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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Interesting inequalities
sqing   1
N 27 minutes ago by lbh_qys
Source: Own
Let $ a,b $ be reals such that $ a^2+ab+b^2 =1$ . Prove that
$$ \frac{8}{ 5 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq 1$$$$ \frac{9}{ 5 }\geq\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq 1$$$$ \frac{27}{ 14 }\geq \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq 1$$Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
1 reply
1 viewing
sqing
3 hours ago
lbh_qys
27 minutes ago
J
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A cyclic inequality
KhuongTrang   11
N 30 minutes ago by KhuongTrang
Source: own-CRUX
IMAGE
Link
11 replies
KhuongTrang
Apr 2, 2025
KhuongTrang
30 minutes ago
J
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Divisibility NT FE
CHESSR1DER   1
N 35 minutes ago by CHESSR1DER
Source: Own
Find all functions $f$ $N \iff N$ such for any $a,b$:
$(a+b)|a^{f(b)} + b^{f(a)}$.
1 reply
CHESSR1DER
Yesterday at 7:07 PM
CHESSR1DER
35 minutes ago
J
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hard problem
Cobedangiu   3
N 37 minutes ago by lbh_qys
Let $x,y>0$ and $\dfrac{1}{x}+\dfrac{1}{y}+1=\dfrac{10}{x+y+1}$. Find max $A$ (and prove):
$A=\dfrac{x^2}{y}+\dfrac{y^2}{x}+\dfrac{1}{xy}$
3 replies
Cobedangiu
Today at 5:19 AM
lbh_qys
37 minutes ago
J
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idk12345678 Math Contest
idk12345678   20
N Today at 4:04 AM by crazydog
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
20 replies
idk12345678
Apr 10, 2025
crazydog
Today at 4:04 AM
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Geo Mock #6
Bluesoul   3
N Today at 3:16 AM by dudade
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
3 replies
Bluesoul
Apr 1, 2025
dudade
Today at 3:16 AM
J
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AoPS Volume 2, Problem 262
Shiyul   12
N Today at 2:49 AM by bhavanal
Given that $\color[rgb]{0.35,0.35,0.35}v_1=2$, $\color[rgb]{0.35,0.35,0.35}v_2=4$ and $\color[rgb]{0.35,0.35,0.35} v_{n+1}=3v_n-v_{n-1}$, prove that $\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$, where the terms $\color[rgb]{0.35,0.35,0.35}F_n$ are the Fibonacci numbers.

Can anyone give me hint on how to solve this (not solve the full problem). I'm not sure how to relate the v series to the Fibonacci sequence.

12 replies
Shiyul
Apr 9, 2025
bhavanal
Today at 2:49 AM
J
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Find the angle
pythagorazz   4
N Today at 2:41 AM by dudade
Let $X$ be a point inside equilateral triangle $ABC$ such that $AX=\sqrt{2},BX=3$, and $CX=\sqrt{5}$. Find the measure of $\angle{AXB}$ in degrees.
4 replies
pythagorazz
Yesterday at 9:07 AM
dudade
Today at 2:41 AM
J
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Inequalities
sqing   12
N Today at 2:11 AM by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
12 replies
sqing
Apr 9, 2025
sqing
Today at 2:11 AM
J
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Probability
Ecrin_eren   4
N Today at 1:20 AM by huajun78
In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

4 replies
Ecrin_eren
Apr 3, 2025
huajun78
Today at 1:20 AM
J
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Algebra book recomndaitons
idk12345678   3
N Today at 12:50 AM by idk12345678
Im currently reading EGMO by Evan Chen and i was wondering if there was a similar book for olympiad algebra. I have egmo for geo and aops intermediate c&p for combo, and the intermediate number thoery transcripts for nt, but i couldnt really find anything for alg
3 replies
idk12345678
Yesterday at 10:45 PM
idk12345678
Today at 12:50 AM
J
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Help with Competitive Geometry?
REACHAW   2
N Today at 12:09 AM by alextheadventurer
Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).

It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).

Do you have any advice, resources, or techniques I should try?
2 replies
REACHAW
Yesterday at 11:51 PM
alextheadventurer
Today at 12:09 AM
J
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Algebra Problems
ilikemath247365   7
N Yesterday at 9:56 PM by alexheinis
Find all real $(a, b)$ with $a + b = 1$ such that

$(a + \frac{1}{a})^{2} + (b + \frac{1}{b})^{2} = \frac{25}{2}$.
7 replies
ilikemath247365
Yesterday at 4:52 PM
alexheinis
Yesterday at 9:56 PM
J
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US Puzzle Championship Scorecards
djmathman   2
N Yesterday at 9:20 PM by zhoujef000
Some of the discussion in the Contests & Programs SMT thread reminded me of Scorecards puzzles from the US Puzzle Championship. They behave similarly to "24 game" puzzles, but the allowable operations are slightly expanded.

[quote="Scorecards rules"]
Operations are limited to addition ("+"), subtraction ("-"), multiplication ("x"), division ("/"), and exponentiation ("^").
Decimal points may be used; ...; use minus sign ("-") to indicate negative values. Use parentheses if needed to disambiguate operator precedence.
[/quote]
As an example, the puzzle $11 \leftarrow 5,8,9$ would have answer $8 + 9^{.5} = 11$.

These have appeared on the USPC three years: 2018, 2022, and 2023. Try your hand at these! Some of them are much more devious than they first appear.

[list=1]
[*] 2018
[list=a]
[*] $30\leftarrow 4, 5, 8$
[*] $25\leftarrow 2,6,9$
[*] $23\leftarrow 4,5,9$
[/list]
[*] 2022
[list=a]
[*] $13\leftarrow 3,6,7$
[*] $24\leftarrow 3,8,8$
[*] $25\leftarrow 3,3,4$
[/list]
[*] 2023
[list=a]
[*] $26\leftarrow 5,5,6$
[*] $15\leftarrow 2,6,8$
[*] $11\leftarrow 2,8,9$
[/list]
[/list]
2 replies
djmathman
Yesterday at 5:52 PM
zhoujef000
Yesterday at 9:20 PM
J
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J
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Bundeswettbewerb Mathematik 1975 Problem 1.3
sqrtX   0
Oct 21, 2022
Source: Bundeswettbewerb Mathematik 1975 Round 1
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
0 replies
sqrtX
Oct 21, 2022
0 replies
J
Bundeswettbewerb Mathematik 1975 Problem 1.3
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Reveal topic tags
quadrilateral
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Source: Bundeswettbewerb Mathematik 1975 Round 1
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sqrtX
675 posts
sqrtX
#1 h  Oct 21, 2022, 11:17 PM • 3 Y
Y by Mango247, Mango247, Mango247
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
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