Inspired by Omerking

by sqing, Apr 16, 2025, 5:11 AM

Let $a,b,c>0$ and $ab+bc+ca\geq \dfrac{1}{3}.$ Prove that
$ka+ b+kc\geq \sqrt{\frac{4k-1}{3}}$ Where $k\geq 1.$ $4a+ b+4c\geq \sqrt{5}$

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

inequalities proposed

algebra

Interesting inequalities

by sqing, Apr 16, 2025, 3:36 AM

Let $a,b,c\geq 0$ and $ab+bc+ca+abc=4$ . Prove that
$k(a+b+c) -ab-bc\geq 4\sqrt{k(k+1)}-(k+4)$ Where $k\geq \frac{16}{9}.$
$\frac{16}{9}(a+b+c) -ab-bc\geq \frac{28}{9}$

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

inequalities proposed

inequalities

A Segment Bisection Problem

by buratinogigle, Apr 16, 2025, 1:36 AM

In triangle $ABC$ , let the incircle $\omega$ touch sides $BC, CA, AB$ at $D, E, F$ , respectively. Let $P$ lie on the line through $D$ perpendicular to $BC$ . Let $Q, R$ be the intersections of $PC, PB$ with $EF$ , respectively. Let $K, L$ be the projections of $R, Q$ onto line $BC$ . Let $M, N$ be the second intersections of $DQ, DR$ with the incircle $\omega$ . Let $S$ be the intersection of $KM$ and $LN$ . Prove that the line $DS$ bisects segment $QR$ .

Attachments:

geometry

Weird Inequality Problem

by Omerking, Apr 15, 2025, 8:56 AM

Following inequality is given:
$3\geq ab+bc+ca\geq \dfrac{1}{3}$ Find the range of values that can be taken by :
$1)a+b+c$
$2)abc$

Where $a,b,c$ are positive reals.

This post has been edited 1 time. Last edited by Omerking, Yesterday at 9:18 AM

NEPAL TST 2025 DAY 2

by Tony_stark0094, Apr 12, 2025, 8:40 AM

Consider an acute triangle $\Delta ABC$ . Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$ , respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$ . Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$ , and by $Q$ the intersection of ray $EB$ with $\Gamma$ .

If $O$ is the circumcenter of $\Delta ABC$ , prove that $O$ , $D$ , and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$

This post has been edited 1 time. Last edited by Tony_stark0094, Apr 13, 2025, 12:37 AM
Reason: typo

geometry

altitudes

cyclic quadrilateral

NEPAL TST DAY 2 PROBLEM 2

by Tony_stark0094, Apr 12, 2025, 8:37 AM

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

$\textbf{Proposed by Shining Sun. USA}$

This post has been edited 2 times. Last edited by Tony_stark0094, Apr 13, 2025, 3:12 AM
Reason: typo

combinatorics

NEPAL TST DAY-2 PROBLEM 1

by Tony_stark0094, Apr 12, 2025, 8:34 AM

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by
$a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}$ Prove that
$a_n^{2025} >n^{2024}$ for all positive integers $n \geq 2$ .

$\textbf{Proposed by Prajit Adhikari, Nepal.}$

This post has been edited 1 time. Last edited by Tony_stark0094, Apr 13, 2025, 12:36 AM
Reason: typo

algebra

Recurrence

Inequality

Hard number theory

by Hip1zzzil, Mar 30, 2025, 5:08 AM

Positive integers $a, b$ satisfy both of the following conditions.
For a positive integer $m$ , if $m^2 \mid ab$ , then $m = 1$ .
There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$ .
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$ , for each integer $n$ .

This post has been edited 4 times. Last edited by Hip1zzzil, Mar 30, 2025, 1:07 PM
Reason: Better

number theory

Diophantine equation

Constant Angle Sum

by i3435, May 11, 2021, 1:06 PM

Let $ABC$ be a triangle with circumcircle $\Omega$ , $A$ -angle bisector $l_A$ , and $A$ -median $m_A$ . Suppose that $l_A$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$ . A line $l$ parallel to $\overline{BC}$ meets $l_A$ , $m_A$ at $G$ , $N$ respectively, so that $G$ is between $A$ and $D$ . The circle with diameter $\overline{AG}$ meets $\Omega$ again at $J$ .

As $l$ varies, show that $\angle AMN + \angle DJG$ is constant.

MP8148

geometry

2017 PAMO Shortlsit: Power of a prime is a sum of cubes

by DylanN, May 5, 2019, 8:46 PM

For which prime numbers $p$ can we find three positive integers $n$ , $x$ and $y$ such that $p^n = x^3 + y^3$ ?

A major change in Cryptography

by fortenforge, Feb 22, 2010, 4:23 AM

So, the time is WWII and in the field of cryptography, the code-breakers have the upper hand. Any code that is made eventually gets broken. The problem was that all of the codes were really simple because all messages were encrypted by hand so complex rules were simply too complex and took too long. Cryptography is about to take a major change in both the fields of code-making and cryptanalysis. The change is basically the idea of mechanizing encryption and decryption, in other words using a machine to encode the message (note that they were using machines not computers, computers make the next drastic change in cryptography). The best example of an encryption machine is the Enigma, a machine developed by the Germans that gave the code-makers the upper hand. For the next few weeks it is what I am going to talk about.

Cryptography in general

Enigma

Submit

Good website!
by bluegoose101, Aug 5, 2021, 6:28 PM
uh-huh, a great place here
by fenchelfen, Sep 1, 2019, 11:30 AM
uh, yeah he is o_O
by SonyWii, Oct 8, 2010, 2:11 PM
dude i think you're my roommate from camp :O
by themorninglighttt, Aug 29, 2010, 10:06 PM
what i'm still not a contrib D:
by SonyWii, Aug 6, 2010, 2:20 PM
I see what you did there
by Jongy, Aug 1, 2010, 11:52 PM
omg, apparently you like cryptography; and apparently I'm not a contribb D:
by SonyWii, Jul 26, 2010, 9:48 PM
Thank You
by fortenforge, Jan 17, 2010, 6:35 PM
Wow this is a really cool blog
by alkjash, Jan 16, 2010, 7:04 PM
Hi
by fortenforge, Jan 7, 2010, 12:12 AM
Hi
by Richard_Min, Jan 5, 2010, 9:29 PM
Hi
by fortenforge, Jan 3, 2010, 10:14 PM
HELLO FORTENFORGE I AM THE PERSON SITTING NEXT TO YOU IN IDEAMATH
by ButteredButNotEaten, Dec 24, 2009, 4:19 AM
@dragon96 Not if you celebrate Christmas with neon lights
@batteredbutnotdefeated Sure, You are now a contributer
by fortenforge, Dec 20, 2009, 4:39 AM
I too share a love for cryptography and cryptanalysis, may I be a contrib?
by batteredbutnotdefeated, Dec 20, 2009, 2:38 AM
The green is too bright for Christmas.
by dragon96, Dec 20, 2009, 2:12 AM
I thought I'd change the colors for the Holidays
by fortenforge, Dec 13, 2009, 10:53 PM
hi, some "simple" cryptography here: http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=317795
by phiReKaLk6781, Dec 12, 2009, 3:46 AM
Yeah, that is binary, for modern cryptography, most text is converted to binary first and then algorithm's for encryption are preformed on the binary rather than the English letters. The text is converted using the ASCII table or UNICODE.
by fortenforge, Oct 13, 2009, 10:33 PM
Whoa, I love your background! Is that binary?
by pianogirl, Oct 13, 2009, 8:34 PM
Sure, I'll add you as a contributer...
by fortenforge, Oct 2, 2009, 4:44 AM
May I make a post on one cipher I made up? (It's a good code for science people! *hint hint*)
by dragon96, Oct 2, 2009, 4:04 AM
Nice blog, this is interesting...

and guess who i am
by Yoshi, Sep 21, 2009, 4:02 AM
Thanks
by fortenforge, Sep 17, 2009, 1:33 AM
Very interesting blog. Nice!
by AIME15, Sep 16, 2009, 5:21 PM
When you mean 'write' do you mean like programming? Much of cryptography has to do with programming and most modern cryptographers are excellent programmers because modern complex ciphers are difficult to implement by hand.

See if you can write a program for the substitution cipher. The user should be able to enter the key and the message. I know it is possible to do it in pretty much any language because I was able to do it in c.
by fortenforge, Aug 7, 2009, 8:17 PM
Hello. I don't know much about advanced cryptography but I did write a Caeser Chipher encrypter and decrypter!
by Poincare, Jul 31, 2009, 8:55 PM

27 shouts