Linetown Mayor Admits Orz

by Rijul saini, Jun 4, 2025, 6:59 PM

Having won the elections in Linetown, Turbo the Snail has become mayor, and one of the most pressing issues he needs to work on is the road network. Linetown can be represented as a configuration of $2025$ lines
in the plane, of which no two are parallel and no three are concurrent.

There is one house in Linetown for each pairwise intersection of two lines. The $2025$ lines are used as roads by the townsfolk. In the past, the roads in Linetown used to be two-way, but this often led to residents accidentally cycling back to where they started.

Turbo wants to make each of the $2025$ roads one-way such that it is impossible for any resident to start at a house, follow the roads in the correct directions, and end up back at the original house. In how many ways can Turbo achieve this?

Proposed by Archit Manas

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:27 PM

combinatorics

graph theory

lmao

Cute geometry

by Rijul saini, Jun 4, 2025, 6:51 PM

Let scalene $\triangle ABC$ have altitudes $BE, CF,$ circumcenter $O$ and orthocenter $H$ . Let $R$ be a point on line $AO$ . The points $P,Q$ are on lines $AB,AC$ respectively such that $RE \perp EP$ and $RF \perp FQ$ . Prove that $PQ$ is perpendicular to $RH$ .

Proposed by Rijul Saini

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:25 PM

geometry

Orthocenters equidistant from circumcenter

by Rijul saini, Jun 4, 2025, 6:31 PM

In triangle $ABC$ , consider points $A_1,A_2$ on line $BC$ such that $A_1,B,C,A_2$ are in that order and $A_1B=AC$ and $CA_2=AB$ . Similarly consider points $B_1,B_2$ on line $AC$ , and $C_1,C_2$ on line $AB$ . Prove that orthocenters of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equidistant from the circumcenter of $ABC$ .

Proposed by Shantanu Nene

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:19 PM

geometry

circumcircle

Beware the degeneracies!

by Rijul saini, Jun 4, 2025, 6:30 PM

Let $a,b,c$ be real numbers satisfying $\max \{a(b^2+c^2),b(c^2+a^2),c(a^2+b^2) \} \leqslant 2abc+1$ Prove that $a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2) \leqslant 6abc+2$ and determine all cases of equality.

Proposed by Shantanu Nene

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:19 PM

inequalities

algebra

My Unsolved Problem

by ZeltaQN2008, Jun 4, 2025, 1:23 PM

Let $ABC$ be an acute triangle inscribed in its circumcircle $(O)$ , and let $(I)$ be its incircle. Let $K$ be the point where the $A-mixtilinear$ incircle of triangle $ABC$ touches $(O)$ . Suppose line $OI$ intersects segment $AK$ at $P$ , and intersects line $BC$ at $Q$ . Let the line through $I$ perpendicular to $BC$ intersect line $KQ$ at $A'$ . Prove that: $AI \parallel PA'.$

This post has been edited 3 times. Last edited by ZeltaQN2008, Yesterday at 2:06 PM
Reason: misleading information

geometry

geometry unsolved

Functional equation: f(xf(y)+f(x)f(y))=xf(y)+f(xy)

by Behappy0918, Jun 3, 2025, 12:24 PM

Find all function $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y\in\mathbb{R}$ , $f(xf(y)+f(x)f(y))=xf(y)+f(xy)$

functional equation

algebra

2024 IMO P6

by IndoMathXdZ, Jul 17, 2024, 12:48 PM

Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$ ,
$f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y).$ Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$ , and find the smallest possible value of $c$ .

algebra

IMO

Painting Beads on Necklace

by amuthup, Jul 12, 2022, 12:24 PM

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.

Carl Schildkraut, USA

This post has been edited 2 times. Last edited by amuthup, Jul 15, 2022, 4:06 PM

IMO Shortlist

pigeonhole principle

ISL 2021

24 convex quadrilaterals

by popcorn1, Jul 20, 2021, 8:37 PM

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that

the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.

Turkey TST 2015 P1

by aloski1687, Apr 1, 2015, 7:22 AM

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

This post has been edited 1 time. Last edited by aloski1687, Apr 1, 2015, 9:03 AM
Reason: typo

Submit

11 year bump
by john0512, Dec 28, 2020, 3:03 AM
Sure.
by googology101, Feb 12, 2009, 12:38 AM
me want contrib
by gauss1181, Feb 11, 2009, 5:49 PM
Sorry, I ran out of time on the last shout...
by googology101, Feb 8, 2009, 11:19 PM
and contrib?
by james4l, Feb 8, 2009, 4:42 AM
I guess...
by googology101, Feb 7, 2009, 11:49 PM
contrib please
by james4l, Feb 7, 2009, 10:18 PM
Yeah, I noticed that too. I'll change it later, but there's a weird glitch for you. The top and left properties should be automatically set to 0 if position is set to absolute or fixed. Go Firefox!
by googology101, Jan 21, 2009, 10:03 PM
i viewed this from ie and it looks like position fixed also ruins the left aligned so put left:0. nice submit buttons
by Sephiroth, Jan 21, 2009, 5:32 AM
Use filter:alpha(opacity=80) to make it semitransparent in IE. And make sure you use -moz-opacity instead of opacity to make sure that older versions of Firefox will cooperate. And by the way, you're free to copy the code. The whole thing is at http://www.artofproblemsolving.com/Forum/templates/blogs/Hyperion/style/c1366.css
by googology101, Jan 19, 2009, 9:50 PM
add filter: alpha(opacity=80) and opacity:0.8
by Sephiroth, Jan 19, 2009, 3:51 AM
Could I use that code in my blog?
by dysfunctionalequations, Jan 19, 2009, 1:51 AM
Watch out! position:fixed ruins the box's 100% width. Here's the code I used:

#navigation_box {
background-color: #005500 !important;
border-bottom-style:solid !important;
-moz-opacity:0.80;
color:white !important;
position:fixed !important;
width:100%;
padding-right:10px;
}

Don't overdo the opacity. 0.80 or 0.90 will do. Sorry, IE and Safari users, you aren't seeing it.
by googology101, Jan 19, 2009, 1:26 AM
what do you mean by floating menus? like how he made navigation_box fixed? he used "#navigation_box{ position: fixed;}" i like how you made it opacity
by Sephiroth, Jan 18, 2009, 4:45 AM
Floating menus? How do you do that?
by dysfunctionalequations, Jan 18, 2009, 1:25 AM
Yup. I'm trying to post daily.
by googology101, Jan 14, 2009, 12:05 AM
3rd post!
by 007math, Jan 13, 2009, 11:57 PM
Do you mean to this blog?
by googology101, Jan 12, 2009, 9:30 PM
Can i be a contributor?
by Wickedestjr, Jan 12, 2009, 5:49 PM

19 shouts

A googol is a tiny dot

by Rijul saini, Jun 4, 2025, 6:59 PM

by Rijul saini, Jun 4, 2025, 6:51 PM

by Rijul saini, Jun 4, 2025, 6:31 PM

by Rijul saini, Jun 4, 2025, 6:30 PM

by ZeltaQN2008, Jun 4, 2025, 1:23 PM

by Behappy0918, Jun 3, 2025, 12:24 PM

by IndoMathXdZ, Jul 17, 2024, 12:48 PM

by amuthup, Jul 12, 2022, 12:24 PM

by popcorn1, Jul 20, 2021, 8:37 PM

by aloski1687, Apr 1, 2015, 7:22 AM