if you find any.

What is IMOC

edit: the FE problems had been fixed.
edit: Fixed a minor typo in G2 and a severe typo in G3.
Attachments:
2022 IMOC.pdf (55kb)
This post has been edited 2 times. Last edited by CrazyInMath, Sep 24, 2022, 1:47 PM
Reason: fixing typos
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ZETA_in_olympiad
2211 posts
ZETA_in_olympiad
#2 h  Sep 5, 2022, 12:59 PM • 1 Y
Y by Mango247
Who will post the problems in HSO threads?
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CrazyInMath
445 posts
CrazyInMath
#3 h  Sep 5, 2022, 1:03 PM
Y by
ZETA_in_olympiad wrote:
Who will post the problems in HSO threads?

I will post the C and N problems today, but maybe later.
The A and G will be posted tomorrow (as long as nobody else post them).
Because I don't want to post too much a day.
This post has been edited 1 time. Last edited by CrazyInMath, Sep 5, 2022, 1:04 PM
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ZETA_in_olympiad
2211 posts
ZETA_in_olympiad
#4 h  Sep 5, 2022, 1:12 PM
Y by
Error: in problems 1.3-1.6 you should mention that "it holds for all $x,y$" etc.

P.S. I may post the A problems then.
This post has been edited 1 time. Last edited by ZETA_in_olympiad, Sep 5, 2022, 1:16 PM
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parmenides51
30630 posts
parmenides51
#5 h  Sep 5, 2022, 1:14 PM • 2 Y
Y by CrazyInMath, Mango247
I shall post the geo problems
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CrazyInMath
445 posts
CrazyInMath
#6 h  Sep 5, 2022, 1:16 PM • 1 Y
Y by Mango247
ZETA_in_olympiad wrote:
Error: in problems 1.3-1.6 you should mention that "it holds for all $x,y$" etc.

I would modify it later. If somebody wants to post them before I modify, please add the statements that ZETA_in_olympiad mentioned.
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parmenides51
30630 posts
parmenides51
#7 h  Sep 5, 2022, 1:20 PM
Y by
Quote:
Problem 3.2 (G2, kyou46). The incenter of triangle ABC is I. the circumcircle of ABC is tangent to BC, CA, AB at T, E, F. R is a point on BC such that ∠CIR = 90^o . Set the C−excenter of CER be L. Prove that LT F is conlinear if and only if BEAR is concyclic
in G2, LTF is conlinear, do you mean, points L,F,T are collinear or something else?
This post has been edited 1 time. Last edited by parmenides51, Sep 5, 2022, 1:21 PM
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CrazyInMath
445 posts
CrazyInMath
#8 h  Sep 5, 2022, 1:24 PM • 1 Y
Y by parmenides51
parmenides51 wrote:
Quote:
Problem 3.2 (G2, kyou46). The incenter of triangle ABC is I. the circumcircle of ABC is tangent to BC, CA, AB at T, E, F. R is a point on BC such that ∠CIR = 90^o . Set the C−excenter of CER be L. Prove that LT F is conlinear if and only if BEAR is concyclic
in G2, LTF is conlinear, do you mean, points L,F,T are collinear or something else?

I mean the points L, T, F are colinear
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supercarry
101 posts
supercarry
#9 h  Sep 5, 2022, 1:24 PM • 1 Y
Y by parmenides51
parmenides51 wrote:
Quote:
Problem 3.2 (G2, kyou46). The incenter of triangle ABC is I. the circumcircle of ABC is tangent to BC, CA, AB at T, E, F. R is a point on BC such that ∠CIR = 90^o . Set the C−excenter of CER be L. Prove that LT F is conlinear if and only if BEAR is concyclic
in G2, LTF is conlinear, do you mean, points L,F,T are collinear or something else?

Yes, point $L, T, F$ are collinear iff $BEAR$ are concyclic
This post has been edited 1 time. Last edited by supercarry, Sep 5, 2022, 1:26 PM
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CrazyInMath
445 posts
CrazyInMath
#10 h  Sep 5, 2022, 1:36 PM
Y by
Can somebody post C3 for me? I don't know how to use Tikz on AoPS
Here is the raw C3 code.
C3 code
I would post the NT and C4~6 later
This post has been edited 1 time. Last edited by CrazyInMath, Sep 5, 2022, 1:37 PM
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parmenides51
30630 posts
parmenides51
#11 h  Sep 5, 2022, 1:39 PM • 1 Y
Y by Mango247
g1, g2, g3, g4, g5, g6 have been posted (all geo IMOC shortlists)

enjoy / start solving

IMOC shortlist collection, I am creating the 2022 post collection by the problems already posted
This post has been edited 3 times. Last edited by parmenides51, Sep 5, 2022, 1:54 PM
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ZETA_in_olympiad
2211 posts
ZETA_in_olympiad
#12 h  Sep 5, 2022, 1:44 PM
Y by
@OP I believe there is a sign error and missing parenthesis in A5. I took the liberty of adding it.
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ZETA_in_olympiad
2211 posts
ZETA_in_olympiad
#13 h  Sep 5, 2022, 1:53 PM
Y by
Just a reminder, all the A problems are posted by me,
https://artofproblemsolving.com/community/u930678h2918265p26069649
https://artofproblemsolving.com/community/u930678h2918269p26069665
https://artofproblemsolving.com/community/u930678h2918270p26069674
https://artofproblemsolving.com/community/u930678h2918274p26069694
https://artofproblemsolving.com/community/u930678h2918279p26069722
https://artofproblemsolving.com/community/u930678h2918281p26069750
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CrazyInMath
445 posts
CrazyInMath
#14 h  Sep 5, 2022, 2:46 PM
Y by
The NT problems had been posted.
N1
N2
N3
N4
N5
N6

somebody please post the combinatorics problems, especially C3 (as I don't know how to use Tikz on AoPS, the raw code is in my last post in this thread).
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Quidditch
818 posts
Quidditch
#15 h  Sep 5, 2022, 3:16 PM
Y by
Ok I post C3: https://artofproblemsolving.com/community/c6h2918316_cover_a_board_with_dominos.

(I use asy instead of tikz :))
This post has been edited 1 time. Last edited by Quidditch, Sep 5, 2022, 3:16 PM
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CrazyInMath
445 posts
CrazyInMath
#16 h  Sep 7, 2022, 2:20 PM
Y by
C4~C6 had been posted
C4
C5
C6
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CrazyInMath
445 posts
CrazyInMath
#17 h  Sep 25, 2022, 4:07 AM
Y by
Forgot to mention, problems from the previous year's IMOC can be found here
USJL wrote:
2017 (with a short intro of what IMOC is): https://artofproblemsolving.com/community/c6h1740077p11309077
2018: https://artofproblemsolving.com/community/c6h1740825p11314688
2019: https://artofproblemsolving.com/community/c6h1958463p13536764
2020: https://artofproblemsolving.com/community/c6h2254883p17398793
2021: https://artofproblemsolving.com/community/c6h2645263p22889979

and here are the qualification problems for the 2021 IMOC.
https://artofproblemsolving.com/community/c2746977_2021imoc_qualification
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