100th post

by MathJedi108, Apr 20, 2025, 10:59 PM

Well I guess this is my 100th post, it would be really funny if it isn't can yall share your favorite experience on AoPS here?

100th Post

Median geometry

by Sedro, Apr 20, 2025, 6:03 PM

In triangle $ABC$ , points $D$ , $E$ , and $F$ are the midpoints of sides $BC$ , $CA$ , and $AB$ , respectively. Prove that the area of the triangle with side lengths $AD$ , $BE$ , and $CF$ has area $\tfrac{3}{4}[ABC]$ .

geometry

median

geometry

by carvaan, Apr 20, 2025, 5:46 PM

The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.

geometry

Angle Chasing

Complex Numbers Question

by franklin2013, Apr 20, 2025, 4:08 PM

Hello everyone! This is one of my favorite complex numbers questions. Have fun!

$f(z)=z^{720}-z^{120}$ . How many complex numbers $z$ are there such that $|z|=1$ and $f(z)$ is an integer.

Hint

complex numbers

VOLUNTEERING OPPORTUNITIES OPEN TO HIGH/MIDDLE SCHOOLERS

by im_space_cadet, Apr 20, 2025, 2:27 PM

Hi everyone!
Do you specialize in contest math? Do you have a passion for teaching? Do you want to help leverage those college apps? Well, I have something for all of you.

I am im_space_cadet, and during the fall of last year, I opened my non-profit DeltaMathPrep which teaches students preparing for contest math the problem-solving skills they need in order to succeed at these competitions. Currently, we are very much understaffed and would greatly appreciate the help of more tutors on our platform.

Each week on Saturday and Wednesday, we meet once for each competition: Wednesday for AMC 8 and Saturday for AMC 10 and we go over a past year paper for the entire class. On both of these days, we meet at 9PM EST in the night.

This is a great opportunity for anyone who is looking to have a solid activity to add to their college resumes that requires low effort from tutors and is very flexible with regards to time.

This is the link to our non-profit for anyone who would like to view our initiative:
https://www.deltamathprep.org/

If you are interested in this opportunity, please send me a DM on AoPS or respond to this post expressing your interest. I look forward to having you all on the team!

Thanks,
im_space_cadet

Three variables inequality

by Headhunter, Apr 20, 2025, 6:58 AM

$\forall a\in R$ , $~\forall b\in R$ , $~\forall c \in R$
Prove that at least one of $(a-b)^{2}$ , $(b-c)^{2}$ , $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$ .

I assume that all are greater than it, but can't go more.

inequalities

weird permutation problem

by Sedro, Apr 20, 2025, 2:09 AM

Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$ . How many possible $\sigma$ exist?

Find all triples

by pedronis, Apr 19, 2025, 12:14 AM

Find all triples of positive integers $(n, r, s)$ such that $n^2 + n + 1$ divides $n^r + n^s + 1$ .

This post has been edited 2 times. Last edited by pedronis, Apr 19, 2025, 12:16 AM

Divisibility

Inequalities

by sqing, Apr 16, 2025, 4:52 AM

Let $a,b$ be reals such that $a^2+ab+b^2 =3$ . Prove that
$\frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$ $\frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$ $\frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$ Let $a,b$ be reals such that $a^2-ab+b^2 =1$ . Prove that
$\frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$ $\frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$ $\frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$

inequalities

Indonesia Regional MO 2019 Part A

by parmenides51, Nov 11, 2021, 10:33 PM

Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules

p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...

p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...

p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$ . The value of $(r-s)^2$ is ...

p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...

p5. A data with four real numbers $2n-4$ , $2n-6$ , $n^2-8$ , $3n^2-6$ has an average of $0$ and a median of $9/2$ . The largest number of such data is ...

p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$ . If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...

p7. Given a triangle $ABC$ , with $AB = 6$ , $AC = 8$ and $BC = 10$ . The points $D$ and $E$ lies on the line segment $BC$ . with $BD = 2$ and $CE = 4$ . The measure of the angle $\angle DAE$ is ...

p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$ . The value of $a_1a_2a_3...a_{2019}$ is ....

p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...

p10. Pairs of natural numbers $(m , n)$ which satisfies $m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$ are as many as ...

p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$ . Point $D$ lies on the side $BC$ so that $AB=CD$ . Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$ . The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$ . The large angle $\angle AEC$ is ...

p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$ . The largest value of $n$ is ....

p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....

p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....

p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....

This post has been edited 1 time. Last edited by parmenides51, Nov 11, 2021, 10:34 PM

Indonesia Regional MO

Submit

hi guys $~~~$
by Yiyj1, Apr 9, 2025, 7:25 AM
You will be remembered
by giangtruong13, Feb 26, 2025, 3:38 PM
2025 shout!
by just_a_math_girl, Jan 12, 2025, 7:25 AM
2024 shout
by bachkieu, Aug 22, 2024, 12:52 AM
helooooooooooo
by owenccc, Sep 27, 2023, 12:59 AM
hullo :<
by gracemoon124, Jul 14, 2023, 2:33 AM
hello $\text{[math]}$
by LeoLionTank, Feb 17, 2023, 9:02 PM
Halo thear
by HoRI_DA_GRe8, Oct 18, 2022, 7:44 AM
hi!!
just found this and I can't wait to read more!
so happy to have found this blog!
by Morrigan_Black, Jan 28, 2022, 1:05 PM
still waiting for the mathlinks camp lol
by CinarArslan, Jan 9, 2022, 1:55 PM
hello
by CyclicISLscelesTrapezoid, Nov 29, 2021, 6:36 PM
Right below the shout box it says how many it has.
by pith0n, May 11, 2021, 5:08 AM
Oh really the blog has 100 posts! I never counted the number of posts here. If I get some free time I will create a new page on my wordpress website and there I will post all the contents of this blog. So, make sure that you check the wordpress site.
by adityaguharoy, Mar 21, 2021, 2:58 PM
Nice blog!
by DCode10, Mar 10, 2021, 7:00 PM
Hi adityaguharoy! Nice blog!
by masadca, Feb 4, 2021, 9:19 PM
Nice blog!
by Mathisfun04, Dec 2, 2016, 12:00 AM
Nice blog!!!
by Ilikeapos, Aug 24, 2016, 2:56 PM
really nice CSS
by StarFrost7, Aug 16, 2016, 3:37 AM
hi!
Post some more!
Also nice css
by Intellectuality123, Aug 12, 2016, 11:35 PM
Hello!!!!!
by MinionLA, Jul 18, 2016, 9:15 PM
hi everyone
by matthMaster, Jul 2, 2016, 1:36 AM
You have a great blog here; just add some more stuff and it will be even better!
by bobert1, Jun 30, 2016, 9:29 PM
yeah sure we must and will discuss real numbers.
by adityaguharoy, Jun 20, 2016, 4:11 AM
Why not real numbers
by skipiano, Jun 17, 2016, 2:35 PM
very interesting topic !! ?? !!
by adityaguharoy, Jun 12, 2016, 6:52 AM
Hello!!!!!!! Very interesting topic, here!
by MinionLA, Jun 9, 2016, 11:43 PM
Should it not start by the discussion of Set Theory.
by alexanderoldspartan, Jun 9, 2016, 11:03 AM
Darn I missed it
by skipiano, Jun 9, 2016, 4:52 AM
first shout >
by doitsudoitsu, Apr 26, 2016, 8:03 PM

118 shouts

An Introduction to the Theory of Mathematics

by MathJedi108, Apr 20, 2025, 10:59 PM

by Sedro, Apr 20, 2025, 6:03 PM

by carvaan, Apr 20, 2025, 5:46 PM

by franklin2013, Apr 20, 2025, 4:08 PM

by im_space_cadet, Apr 20, 2025, 2:27 PM

by Headhunter, Apr 20, 2025, 6:58 AM

by Sedro, Apr 20, 2025, 2:09 AM

by pedronis, Apr 19, 2025, 12:14 AM

by sqing, Apr 16, 2025, 4:52 AM

by parmenides51, Nov 11, 2021, 10:33 PM