Complex number

by ronitdeb, May 29, 2025, 6:13 PM

Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$. Find all possible values of $z_1^2+z_2^2+...+z_5^2$
This post has been edited 1 time. Last edited by ronitdeb, 3 hours ago

Cup of Combinatorics

by M11100111001Y1R, May 27, 2025, 7:24 AM

There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
This post has been edited 1 time. Last edited by M11100111001Y1R, May 27, 2025, 7:26 AM

Elementary Problems Compilation

by Saucepan_man02, May 26, 2025, 1:44 PM

Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2  -ab-bc-cd-d +2/5=0$$

Arc Midpoints Form Cyclic Quadrilateral

by ike.chen, Jul 9, 2023, 4:36 AM

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
This post has been edited 4 times. Last edited by ike.chen, Jul 9, 2023, 5:15 PM

Odd digit multiplication

by JuanDelPan, Oct 6, 2021, 10:21 PM

Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.

$\textit{Proposed by Pablo Serrano, Ecuador}$
This post has been edited 2 times. Last edited by JuanDelPan, Oct 7, 2021, 12:36 AM

Drawing Triangles Against Your Clone

by pieater314159, Jun 27, 2019, 9:38 PM

Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)

Proposed by Milan Haiman

Inequality

by knm2608, Jun 25, 2017, 1:04 PM

If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$When does the equality occur?

Proposed by Dorlir Ahmeti, Albania
This post has been edited 4 times. Last edited by v_Enhance, Jun 25, 2017, 2:33 PM

Find all possible values of BT/BM

by va2010, Jul 7, 2016, 8:59 PM

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
This post has been edited 1 time. Last edited by v_Enhance, Jul 7, 2016, 9:23 PM
Reason: improve title

Circumcircle of XYZ is tangent to circumcircle of ABC

by mathuz, May 22, 2013, 3:39 PM

Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.

n-term Sequence

by MithsApprentice, Oct 22, 2005, 11:53 PM

An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

life crisis

avatar

pupitrethebean
Archives
+ November 2024
+ April 2023
Shouts
Submit
  • omg tysmmmm

    by pupitrethebean, May 10, 2025, 3:16 AM

  • hola bro

    by Staragon, May 10, 2025, 3:16 AM

  • haiiiiiii cute blog <3

    by yaxuan, May 8, 2025, 8:29 AM

  • 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

689 shouts
Tags
idk
poll
random
Taiwan
school
milestone unlocked
yay
Life
reveal
sad
Photo
spam
drawing
rant
food
no sleep
Halp
Scary
sleep
Bruh
lol
Weird
Happy
Long Post
rev
cry
dream
ew
excited
art
cool
CSS
ded
Japan
L L
math
Meme
PUPITRE
AAA
Halloween
hehe
help
Hi
interesante
kendo
Viola
woah
aaaaa
adios taiwan
Bleh
bye
Fun
funny
game
GET A LIFE
heppy
hungry
oof
OW
Plane
procrastination
scared
yummy
10K
a year in review
Adoptme
Bad
Baseball
brain
burp
Flight
fun facts abt me
GROSS
haha
happybirthday
I believe I can fly
insults
jetlag
Joke
music
nervous
new
no
nolife
noo
Orchestra
ouch
question
random lol
So cool
sob
stars
tics
touch grass
Why
Yes
yum
100
15th birthday
2.5k
4th of July
a post
AAAA
AAAAAAAAAAAAAAA
aaah
achievement
addy dum
addy is the best
adios
Airplane
airplaneee
airport
aldksaghag
Alpha
Annoying
Beach
bird
blog
BLUEY
boooooo
Bored
brain fart
broccoli
bus
candy
careful
Celebration
cheddar
chicken
chucken
Concert
congested
costco
Cousin
Crazy
creepy
cries
crying
dance
darn
din tai fung
Doctors
doktor
Donut
duck
ducky
DUM
Egg
epik
espanol
Evil
fact
failure
Fancy
fjkaljfak
FLEX
Fly
FUNZIES
Gecko
geometry
gingerbreadman
go
goals
goldenbirthday
goodnight
google
Grandma
GRRRR
h mart
HACKS
hahaha
Hand
HBD DUCK
heat
Hehehehe
hehehehehe
hmm
hmmmmm
hola
hop
HoW
Huh
huhh
hw
i am hot
i might take this seriously
i need friends
icecream
idrk
im going insane what is wrong
im hungry
Important
IS
itch
itsnight
Life stuff
like REALLY random
live laugh love addy
llws
mahjong
May
mean
Memory
Meow
miracle
more itch
Mosquito
my brother
Name
Netflix
nice
no ac
no comprendo
no school
Nooooo
Oh
oh no
oh no children
Old
oops
ootd
orange
panera
Photos
picture
pizza
Power
project
qotd
questionable
Rabbit
rating
remember kids
reminder
RIP
ruff ruff
sadness
sajfdjasf
seansuksd
send help
Sick
skip
smort
so random
sobbing
sobs
song
SOUP
Spanish
springbreak
stairs
Stinky
story
stuff
taipei
taipei 101
teasuckspp
this is a bad idea
Thoughts
Tired
Travel
twoset
uh oh
uju dont say smooth hand
update
Vacation
vans
vansisstinky
Warm
wedding
wevideo
what
what an L
whenitsnotday
work
WWI
y are there 24 posts in 14 day
ye
YEEE
About Owner
  • Posts: 4
  • Joined: Jan 19, 2023
Blog Stats
  • Blog created: Mar 31, 2023
  • Total entries: 681
  • Total visits: 26600
  • Total comments: 2446
Search Blog
a