$$ac=bd$$

by sqing, Mar 30, 2025, 2:25 PM

Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ abcd\ge 9.$ Prove that$$ac=bd$$Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ ad+bc \ge 6.$ Prove that$$ac=bd$$Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ab+cd \geq \frac{13}{2}.$ Prove that$$ac=bd$$
algebra
L

Polynomials and their shift with all real roots and in common

by Assassino9931, Mar 30, 2025, 1:12 PM

We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials \( P(x), Q(x) \) and a constant \( C \in \mathbb{R}, C \neq 0 \), it is given that \( P(x) + C \) and \( Q(x) + C \) are also friendly polynomials. Prove that \( P(x) \equiv Q(x) \).
This post has been edited 1 time. Last edited by Assassino9931, Yesterday at 1:13 PM
algebra
polynomial
L

Thanks u!

by Ruji2018252, Mar 30, 2025, 11:07 AM

Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$.
$1,$ Prove $xz=yt$
$2,$ Find maximum $P=x+z$
inequalities
L

VERY HARD MATH PROBLEM!

by slimshadyyy.3.60, Mar 29, 2025, 10:49 PM

Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.
Inequality
inequalities
L

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
L

Inspired by old results

by sqing, Mar 27, 2025, 12:35 PM

Let $ a,b,c > 0 $ and $ a+b+c +abc =4. $ Prove that
$$ a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$$Let $ a,b,c > 0 $ and $ ab+bc+ca+abc=4. $ Prove that
$$ a^2 + b^2 + c^2 + 2abc \geq 5$$
This post has been edited 1 time. Last edited by sqing, Mar 27, 2025, 12:44 PM
inequalities proposed
algebra
inequalities
L

The reflection of AD intersect (ABC) lies on (AEF)

by alifenix-, Jan 27, 2020, 5:00 PM

Let $ABC$ be a triangle. Distinct points $D$, $E$, $F$ lie on sides $BC$, $AC$, and $AB$, respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$. Let $Q$ denote the reflection of $P$ across $BC$. Show that $Q$ lies on the circumcircle of $\triangle AEF$.

Proposed by Ankan Bhattacharya
This post has been edited 2 times. Last edited by alifenix-, Jan 27, 2020, 7:03 PM
geometry
Angle Chasing
auyesl
L

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.
geometry
circumcircle
trigonometry
geometry solved
Inversion
ELMO Shortlist
USA
L

x is rational implies y is rational

by pohoatza, Jun 28, 2007, 7:24 PM

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.

Proposed by J.P. Grossman, Canada
number theory
rational
decimal representation
IMO Shortlist
L

USAMO 1995

by paul_mathematics, Dec 31, 2004, 1:01 PM

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.
AMC
USA(J)MO
USAMO
geometry
circumcircle
trigonometry
perpendicular bisector
L

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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

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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
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
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  • Blog created: Mar 31, 2023
  • Total entries: 677
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