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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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RGB chessboard
BR1F1SZ   0
5 minutes ago
Source: 2025 Argentina TST P3
A $100 \times 100$ board has some of its cells coloured red, blue, or green. Each cell is coloured with at most one colour, and some cells may remain uncoloured. Additionally, there is at least one cell of each colour. Two coloured cells are said to be friends if they have different colours and lie in the same row or in the same column. The following conditions are satisfied:
[list=i]
[*]Each coloured cell has exactly three friends.
[*]All three friends of any given coloured cell lie in the same row or in the same column.
[/list]
Determine the maximum number of cells that can be coloured on the board.
0 replies
BR1F1SZ
5 minutes ago
0 replies
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[ELMO2] The Multiplication Table
v_Enhance   26
N 38 minutes ago by de-Kirschbaum
Source: ELMO 2015, Problem 2 (Shortlist N1)
Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \]
Proposed by Yang Liu
26 replies
1 viewing
v_Enhance
Jun 27, 2015
de-Kirschbaum
38 minutes ago
J
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Geometry
Whatisthepurposeoflife   0
an hour ago
Source: Lithuania TST 2013
From the point of intersection of the bisectors of the acute triangle ABC (where AB > AC), the base of the perpendicular descending from the point of intersection of the bisectors of the acute triangle ABC to the side BC is the point D. Find the ratio BD : BA if AD is the bisector of the angle BAC.
0 replies
Whatisthepurposeoflife
an hour ago
0 replies
J
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Sets with ab+1-closure
pieater314159   28
N an hour ago by john0512
Source: ELMO 2019 Problem 5, 2019 ELMO Shortlist N3
Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite.

Proposed by Carl Schildkraut
28 replies
pieater314159
Jun 25, 2019
john0512
an hour ago
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realanalysis
ay19bme   1
N Today at 3:42 PM by alexheinis
........
1 reply
ay19bme
Today at 2:22 PM
alexheinis
Today at 3:42 PM
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real analysis
ay19bme   4
N Today at 1:55 PM by ay19bme
.............
4 replies
ay19bme
Yesterday at 7:11 PM
ay19bme
Today at 1:55 PM
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An interesting limit
Alphaamss   6
N Today at 1:54 PM by solyaris
Suppose $$x_1=\frac\pi2,\quad x_{n+1}=x_n-\frac{\sin x_n}{n+1},$$I can prove that the sequence $\{nx_n\}$ is convergent by monotone bounded convergence theorem.
Is there any method to compute the limit of $\{nx_n\}$, or give the asymptotic representation of $\{nx_n\}$? Any help and hints will welcome!
6 replies
Alphaamss
Apr 3, 2025
solyaris
Today at 1:54 PM
J
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Unique global minimum points
chirita.andrei   1
N Today at 9:39 AM by chirita.andrei
Source: Own. Proposed for Romanian National Olympiad 2025.
Let $f\colon[0,1]\rightarrow \mathbb{R}$ be a continuous function. Suppose that for each $t\in(0,1)$, the function \[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]has an unique global minimum point, which we will denote by $g(t)$. Prove that if $\lim\limits_{t\to 0}g(t)=0$, then $g$ is constant zero.
1 reply
chirita.andrei
Apr 2, 2025
chirita.andrei
Today at 9:39 AM
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abc=a+b+c in ring
Miquel-point   2
N Today at 9:31 AM by RobertRogo
Source: RNMO SHL 2025, grade 12
In which finite rings can we find three (not necessarily distinct) nonzero elements so that their sum equals their product?

David-Andrei Anghel
2 replies
Miquel-point
Apr 6, 2025
RobertRogo
Today at 9:31 AM
J
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Determine the chromatic number of the Euclidean plane
paxtonw   1
N Today at 7:33 AM by solyaris
What is the smallest number of colors required to color every point in ℝ² so that no two points exactly 1 unit apart share the same color?
1 reply
paxtonw
Yesterday at 3:03 PM
solyaris
Today at 7:33 AM
J
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Simple limit with standard recurrence
AndreiVila   2
N Today at 7:33 AM by Tung-CHL
Source: Romanian District Olympiad 2025 11.1
Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$Mathematical Gazette
2 replies
AndreiVila
Mar 8, 2025
Tung-CHL
Today at 7:33 AM
J
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Inequality
Snoop76   2
N Today at 6:21 AM by Snoop76
Source: Own
Show that:$$(2n+1)!!\left(1+\frac 1 {2n}\right)^n>\sum_{k=0}^n (2k+1)!!{n\choose k}, n>0$$
2 replies
Snoop76
Feb 28, 2025
Snoop76
Today at 6:21 AM
J
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Matrices and Determinants
Saucepan_man02   4
N Today at 5:45 AM by Saucepan_man02
Hello

Can anyone kindly share some problems/handouts on matrices & determinants (problems like Putnam 2004 A3, which are simple to state and doesnt involve heavy theory)?

Thank you..
4 replies
Saucepan_man02
Apr 4, 2025
Saucepan_man02
Today at 5:45 AM
J
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Galois theory
ILOVEMYFAMILY   3
N Today at 5:34 AM by yofro
Prove that there does not exist a positive integer \( n \) such that the \( n \)th cyclotomic field over \( \mathbb{Q} \) is an extension of the field \( \mathbb{Q}(\sqrt[3]{5}) \).
3 replies
ILOVEMYFAMILY
Apr 4, 2025
yofro
Today at 5:34 AM
J
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J
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Angle BDC is 24 degrees
Ankoganit   3
N Jan 18, 2017 by AlexLewandowski
Source: India Postal Set 2 P1 2016
Let $ABCD$ be a convex quadrilateral in which $$\angle BAC = 48^{\circ}, \angle CAD = 66^{\circ}, \angle CBD = \angle DBA.$$Prove that $\angle BDC = 24^{\circ}$.
3 replies
Ankoganit
Jan 18, 2017
AlexLewandowski
Jan 18, 2017
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Angle BDC is 24 degrees
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Reveal topic tags
geometry
angle
Angle Chasing
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G H BBookmark kLocked kLocked NReply
Source: India Postal Set 2 P1 2016
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Ankoganit
3070 posts
Ankoganit
#1 h  Jan 18, 2017, 2:59 AM • 1 Y
Y by Adventure10
Let $ABCD$ be a convex quadrilateral in which $$\angle BAC = 48^{\circ}, \angle CAD = 66^{\circ}, \angle CBD = \angle DBA.$$Prove that $\angle BDC = 24^{\circ}$.
This post has been edited 1 time. Last edited by Ankoganit, Jan 18, 2017, 3:08 AM
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AlexLewandowski
157 posts
AlexLewandowski
#2 h  Jan 18, 2017, 7:12 AM • 2 Y
Y by Adventure10, Mango247
Let ABD = CBD = a. Extend ray BA to X and BC to Y. Simple angle chasing gives us DAX = DAC = 66, or AD is the exterior angle bisector of BAC.
So, D is the B-excenter of triangle ABC. Thus, DCA = DCY = b (say).
Now let us perform some trivial angle chasing.
Using the exterior angle sum property, we get b = a + BDC and 66 = a + ADB.
So, b + 66 = 2a + ADC,
or, 180 - ADC = 2a + ADC,
or, a + ADC = 90.
Hence, we get BDC = ADC - ADB = 90 - a - ADB = 90 - 66 = 24. :) :) :)
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sunken rock
4380 posts
sunken rock
#3 h  Jan 18, 2017, 2:19 PM • 2 Y
Y by Adventure10, Mango247
Excellent observation, Alex; to complete, if $I$ was the incenter of $\triangle ABC$, then $AICD$ is cyclic, and $\angle BDC=\angle IAC=24^\circ$, done.

Best regards,
sunken rock
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AlexLewandowski
157 posts
AlexLewandowski
#4 h  Jan 18, 2017, 4:06 PM • 2 Y
Y by Adventure10, Mango247
Oh...that was even better
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