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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 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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Where are the Circles?
luminescent   43
N 38 minutes ago by Amkan2022
Source: EGMO 2022/1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
43 replies
luminescent
Apr 9, 2022
Amkan2022
38 minutes ago
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Divisibilty...
Sadigly   0
an hour ago
Source: Azerbaijan Junior NMO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.
0 replies
Sadigly
an hour ago
0 replies
J
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Quadratic system
juckter   35
N 2 hours ago by shendrew7
Source: Mexico National Olympiad 2011 Problem 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

\[a_1^2 + a_1 - 1 = a_2\]\[ a_2^2 + a_2 - 1 = a_3\]\[\hspace*{3.3em} \vdots \]\[a_{n}^2 + a_n - 1 = a_1\]
35 replies
juckter
Jun 22, 2014
shendrew7
2 hours ago
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IMO Shortlist 2012, Geometry 3
lyukhson   75
N 3 hours ago by numbertheory97
Source: IMO Shortlist 2012, Geometry 3
In an acute triangle $ABC$ the points $D,E$ and $F$ are the feet of the altitudes through $A,B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1$ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
75 replies
1 viewing
lyukhson
Jul 29, 2013
numbertheory97
3 hours ago
J
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Another integral limit
RobertRogo   2
N Today at 4:02 PM by Gauler
Source: "Traian Lalescu" student contest 2025, Section A, Problem 3
Let $f \colon [0, \infty) \to \mathbb{R}$ be a function differentiable at 0 with $f(0) = 0$. Find
$$\lim_{n \to \infty} \frac{1}{n} \int_{2^n}^{2^{n+1}} f\left(\frac{\ln x}{x}\right) dx$$
2 replies
1 viewing
RobertRogo
Yesterday at 2:28 PM
Gauler
Today at 4:02 PM
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Numerical methods problems
jjfgtuuu   0
Today at 3:18 PM
Given that $x_1 = \dfrac{1}{\sqrt{2}}$, $x_2 = \dfrac{1}{\sqrt{6}}$, $x_3 = \dfrac{1}{\sqrt{8}}$, $x_4 = \dfrac{1}{\sqrt{10}}$.
Find the approximate value of $\mathrm{A} = \sum\limits_{i=1}^{4}x_i $ and its absolute and relative error, known that its absolute error is equal or lower than $10^{-5}.$
0 replies
jjfgtuuu
Today at 3:18 PM
0 replies
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Group Theory
Stephen123980   2
N Today at 2:23 PM by BadAtMath23
Let G be a group of order $45.$ If G has a normal subgroup of order $9,$ then prove that $G$ is abelian without using Sylow Theorems.
2 replies
Stephen123980
Yesterday at 5:32 PM
BadAtMath23
Today at 2:23 PM
J
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Double integrals
fermion13pi   0
Today at 1:58 PM
Source: Apostol, vol 2
Evaluate the double integral by converting to polar coordinates:

\[ \int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx \]
Change the order of integration and then convert to polar coordinates.

0 replies
fermion13pi
Today at 1:58 PM
0 replies
J
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D1028 : A strange result about linear algebra
Dattier   0
Today at 1:49 PM
Source: les dattes à Dattier
Let $p>3$ a prime number, with $H \subset M_p(\mathbb R), \dim(H)\geq 2$ and $H-\{0\} \subset GL_p(\mathbb R)$, $H$ vector space.

Is it true that $H-\{0\}$ is a group?
0 replies
Dattier
Today at 1:49 PM
0 replies
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Preparing for Putnam level entrance examinations
Cats_on_a_computer   4
N Today at 1:16 PM by Cats_on_a_computer
Non American high schooler in the equivalent of grade 12 here. Where I live, two the best undergraduates program in the country accepts students based on a common entrance exam. The first half of the exam is “screening”, with 4 options being presented per question, each of which one has to assign a True or False. This first half is about the difficulty of an average AIME, or JEE Adv paper, and it is a requirement for any candidate to achieve at least 24/40 on this half for the examiners to even consider grading the second part. The second part consists of long form questions, and I have, no joke, seen them literally rip off, verbatim, Putnam A6s. Some of the problems are generally standard textbook problems in certain undergrad courses but obviously that doesn’t translate it to being doable for high school students. I’ve effectively got to prepare for a slightly nerfed Putnam, if you will, and so I’ve been looking for resources (not just problems) for Putnam level questions. Does anyone have any suggestions?
4 replies
Cats_on_a_computer
Yesterday at 8:32 AM
Cats_on_a_computer
Today at 1:16 PM
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Marginal Profit
NC4723   1
N Today at 10:09 AM by Juno_34
Please help me solve this
1 reply
NC4723
Dec 11, 2015
Juno_34
Today at 10:09 AM
J
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Romania NMO 2023 Grade 11 P1
DanDumitrescu   15
N Today at 5:46 AM by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]
15 replies
DanDumitrescu
Apr 14, 2023
anudeep
Today at 5:46 AM
J
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Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas   11
N Today at 5:27 AM by cappucher
Source: Putnam 1990 A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
11 replies
ahaanomegas
Jul 12, 2013
cappucher
Today at 5:27 AM
J
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Putnam 2000 B4
ahaanomegas   6
N Today at 1:53 AM by mqoi_KOLA
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.
6 replies
ahaanomegas
Sep 6, 2011
mqoi_KOLA
Today at 1:53 AM
J
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J
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A simple evaluation
jayme   2
N Apr 18, 2025 by jayme
Dear Mathlinkers,
a very simple question:

evaluate Arctan (-1 +√2) in degree

Sincerely
Jean-Louis
2 replies
jayme
Apr 18, 2025
jayme
Apr 18, 2025
J
A simple evaluation
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jayme
9792 posts
jayme
#1 h  Apr 18, 2025, 1:57 PM
Y by
Dear Mathlinkers,
a very simple question:

evaluate Arctan (-1 +√2) in degree

Sincerely
Jean-Louis
Z K Y
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zhoujef000
317 posts
zhoujef000
#2 h  Apr 18, 2025, 2:08 PM
Y by
sol
This post has been edited 2 times. Last edited by zhoujef000, Apr 18, 2025, 2:09 PM
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jayme
9792 posts
jayme
#3 h  Apr 18, 2025, 2:18 PM
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Thank you

Jean-Louis
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