Hard Combi Geo

by AbbyWong, Mar 30, 2025, 11:03 PM

A (possibly non-convex) planar polygon P is good if no two sides of P are parallel.
For any good polygon P, we may take any three sides of P and extend them into lines. These lines
intersect to form a triangle. Such a triangle is called a peritriangle of P. Let f(P) denote the minimal
number of peritriangles of P whose union completely cover P.
For each positive integer n, find all possible values of f(P) as P ranges over all good n-gons.

combinatorial geometry

combinatorics

ded
comment

Collinearity with orthocenter

by Retemoeg, Mar 30, 2025, 4:42 PM

Given scalene triangle $ABC$ with circumcenter $(O)$ . Let $H$ be a point on $(BOC)$ such that $\angle AOH = 90^{\circ}$ . Denote $N$ the point on $(O)$ satisfying $AN \parallel BC$ . If $L$ is the projection of $H$ onto $BC$ , show that $LN$ passes through the orthocenter of $\triangle ABC$ .

geometry

Easy problem

by Hip1zzzil, Mar 30, 2025, 1:18 PM

$(C,M,S)$ is a pair of real numbers such that

$2C+M+S-2C^{2}-2CM-2MS-2SC=0$
$C+2M+S-3M^{2}-3CM-3MS-3SC=0$
$C+M+2S-4S^{2}-4CM-4MS-4SC=0$

Find $2C+3M+4S$ .

This post has been edited 1 time. Last edited by Hip1zzzil, Yesterday at 1:20 PM
Reason: Rr

algebra

Another "OR" FE problem

by pokmui9909, Mar 29, 2025, 10:16 AM

Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$

(Condition) For all $x, y \in \mathbb{R}$ , $f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$

This post has been edited 1 time. Last edited by pokmui9909, Saturday at 10:25 AM

algebra

function

functional equation

fifth power

by mathbetter, Mar 25, 2025, 1:11 PM

$\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}$

number theory

prime numbers

Incenter and midpoint geom

by sarjinius, Jul 17, 2024, 12:41 PM

Let $ABC$ be a triangle with $AB < AC < BC$ . Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$ , respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$ . Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$ . Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$ .

Proposed by Dominik Burek, Poland

This post has been edited 4 times. Last edited by sarjinius, Jul 17, 2024, 4:20 PM

parallel tangent line

Bishops and permutations

by Assassino9931, Feb 29, 2024, 7:57 PM

Let $n$ be a positive integer. Initially, a bishop is placed in each square of the top row of a $2^n \times 2^n$
chessboard; those bishops are numbered from $1$ to $2^n$ from left to right. A jump is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row.

Find the total number of permutations $\sigma$ of the numbers $1, 2, \ldots, 2^n$ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order $\sigma(1), \sigma(2), \ldots, \sigma(2^n)$ , from left to right.

Israel

This post has been edited 1 time. Last edited by Assassino9931, Mar 4, 2024, 10:59 AM

Orthocenter madness once again!

by MathLuis, Oct 22, 2023, 10:58 PM

Let $ABC$ be an acute triangle with orthocenter $H$ . Points $A_1$ , $B_1$ , $C_1$ are chosen in the interiors of sides $BC$ , $CA$ , $AB$ , respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$ . Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$ , $B_2 = \overline{BH} \cap \overline{C_1A_1}$ , and $C_2 = \overline{CH} \cap \overline{A_1B_1}$ .

Prove that triangle $A_2B_2C_2$ has orthocenter $H$ .

Ankan Bhattacharya

This post has been edited 2 times. Last edited by v_Enhance, Oct 22, 2023, 11:44 PM

Number of times to do Euclidean GCD.

by MarkBcc168, Jul 10, 2018, 11:21 AM

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:

$f(1,1)=0$ .
$f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
$f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$ .

Prove that
$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$

This post has been edited 4 times. Last edited by MarkBcc168, Feb 8, 2020, 2:01 PM

IMO Shortlist

number theory

Euclidean algorithm

Similar triangles and complementary angles

by math154, Jul 2, 2012, 3:16 AM

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$ , let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$ , and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$ . Show that $\angle QPA + \angle OQB = 90^{\circ}$ .

Alex Zhu.

Submit

done!
$~~~~$
by pupitrethebean, Mar 19, 2025, 2:05 PM
Hi! Contrib?
by RandomeMids, Mar 13, 2025, 2:22 PM
aww tysmmmm
by pupitrethebean, Feb 3, 2025, 10:19 PM
revive the shoutbox lol

i like your blog : )
by witchofblackbirdpond, Feb 3, 2025, 7:56 PM
revive the shoutbox
by pupitrethebean, Jan 20, 2025, 4:42 AM
d
e
d
. $~~$
by ujulee, Nov 26, 2024, 11:37 PM
bruhhh nahhhh
by pupitrethebean, Oct 13, 2024, 3:22 PM
this blog is so aesthetic it's crazy
by Technodoggo, Oct 12, 2024, 11:16 PM
ehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh soon :>
by pupitrethebean, Oct 12, 2024, 1:29 AM
pupitre you should totally promote meee
by CC_chloe, Oct 11, 2024, 7:34 PM
admins and contribs r different colors
by pupitrethebean, Oct 8, 2024, 3:01 PM
why are certain usernames different colors :thonk:
by Technodoggo, Oct 8, 2024, 12:44 AM
spoooooky
by blair_givenchy, Oct 3, 2024, 12:24 PM
okie dokes
by pupitrethebean, Oct 3, 2024, 1:55 AM
could i pls contrib as well? and if it's alright could i have display name of "Lydia"?
by Hestu_the_Bestu, Oct 2, 2024, 5:52 PM
oh my good lord
by pupitrethebean, Apr 11, 2023, 6:43 PM
bALLSS
in myyyyyyy
by teasang, Apr 11, 2023, 6:38 PM
schoooolllllll
by pupitrethebean, Apr 11, 2023, 4:21 PM
NOOO DONT REMIND ME
by addyc, Apr 11, 2023, 2:18 PM
yes happy new day
congrats on grades being teleported to powerschool
by pupitrethebean, Apr 11, 2023, 4:29 AM
HAPPY NEW DAY
by addyc, Apr 11, 2023, 4:18 AM
yoda, i am not
by pupitrethebean, Apr 11, 2023, 2:59 AM
yoda are you???
by addyc, Apr 10, 2023, 11:57 PM
correct you are
by pupitrethebean, Apr 10, 2023, 7:17 PM
poop $\text{[math]}$
by addyc, Apr 9, 2023, 3:25 AM
hmm how can i make this shout box more interesting
by pupitrethebean, Apr 8, 2023, 3:42 PM
shhhhh we dont talk about that
that was before everything was done so it never happened
by pupitrethebean, Apr 5, 2023, 10:41 PM
NO U DELTED MY FIRST SHOUT, COMMENT, AND POST
by addyc, Apr 5, 2023, 9:52 PM
lets gooooo first shout
by pupitrethebean, Apr 5, 2023, 7:12 AM

686 shouts

pupitre

by AbbyWong, Mar 30, 2025, 11:03 PM

by Retemoeg, Mar 30, 2025, 4:42 PM

by Hip1zzzil, Mar 30, 2025, 1:18 PM

by pokmui9909, Mar 29, 2025, 10:16 AM

by mathbetter, Mar 25, 2025, 1:11 PM

by sarjinius, Jul 17, 2024, 12:41 PM

by Assassino9931, Feb 29, 2024, 7:57 PM

by MathLuis, Oct 22, 2023, 10:58 PM

by MarkBcc168, Jul 10, 2018, 11:21 AM

by math154, Jul 2, 2012, 3:16 AM