chat gpt

by fuv870, Mar 16, 2025, 9:51 PM

The chat gpt alreadly knows how to solve the problem of IMO USAMO and AMC?

Perfect Squares, Infinite Integers and Integers

by steven_zhang123, Mar 16, 2025, 12:06 PM

For which integer $h$ , are there infinitely many positive integers $n$ such that $\lfloor \sqrt{h^2 + 1} \cdot n \rfloor$ is a perfect square? (Here $\lfloor x \rfloor$ denotes the integer part of the real number $x$ ?

number theory

Perfect Squares

Unsolved Diophantine(I think)

by Nuran2010, Mar 14, 2025, 4:41 PM

Find all solutions for the equation $2^n=p+3^p$ where $n$ is a positive integer and $p$ is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)

number theory

function

Another NT FE

by nukelauncher, Sep 22, 2020, 11:58 PM

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$ .

another geometry problem with sharky-devil point

by anyone__42, Jun 27, 2020, 6:59 PM

Let $ABC$ be an acute triangle with $AC>AB$ , Let $DEF$ be the intouch triangle with $D \in (BC)$ , $E \in (AC)$ , $F \in (AB)$ ,, let $G$ be the intersecttion of the perpendicular from $D$ to $EF$ with $AB$ , and $X=(ABC)\cap (AEF)$ .
Prove that $B,D,G$ and $X$ are concylic

This post has been edited 1 time. Last edited by anyone__42, Jun 28, 2020, 8:35 PM

Foot from vertex to Euler line

by cjquines0, Jul 19, 2017, 4:36 PM

Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$ . A circle $\omega$ with centre $S$ passes through $A$ and $D$ , and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$ , and let $M$ be the midpoint of $BC$ . Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$ .

geometry

IMO Shortlist

Euler Line

Line Perpendicular to Euler Line

by tastymath75025, Jun 29, 2017, 2:55 AM

Let $ABC$ be a triangle with circumcircle $\Gamma$ , circumcenter $O$ , and orthocenter $H$ . Assume that $AB\neq AC$ and that $\angle A \neq 90^{\circ}$ . Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$ , respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\triangle ABC$ , respectively. Let $P$ be the intersection of line $MN$ with the tangent line to $\Gamma$ at $A$ . Let $Q$ be the intersection point, other than $A$ , of $\Gamma$ with the circumcircle of $\triangle AEF$ . Let $R$ be the intersection of lines $AQ$ and $EF$ . Prove that $PR\perp OH$ .

Proposed by Ray Li

IMO Shortlist 2011, Algebra 5

by orl, Jul 11, 2012, 10:23 PM

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.

Proposed by Canada

p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21

by who, Jul 8, 2006, 4:51 PM

Four real numbers $p$ , $q$ , $r$ , $s$ satisfy $p+q+r+s = 9$ and $p^{2}+q^{2}+r^{2}+s^{2}= 21$ . Prove that there exists a permutation $\left(a,b,c,d\right)$ of $\left(p,q,r,s\right)$ such that $ab-cd \geq 2$ .

Inequality => square

by Rushil, Oct 7, 2005, 5:12 AM

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$ , prove that $ABCD$ is a square.

Submit

this guy is an absolute legend. much love wherever you are Michael
by LeonidasTheConquerer, Aug 5, 2024, 9:37 PM
amazing blog
by anurag27826, Jun 17, 2023, 7:20 AM
hi i randomly found this
by purplepenguin2, Mar 1, 2023, 8:43 AM
can i be a contributor please?
by cinnamon_e, Mar 10, 2022, 6:58 PM
orzorzorzorzorzorozo
by samrocksnature, Jul 16, 2021, 8:25 PM
2021 post
by the_mathmagician, May 5, 2021, 3:28 PM
Let $ABC$ be an equilateral triangle of side length $1$ . Let $D$ be the point such that $C$ is the midpoint of $BD$ , and let $I$ be the incenter of triangle $ACD$ . Let $E$ be the point on line $AB$ such that $DE$ and $BI$ are perpendicular. $ \
by ARay10, Aug 25, 2020, 5:55 PM
Nice blog!
by User526797, Jan 12, 2020, 4:48 PM
oh my gosh it's been so longggggg.... contrib? what does that mean?
by adiarasel, Dec 1, 2019, 8:31 PM
2019 post
by piphi, Aug 10, 2019, 6:32 AM
hi contrib please
by Emathmaster, Dec 27, 2018, 5:38 PM
hihihihihi contrib plzzzzz
by haha0201, Aug 20, 2018, 3:58 PM
contrib please
by Max0815, Aug 1, 2018, 12:35 AM
contrib /charmander
by mathmaster2000, Apr 16, 2017, 4:59 PM
for contrib
by SomethingNeutral, Mar 30, 2017, 7:57 PM
all contrib requests added I usually get to posting every night after running on the treadmill and showering, posting my favorite problem(s) I did during the day.
by shiningsunnyday, Jun 16, 2016, 4:37 AM
Since problems is plural, he might post multiple problems through different posts in one day...
by zephyrcrush78, Jun 15, 2016, 8:37 PM
hi can i have contrib?
by wu2481632, Jun 15, 2016, 8:36 PM
Hello it is called problems of the DAY
by Temp456, Jun 15, 2016, 8:17 PM
10th shout lol
How often will this blog be updated?
by zephyrcrush78, Jun 15, 2016, 5:24 PM
why are we counting shouts

contrib pls
by MathAwesome123, Jun 15, 2016, 2:55 PM
eighth shout
contrib pls
by skipiano, Jun 15, 2016, 2:21 PM
seventh shout
Can I be contrib?
Though probably not because it's not like I'll post anything anyways :/ :/
by zephyrcrush78, Jun 15, 2016, 1:20 AM
sixth shout

new goal: get the shouts that are congruent to 1 mod 5
by DeathLlama9, Jun 15, 2016, 12:33 AM
fifth shout >
by budu, Jun 14, 2016, 10:04 PM
4th yay
by EpicSkills32, Jun 14, 2016, 9:10 PM
third shout
by skipiano, Jun 14, 2016, 2:35 PM
Second shout
by zephyrcrush78, Jun 14, 2016, 12:06 AM
omg yesssss
by DeathLlama9, Jun 13, 2016, 2:19 PM