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3 var inequality
SunnyEvan   3
N a few seconds ago by JARP091
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{53}{2}-9\sqrt{14} \leq \frac{8(a^3b+b^3c+c^3a)}{27(a^2+b^2+c^2)^2} \leq \frac{53}{2}+9\sqrt{14} $$
3 replies
SunnyEvan
5 hours ago
JARP091
a few seconds ago
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Stanford Math Tournament (SMT) 2025
stanford-math-tournament   6
N Today at 5:47 AM by techb
[center] :trampoline: :first: Stanford Math Tournament :first: :trampoline: [/center]

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[center]IMAGE[/center]

We are excited to announce that registration is now open for Stanford Math Tournament (SMT) 2025!

This year, we will welcome 800 competitors from across the nation to participate in person on Stanford’s campus. The tournament will be held April 11-12, 2025, and registration is open to all high-school students from the United States. This year, we are extending registration to high school teams (strongly preferred), established local mathematical organizations, and individuals; please refer to our website for specific policies. Whether you’re an experienced math wizard, a puzzle hunt enthusiast, or someone looking to meet new friends, SMT has something to offer everyone!

Register here today! We’ll be accepting applications until March 2, 2025.

For those unable to travel, in middle school, or not from the United States, we encourage you to instead register for SMT 2025 Online, which will be held on April 13, 2025. Registration for SMT 2025 Online will open mid-February.

For more information visit our website! Please email us at stanford.math.tournament@gmail.com with any questions or reply to this thread below. We can’t wait to meet you all in April!

6 replies
stanford-math-tournament
Feb 1, 2025
techb
Today at 5:47 AM
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2016 Sets
NormanWho   111
N Yesterday at 11:59 PM by Amkan2022
Source: 2016 USAJMO 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
111 replies
NormanWho
Apr 20, 2016
Amkan2022
Yesterday at 11:59 PM
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camp/class recommendations for incoming freshman
walterboro   14
N Yesterday at 11:55 PM by jb2015007
hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty
14 replies
walterboro
May 10, 2025
jb2015007
Yesterday at 11:55 PM
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Metamorphosis of Medial and Contact Triangles
djmathman   102
N Yesterday at 8:40 PM by Mathandski
Source: 2014 USAJMO Problem 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$.

(a) Prove that $I$ lies on ray $CV$.

(b) Prove that line $XI$ bisects $\overline{UV}$.
102 replies
djmathman
Apr 30, 2014
Mathandski
Yesterday at 8:40 PM
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ranttttt
alcumusftwgrind   41
N Yesterday at 7:58 PM by meyler
rant
41 replies
alcumusftwgrind
Apr 30, 2025
meyler
Yesterday at 7:58 PM
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high tech FE as J1?!
imagien_bad   62
N Yesterday at 7:44 PM by jasperE3
Source: USAJMO 2025/1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
62 replies
imagien_bad
Mar 20, 2025
jasperE3
Yesterday at 7:44 PM
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Holy garbanzo
centslordm   13
N Yesterday at 5:55 PM by daijobu
Source: 2024 AMC 12A #23
What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]
$\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84$
13 replies
centslordm
Nov 7, 2024
daijobu
Yesterday at 5:55 PM
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2v2 (bob lost the game)
GoodMorning   85
N Yesterday at 1:18 PM by maromex
Source: 2023 USAJMO Problem 5/USAMO Problem 4
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.

After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
85 replies
GoodMorning
Mar 23, 2023
maromex
Yesterday at 1:18 PM
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Suggestions for preparing for AMC 12
peppermint_cat   3
N Yesterday at 7:14 AM by Konigsberg
So, I have decided to attempt taking the AMC 12 this fall. I don't have any experience with math competitions, and I thought that here might be a good place to see if anyone who has taken the AMC 12 (or done any other math competitions) has any suggestions on what to expect, how to prepare, etc. Thank you!
3 replies
peppermint_cat
Yesterday at 1:04 AM
Konigsberg
Yesterday at 7:14 AM
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Harmonic Mean
Happytycho   4
N Yesterday at 4:42 AM by elizhang101412
Source: Problem #2 2016 AMC 12B
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 504 \qquad \textbf{(D)}\ 1008 \qquad \textbf{(E)}\ 2015 $
4 replies
Happytycho
Feb 21, 2016
elizhang101412
Yesterday at 4:42 AM
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Normal but good inequality
giangtruong13   4
N Apr 23, 2025 by IceyCold
Source: From a province
Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$
4 replies
giangtruong13
Mar 31, 2025
IceyCold
Apr 23, 2025
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Normal but good inequality
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Source: From a province
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giangtruong13
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giangtruong13
#1 h  Mar 31, 2025, 4:04 PM
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Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$
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arqady
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arqady
#2 h  Apr 1, 2025, 4:11 AM
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giangtruong13 wrote:
Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$
After using Bergstrom we obtain something trivial.
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giangtruong13
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giangtruong13
#3 h  Apr 7, 2025, 2:36 PM
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Anyone here?
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Sadigly
224 posts
Sadigly
#4 h  Apr 7, 2025, 3:32 PM • 1 Y
Y by MuradSafarli
$\sum\frac{1}{ab}=3$. Let $x=\frac1{a}$. Define $y,z$ similarly. We got $xy+yz+zx=3$. Problem asks us to prove $$\sum\frac{z}{3xy+z+1}\geq\frac35$$
$3=xy+yz+zx\stackrel{\text{AM-GM}}{\geq}3\sqrt{x^2y^2z^2}\Rightarrow xyz\leq1$

$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)\geq3(xy+yz+zx)\Rightarrow x+y+z\geq3$

$(x+y+z)(x+y+z)\stackrel{\text{C-S}}{\leq} 3(x^2+y^2+z^2)$

$$\sum\frac{z}{3xy+z+1}=\sum\frac{z^2}{3xyz+z^2+z}\stackrel{\text{Titu}}{\geq}\frac{(x+y+z)^2}{x^2+y^2+z^2+x+y+z+9xyz}=\frac{x^2+y^2+z^2+6}{x^2+y^2+z^2+x+y+z+9xyz}=1-\frac{x+y+z+9xyz-6}{x^2+y^2+z^2+x+y+z+9xyz}\stackrel{?}{\geq}\frac35$$
$$\frac{x+y+z+9xyz-6}{x^2+y^2+z^2+x+y+z+9xyz}\stackrel{?}{\leq}\frac25$$
Cross multiplying gives $$2x^2+2y^2+2z^2-3x-3y-3z-27xyz\stackrel{?}{\geq}-30$$$$2x^2+2y^2+2z^2-3x-3y-3z-27xyz\geq 2x^2+2y^2+2z^2-3x-3y-3z-27\stackrel{?}{\geq}-30\Rightarrow 2x^2+2y^2+2z^2-3x-3y-3z\stackrel{?}{\geq}-3$$$$6(x^2+y^2+z^2)-9(x+y+z)\stackrel{?}{\geq}-9$$
$$6(x^2+y^2+z^2)-9(x+y+z)\geq 2(x+y+z)(x+y+z)-9(x+y+z)=(x+y+z)(2x+2y+2z-9)\stackrel{?}{\geq}-9$$
Let $u=x+y+z$. This equation becomes $2u^2-9u+9\stackrel{?}{\geq}0$ or $(u-3)(2u-3)\stackrel{?}{\geq}0$. But this is obviously true since $u=x+y+z\geq3$
This post has been edited 1 time. Last edited by Sadigly, Apr 7, 2025, 3:36 PM
Reason: Sometimes I look in a mirror and ask myself: Am I really scared of passing away? If it's today, I hope I hear a Cry out from heaven so loud it can water down a demon With the holy ghost 'til it drown in the blood of Jesus
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IceyCold
208 posts
IceyCold
#5 h  Apr 23, 2025, 3:56 PM
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giangtruong13 wrote:
Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$

From Ha Noi last year?
The inequality seems nice,somewhat easy for said test,but it can be generalized.
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