Find the probability

by ali3985, Apr 2, 2025, 2:20 PM

Let $A$ be a set of Natural numbers from $1$ to $N$.
Now choose $k$ ($k \geq 3$) distinct elements from this set.

What is the probability of these numbers to be an increasing geometric progression ?
probability
combinatorics
algebra
Sets
Natural Numbers
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3 var inquality

by sqing, Apr 2, 2025, 2:04 PM

Let $a,b,c>0$. Prove that
$$ \dfrac{ab}{a^2-ab+3b^2} + \dfrac{bc}{b^2-bc+3c^2} + \dfrac{ca}{c^2-ca+3a^2} \le1$$$$ \dfrac{ab}{a^2-ab+ 2b^2} + \dfrac{bc}{b^2-bc+2 c^2} + \dfrac{ca}{c^2-ca+ 2a^2}\le \dfrac{3}{2}$$$$ \dfrac{ab}{2a^2-ab+3b^2} + \dfrac{bc}{2b^2-bc+3c^2} + \dfrac{ca}{2c^2-ca+3a^2} \le \dfrac{3}{4}$$
inequalities proposed
inequalities
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Finding pairs of complex numbers with a certain property

by Ciobi_, Apr 2, 2025, 1:22 PM

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]holds for all positive integers $n$.
complex numbers
algebra
modulus
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Olympiad Geometry problem-second time posting

by kjhgyuio, Apr 2, 2025, 1:03 AM

In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
geometry
square roots
SMO
Olympiad Geometry
Olympiad
trapezium
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April Fools Geometry

by awesomeming327., Apr 1, 2025, 2:52 PM

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
geometry
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inequalities

by Cobedangiu, Mar 31, 2025, 5:20 AM

problem
Attachments:
inequalities
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Very hard FE problem

by steven_zhang123, Mar 30, 2025, 1:47 PM

Given a real number \(C\) such that \(x + y + z = C\) (where \(x, y, z \in \mathbb{R}\)), and a functional equation \(f: \mathbb{R} \rightarrow \mathbb{R}\) that satisfies \((f^x(y) + f^y(z) + f^z(x))((f(x))^y + (f(y))^z + (f(z))^x) \geq 2025\) for all \(x, y, z \in \mathbb{R}\), has a finite number of solutions. Find such \(C\).
(Here, $f^{n}(x)$ is the function obtained by composing $f(x)$ $n$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{n \ \text{times}})(x)$)
This post has been edited 2 times. Last edited by steven_zhang123, Mar 30, 2025, 2:03 PM
functional equation
algebra
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The Sums of Elements in Subsets

by bobaboby1, Mar 12, 2025, 3:23 PM

Given a finite set \( X = \{x_1, x_2, \ldots, x_n\} \), and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to \( \binom{2}{2^n} \) inequalities, these given comparisons satisfy the following three constraints:

1. The sum of elements of any non-empty subset is greater than 0.
2. For any two subsets, removing or adding the same elements does not change their comparison of the sums of elements.
3. For any two disjoint subsets \( A \) and \( B \), if the sums of elements of \( A \) and \( B \) are greater than those of subsets \( C \) and \( D \) respectively, then the sum of elements of the union \( A \cup B \) is greater than that of \( C \cup D \).

The question is: Does there necessarily exist a positive solution \( (x_1, x_2, \ldots, x_n) \) that satisfies all these conditions?
This post has been edited 2 times. Last edited by bobaboby1, Mar 12, 2025, 3:28 PM
Reason: Wrong typing.
algebra
hard algebra
algebra unsolved
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IMOC 2017 G2 , (ABC) <= (DEF) . perpendiculars related

by parmenides51, Mar 20, 2020, 9:08 AM

Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
This post has been edited 3 times. Last edited by parmenides51, Jan 2, 2022, 12:07 PM
Reason: huge typo, corrected after #3
geometry
area of a triangle
geometric inequality
areas
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Show that AB/AC=BF/FC

by syk0526, Apr 2, 2012, 3:06 PM

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
This post has been edited 5 times. Last edited by syk0526, Apr 4, 2012, 6:48 AM
geometry
circumcircle
trigonometry
parallelogram
geometric transformation
reflection
Asymptote
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Random Math Tidbits

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  • i searched up moving points and found this
    what the actual orz

    by balllightning37, Mar 24, 2024, 9:24 PM

  • what the orz have I seen here

    by avisioner, Feb 7, 2024, 2:50 PM

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    by 554183, Oct 18, 2021, 3:32 PM

  • One of the best blogs I have come across :omighty:

    by lneis1, Jul 26, 2021, 2:17 PM

  • Are u surprised by him making IMO
    looking at his posts, it was very likely any way

    by 554183, Jul 8, 2021, 6:05 AM

  • WAIT WHAT HE MADE IMO 2020

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    by OlympusHero, Jun 6, 2021, 3:00 AM

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    by RedFireTruck, May 5, 2021, 5:16 PM

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    by yofro, Apr 13, 2021, 1:44 AM

  • @below He is contestant 2 :omighty:

    by Gaussian_cyber, Sep 20, 2020, 10:37 AM

  • did you make IMO 2020? :)
    which contestant are you?

    by Orestis_Lignos, Sep 18, 2020, 2:27 PM

  • yayups IMO 2020 :omighty:

    by fukano_2, Sep 10, 2020, 6:30 AM

  • how do u know he made IMO?

    by Puffer13, Sep 6, 2020, 12:12 PM

  • Congrats on USA IMO!

    by Imayormaynotknowcalculus, Aug 15, 2020, 4:52 PM

  • IMO 2020 :o :omighty:

    by cmsgr8er, Aug 7, 2020, 8:16 PM

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