Inspired by old results

by sqing, Apr 30, 2025, 10:16 AM

Let $a , b , c>0$ and $abc=1$ . Prove that
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3 \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ h

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

inequalities proposed

algebra

Invariant board combi style

by ItzsleepyXD, Apr 30, 2025, 9:25 AM

Oh write $2025^{2025^{2025}}$ real number on the board such that each number is more than $2025^{2025}$ .
Oh erase 2 number $x,y$ on the board and write $\frac{xy-2025}{x+y-90}$ .
Prove that the last number will always be the same regardless the order of number that Oh pick .

Invariance

combinatorics

Parallel condition and isogonal

by ItzsleepyXD, Apr 30, 2025, 9:18 AM

Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .

geometry

circumcircle

problem interesting

by Cobedangiu, Apr 30, 2025, 5:06 AM

Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

This post has been edited 1 time. Last edited by Cobedangiu, Today at 5:06 AM

algebra

D1025 : Can you do that?

by Dattier, Apr 29, 2025, 8:24 PM

Let $x_{n+1}=x_n^3$ and $x_0=3$ .

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$ ?

Computer solutions

number theory

Something about (BIC)

by flower417477, Apr 28, 2025, 3:58 PM

Given $\triangle ABC$ with incenter $I$ , $D$ is a point on $BC$ ,the bisector of $\angle ADB$ meet $(BIC)$ at $E,F$ .Prove that $\angle EAD=\angle IAF$

geometry

incenter

Olympiad

amazing balkan combi

by egxa, Apr 27, 2025, 1:57 PM

There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$ , $k \geq 0$ , such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$ ;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$ ). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$ , find all possible values $F$

Proposed by David-Andrei Anghel, Romania.

This post has been edited 6 times. Last edited by egxa, Apr 27, 2025, 10:59 PM

Combi

combinatorics

weird Condition

by B1t, Apr 27, 2025, 1:37 PM

deleted for a while

This post has been edited 5 times. Last edited by B1t, 5 hours ago

geometry

#14) IMO 2014 P4

by sa2001, Feb 19, 2018, 3:10 PM

Problem statement

A proof using phantom point N' and Pascal's Theorem -

This post has been edited 2 times. Last edited by pro_4_ever, Feb 19, 2018, 4:42 PM

IMO

phantom points

Pascal

Functional equation

by Amin12, Aug 7, 2017, 8:35 AM

Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that
$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$ for all positive real numbers $x$ and $y$ .

This post has been edited 2 times. Last edited by Amin12, Aug 7, 2017, 8:40 AM

algebra

functional equation

RMM 2013 Problem 1

by dr_Civot, Mar 2, 2013, 10:36 AM

For a positive integer $a$ , define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$ . Let $y_n=2^{x_n}-1$ . Determine the largest possible $k$ such that, for some positive integer $a$ , the numbers $y_1,\ldots,y_k$ are all prime.

Submit

Hey, what a nice blog!!!!!
by Power_Set, Jun 24, 2018, 9:53 PM
Is your blog dead ? No new post in almost 2 months.

My blog has died due to malnutrition of problems.
by integrated_JRC, May 2, 2018, 5:52 AM
@ below, I will do it soon
by pro_4_ever, Mar 1, 2018, 10:37 AM
Hi, I think we should 'hide' our solutions, as it is difficult to go up and down the blog
by Drunken_Master, Mar 1, 2018, 10:36 AM
Yeah, I created a test blog, it seems to be an AoPS problem not CSS, can't do anything about it. I'll post it in site support forum later.
by Vrangr, Feb 20, 2018, 5:48 AM
The problem is not in the CSS, I believe, as the CSS only looks after the appearance of the blog, and not the $\text{\LaTeX}$ rendering.
by WizardMath, Feb 19, 2018, 11:34 PM
I'll fix the CSS after boards of need be (I'll probably be starting my own blog too then)
by Vrangr, Feb 19, 2018, 4:41 PM
@ below,
Pls ask WizardMath...
The CSS is taken from his amazing Blog, "An Olympiad Journey"...
I know nothing abt programming.
(BTW I got permission for taking the CSS. No Copyright Issues!)
by pro_4_ever, Feb 19, 2018, 4:16 PM
Huge flaw in website design: comment gets erased if while typing $\LaTeX$ has an error and needs to be fixed.
by Vrangr, Feb 19, 2018, 12:26 PM
Upon WizardMath's Request, let us all post synthetic solutions. Well, let's declare that this blog is intended to give the reader some good problems with simple synthetic solutions. All the best. Not trying to be rude,but I might delete upcoming Bashes
by pro_4_ever, Feb 13, 2018, 4:50 PM
Here's an idea. Ban all bashes on this blog from now on. Also try posting your own previous geo solutions as storage (I hope this is allowed). This would make the quality of the blog better. Bashes are basically write-once, read-never things. Why bother?
by WizardMath, Feb 13, 2018, 4:33 PM
Somebody type Egmo 2013 Problem 1.
No bash,purely troll problem.
by QWERTYphysics, Feb 13, 2018, 4:11 PM
I hate having to see a pure geometry blog being swamped to death with bashes.

Obviously a bash is a super terrible idea. Don't do one.
Unless you are bad at synthetic.
by WizardMath, Feb 13, 2018, 2:58 PM
@ayan Done:)
by pro_4_ever, Feb 13, 2018, 1:38 PM
9th shout
by MEGAKNIGHT, Feb 13, 2018, 11:23 AM
@below Pls don't post a bash if you are thinking about posting a new entry!
by pro_4_ever, Feb 13, 2018, 11:11 AM
Wait, I thought bash is frowned upon in this blog.
by WizardMath, Feb 12, 2018, 6:57 PM
Hmmm...
Combi Geometry is accepted...
by pro_4_ever, Feb 10, 2018, 7:32 AM
Can we post Combi here?
by ayan.nmath, Feb 10, 2018, 7:31 AM
Parag dey
by Paragdey12, Feb 6, 2018, 1:17 PM
More posts please
by AnArtist, Feb 6, 2018, 2:38 AM
Second Shout!
by ccx09, Feb 3, 2018, 9:20 PM
1st shout!
by AnArtist, Jan 31, 2018, 9:20 AM