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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]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
Every popular person is the best friend of a popular person?
yunxiu   8
N a few seconds ago by HHGB
Source: 2012 European Girls’ Mathematical Olympiad P6
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.)
Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.
(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.

Romania (Dan Schwarz)
8 replies
yunxiu
Apr 13, 2012
HHGB
a few seconds ago
2021 EGMO P2: f(xf(x)+y) = f(y) + x^2 for rational x, y
anser   80
N 8 minutes ago by math-olympiad-clown
Source: 2021 EGMO P2
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.

Here, $\mathbb{Q}$ denotes the set of rational numbers.
80 replies
anser
Apr 13, 2021
math-olympiad-clown
8 minutes ago
D1033 : A problem of probability for dominoes 3*1
Dattier   1
N 8 minutes ago by Dattier
Source: les dattes à Dattier
Let $G$ a grid of 9*9, we choose a little square in $G$ of this grid three times, we can choose three times the same.

What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
1 reply
Dattier
May 15, 2025
Dattier
8 minutes ago
2010 Japan MO Finals
parkjungmin   4
N 17 minutes ago by parkjungmin
Is there anyone who can solve question problem 5?
4 replies
parkjungmin
May 15, 2025
parkjungmin
17 minutes ago
Inequalities
sqing   10
N 3 hours ago by sqing
Let $a,b,c >2 $ and $ ab+bc+ca \leq 75.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$$Let $a,b,c >2 $ and $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$$
10 replies
sqing
May 13, 2025
sqing
3 hours ago
Concurrent in a pyramid
vanstraelen   1
N Yesterday at 7:53 PM by vanstraelen

Given a pyramid $(T,ABCD)$ where $ABCD$ is a parallelogram.
The intersection of the diagonals of the base is point $S$.
Point $A$ is connected to the midpoint of $[CT]$, point $B$ to the midpoint of $[DT]$,
point $C$ to the midpoint of $[AT]$ and point $D$ to the midpoint of $[BT]$.
a) Prove: the four lines are concurrent in a point $P$.
b) Calulate $\frac{TS}{TP}$.
1 reply
vanstraelen
May 10, 2025
vanstraelen
Yesterday at 7:53 PM
bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2
parmenides51   3
N Yesterday at 7:49 PM by LeYohan
Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.
3 replies
parmenides51
Sep 20, 2021
LeYohan
Yesterday at 7:49 PM
Pertenacious Polynomial Problem
BadAtCompetitionMath21420   4
N Yesterday at 7:40 PM by soryn
Let the polynomial $P(x) = x^3-x^2+px-q$ have real roots and real coefficients with $q>0$. What is the maximum value of $p+q$?

This is a problem I made for my math competition, and I wanted to see if someone would double-check my work (No Mike allowed):

solution
Is this solution good?
4 replies
BadAtCompetitionMath21420
Yesterday at 3:13 AM
soryn
Yesterday at 7:40 PM
shadow of a cylinder, shadow of a cone
vanstraelen   3
N Yesterday at 7:35 PM by vanstraelen

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
3 replies
vanstraelen
May 9, 2025
vanstraelen
Yesterday at 7:35 PM
Challenge Problem: triangle inequality
Bottema   7
N Yesterday at 6:02 PM by Speedysolver1
Prove that in any triangle we have:

a^{2}+b^{2}+c^{2} \geq 4sqrt{3}S
7 replies
Bottema
May 12, 2004
Speedysolver1
Yesterday at 6:02 PM
2022 MARBLE - Mock ARML I -8 \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32
parmenides51   2
N Yesterday at 4:33 PM by Kempu33334
Let $a,b,c$ complex numbers with $ab+ +bc+ca = 61$ such that
$$\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}= 5$$$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32.$$Find the value of $abc$.
2 replies
parmenides51
Jan 14, 2024
Kempu33334
Yesterday at 4:33 PM
Geomettry ez
AnhIsGod   1
N Yesterday at 1:17 PM by Soupboy0
Let two circles (O) and (O') intersect at two points (one of which is called A). The common tangent CD (with C belonging to (O) and D belonging to (O')) lies on the same side as A with respect to the line OO', intersecting OO' at S. The line segment SA intersects circle (O) at E (different from A). Prove that EC is parallel to AD.
1 reply
AnhIsGod
Yesterday at 12:43 PM
Soupboy0
Yesterday at 1:17 PM
Minimum and Maximum of Complex Numbers
pythagorazz   1
N Yesterday at 8:39 AM by alexheinis
Let $a,b,$ and $c$ be complex numbers. For a complex number $z=p+qi$ where $i=\sqrt(-1)$, define the norm $|z|$ to be the distance of $z$ from the origin, or $|z|=\sqrt(p^2+q^2 )$. Let $m$ be the minimum value and $M$ be the maximum value of $\frac{(|a+b|+|b+c|+|c+a|)}{(|a|+|b|+|c| )}$ for all complex numbers $a,b,c$ where $|a|+|b|+|c|\ne 0$. Find $M+m$.
1 reply
pythagorazz
Apr 14, 2025
alexheinis
Yesterday at 8:39 AM
Folklore
Osim_09   2
N Yesterday at 8:36 AM by pigeon123
Let ABCD be a circumscribed quadrilateral, which is also cyclic. Let I be the incenter, O the circumcenter, and E the intersection point of the diagonals of the quadrilateral. Prove that the points O, I, and E are collinear.
2 replies
Osim_09
Jan 21, 2025
pigeon123
Yesterday at 8:36 AM
Hard inequality
JK1603JK   4
N Apr 30, 2025 by JK1603JK
Source: unknown?
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
4 replies
JK1603JK
Apr 29, 2025
JK1603JK
Apr 30, 2025
Hard inequality
G H J
G H BBookmark kLocked kLocked NReply
Source: unknown?
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JK1603JK
53 posts
#1
Y by
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
Z K Y
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xytunghoanh
39 posts
#2
Y by
Put $t=a+b$ then
\[2t+2c \ge \frac{t^2}{4}+c^2 \]Note that
\[P\ge (t+c)(\frac{4}{t}+\frac{1}{c})=\frac{4c}{t}+\frac{t}{c}+5\]It can done just by Cauchy
This post has been edited 1 time. Last edited by xytunghoanh, Apr 29, 2025, 6:21 AM
Reason: fix
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sqing
42227 posts
#3
Y by
JK1603JK wrote:
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
$$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9 $$
Z K Y
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GeoMorocco
44 posts
#4
Y by
JK1603JK wrote:
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$

Using Cauchy-Schwarz, we get:
$$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 9 $$
With equality at $a=b=c=2$.
Z K Y
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JK1603JK
53 posts
#5
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Also, find the min $$P=a+b+c+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
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