Inequality

by srnjbr, Mar 21, 2025, 4:32 PM

a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2

inequalities

Graph Theory

by JetFire008, Mar 21, 2025, 4:22 PM

Prove that for any Hamiltonian cycle, if it contain edge $e$ , then it must not contain edge $e'$ .

graph theory

hamiltonian graph

Inspired by hunghd8

by sqing, Mar 21, 2025, 4:08 PM

Let $a,b,c\geq 0$ and $a+b+c\geq 2+abc .$ Prove that
$a^2+b^2+c^2- abc\geq \frac{7}{4}$ $a^2+b^2+c^2-2abc \geq 1$ $a^2+b^2+c^2- \frac{1}{2}abc\geq \frac{31}{16}$ $a^2+b^2+c^2- \frac{8}{5}abc\geq \frac{34}{25}$

inequalities proposed

algebra

Inspired by hunghd8

by sqing, Mar 21, 2025, 3:36 PM

Let $a,b,c\geq 0$ and $a+b+c\geq 2+abc .$ Prove that
$a^2+b^2+c^2-\frac{1}{2}a^2b^2c^2\geq 2$ $a^2+b^2+c^2-abc-\frac{1}{2}a^2b^2c^2\geq \frac{3}{2}$ $a^2+b^2+c^2- \frac{19}{10}abc-\frac{1}{2}a^2b^2c^2\geq -\frac{12}{25}$ $a^2+b^2+c^2- \frac{3}{2}abc-\frac{1}{2}a^2b^2c^2\geq \frac{17\sqrt{17}-71}{16}$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

algebra

Prime for square numbers

by giangtruong13, Mar 21, 2025, 2:43 PM

Given that $a,b$ are natural numbers satisfy that: $\frac{a^3}{a+b}$ and $\frac{b^3}{a+b}$ are prime numbers. Prove that $a^2+3ab+3a+b+1$ is a perfect squared number

This post has been edited 1 time. Last edited by giangtruong13, 3 hours ago

number theory

prime numbers

Find min

by hunghd8, Mar 21, 2025, 12:10 PM

Let $a,b,c$ be nonnegative real numbers such that $a+b+c\geq 2+abc$ . Find min
$P=a^2+b^2+c^2.$

inequalities

Problem 4

by blug, Mar 15, 2025, 7:57 PM

In a rhombus $ABCD$ , angle $\angle ABC=100^{\circ}$ . Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$ . Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$ . Prove that $BP=AQ$ .

geometry

Assisted perpendicular chasing

by sarjinius, Mar 9, 2025, 3:41 PM

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$ , let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$ . Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$ , and let $F$ be the point on line $AC$ such that $FA = FE$ . Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$ .
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.

Interesting inequality

by sqing, Feb 26, 2025, 9:30 AM

Let $a,b >0.$ Prove that
$\frac{1}{\frac{a}{a+b}+\frac{a}{2b}} +\frac{1}{\frac{b}{a+b}+\frac{1}{2}} +\frac{a}{2b} \geq \frac{5}{2}$

inequalities proposed

inequalities

algebra

a! + b! = 2^{c!}

by parmenides51, Mar 26, 2024, 3:44 PM

Determine all triples $(a, b, c)$ of positive integers such that
$a! + b! = 2^{c!}.$
(Walther Janous)

number theory

Submit

The blog is locked right?
by First, Apr 14, 2018, 6:00 PM
Great, amazing, inspiring blog. Good luck in life, and just know I aspire to succeed as you will in the future.
by mgrimalo, Apr 7, 2018, 6:19 PM
Yesyesyes
by shiningsunnyday, Mar 29, 2018, 5:30 PM
:O a new background picture
by MathAwesome123, Mar 29, 2018, 3:39 PM
did you get into MIT?
by 15Pandabears, Mar 15, 2018, 10:42 PM
wait what new site?
by yegkatie, Feb 11, 2018, 1:49 AM
Yea, doing a bit of cleaning before migrating to new site
by shiningsunnyday, Jan 21, 2018, 2:43 PM
Were there posts made in December 2017 for this blog and then deleted?

I ask because I was purging my thunderbird inbox and I found emails indicating new blog posts of yours.

email do not lie
by jonlin1000, Jan 21, 2018, 12:12 AM
@below sorry not accepting contribs
by shiningsunnyday, Dec 11, 2017, 11:15 AM
contrib plez?
also wow this blog is very popular
by DavidUsa, Dec 10, 2017, 7:53 PM
@First: lol same

first shout of december
by coolmath34, Dec 6, 2017, 2:32 PM
XD this blog is hilarious
by Mitsuku, Nov 21, 2017, 7:40 PM
@wu2481632: stop encouraging SSD to procrastinate(blog entries are fun but procrastination isn't).
by First, Aug 7, 2017, 5:02 PM
3.5 weeks without a post
by Flash12, Aug 4, 2017, 8:10 AM
First august shout!!
by adik7, Aug 1, 2017, 6:52 AM
"Also, I figured that whenever"

Darn, whenever what?

Belated (oops sorry) well-wishes on the 2016 AMC10A!
by DeathLlama9, Feb 3, 2016, 3:23 AM
what is a 8char?
lol 8char
by whiteawesomesun, Jan 30, 2016, 9:43 AM
lol 8char
by chocopuff, Jan 30, 2016, 3:38 AM
yadayadayadayadayadayada amc 10 is near... hope you get into JMO...
by whiteawesomesun, Jan 30, 2016, 2:45 AM
so much hype and stress!
by chocopuff, Jan 29, 2016, 2:14 PM
Lol lets reconcile man
by Konigsberg, Jan 20, 2016, 11:36 PM
DUDUUDDUDUUDE STAPHHHHH. war? war? TF boys
by whiteawesomesun, Jan 19, 2016, 8:50 AM
dude dude stop stop war war
by Konigsberg, Jan 18, 2016, 3:44 AM
I'm excited!
by paperplates, Jan 15, 2016, 6:55 AM
nice! \8char
by adihaya, Jan 14, 2016, 6:47 AM
yay, looks like this is going to be a cool blog!
by jh235, Jan 7, 2016, 8:15 PM
Post something!
by whiteawesomesun, Jan 4, 2016, 3:54 PM
at least I claim second shot
by Kline, Dec 29, 2015, 1:45 AM
first shout >
by oceanair, Dec 28, 2015, 6:00 AM

416 shouts

SSD's Musings and Dreams

by srnjbr, Mar 21, 2025, 4:32 PM

by JetFire008, Mar 21, 2025, 4:22 PM

by sqing, Mar 21, 2025, 4:08 PM

by sqing, Mar 21, 2025, 3:36 PM

by giangtruong13, Mar 21, 2025, 2:43 PM

by hunghd8, Mar 21, 2025, 12:10 PM

by blug, Mar 15, 2025, 7:57 PM

by sarjinius, Mar 9, 2025, 3:41 PM

by sqing, Feb 26, 2025, 9:30 AM

by parmenides51, Mar 26, 2024, 3:44 PM