A4 BMO SHL 2024

by mihaig, May 15, 2025, 5:33 AM

Let $a\ge b\ge c\ge0$ be real numbers such that $ab+bc+ca=3.$
Prove
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+\left(\sqrt 3-1\right)c}\leq a+b+c.$$Prove that if $k<\sqrt 3-1$ is a positive constant, then
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+kc}\leq a+b+c$$is not always true
inequalities
L

Geometry

by AlexCenteno2007, May 15, 2025, 4:42 AM

Let ABC be an acute triangle. The altitudes from B and C intersect the sides AC and AB at E and F, respectively. The internal bisector of ∠A intersects BE and CF at T and S, respectively. The circles with diameters AT and AS intersect the circumcircle of ABC at X and Y, respectively. Prove that XY, EF, and BC meet at the exsimilicenter of BTX and CSY
geometry
L

Inspired by xytunghoanh

by sqing, May 15, 2025, 3:04 AM

Let $ a,b,c\ge 0, a^2 +b^2 +c^2 =3. $ Prove that
$$ \sqrt 3 \leq a+b+c+ ab^2 + bc^2+ ca^2\leq 6$$Let $ a,b,c\ge 0, a+b+c+a^2 +b^2 +c^2 =6. $ Prove that
$$ ab+bc+ca+ ab^2 + bc^2+ ca^2 \leq 6$$
inequalities proposed
algebra
L

inequality

by danilorj, May 14, 2025, 9:08 PM

Let $a, b, c$ be nonnegative real numbers such that $a + b + c = 3$. Prove that
\[ \frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1, \]and determine all such triples $(a, b, c)$ where the equality holds.
inequalities
L

Based on IMO 2024 P2

by Miquel-point, May 14, 2025, 6:15 PM

Prove that for any positive integers $a$, $b$, $c$ and $d$ there exists infinitely many positive integers $n$ for which $a^n+bc$ and $b^{n+d}-1$ are not relatively primes.

Proposed by Géza Kós
komal
number theory
L

Nice one

by imnotgoodatmathsorry, May 2, 2025, 2:10 PM

With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
inequalities
inequalities proposed
inequalities unsolved
L

Dou Fang Geometry in Taiwan TST

by Li4, Apr 26, 2025, 5:03 AM

Let $\omega$ and $\Omega$ be the incircle and circumcircle of the acute triangle $ABC$, respectively. Draw a square $WXYZ$ so that all of its sides are tangent to $\omega$, and $X$, $Y$ are both on $BC$. Extend $AW$ and $AZ$, intersecting $\Omega$ at $P$ and $Q$, respectively. Prove that $PX$ and $QY$ intersects on $\Omega$.

Proposed by kyou46, Li4, Revolilol.
geometry
circumcircle
L

functional equation

by hanzo.ei, Apr 6, 2025, 6:08 PM

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
functional equation
algebra
L

Equal segments in a cyclic quadrilateral

by a_507_bc, Jul 29, 2023, 12:15 PM

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.
geometry
cyclic quadrilateral
L

Iran geometry

by Dadgarnia, Apr 7, 2018, 3:26 PM

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.

Proposed by Iman Maghsoudi
This post has been edited 2 times. Last edited by Dadgarnia, Apr 8, 2018, 10:46 AM
geometry
Pole and Polar
Pappus
L

Fun with math!

avatar

aoum
Archives
- May 2025
Pascal's Triangle
Pascal's Law
Lucas-Lehmer Numbers
Centroids!
Medoids!
Finding the Center of a Triangle
Squaring the Circle
Euler's Conjecture on Sums of Like Powers
The Taylor Series
The Euler Characteristic
Higher Dimensions
The Fourth Dimension
The Third Dimension
The Second Dimension
The First Dimension
Laplace's Equation
The Platonic Solids
+ April 2025
Bernoulli Numbers
The Riemann Zeta Function
Matrices
Hilbert’s Hotel
The Chicken McNugget Theorem
Diophantine Equations
Recursion
Ptolemy’s Theorem
The Bell Curve
The Sierpiński Carpet
The Sierpiński Triangle
The Graph Minor Theorem
Kruskal’s Tree Theorem
Graham's Number
Problem of the Day
Legendre's Theorem
Dirichlet’s Approximation Theorem
The Pigeonhole Prinicple
Problem of the Day
Graph Theory
Cyclic Sieving Phenomenon
Price's Equation
Exponents
The Ideal Gas Law
Sequences of Binomial Type
Umbral Calculus
The Birthday Attack
The Lost Boarding Pass Problem
The Hales–Jewett Theorem
Van der Waerden's Theorem
Greedoids
Matroids
Ramsey's Theorem
The Gamma Function
+ March 2025
Double Factorials!!
Catalan Numbers
Buffon's Needle
Logarithms
Modular Arithmetic
Irrational Numbers
The Principle of Inclusion-Exclusion (PIE)
The Basic Reproduction Number
Factorials!
Cyclic Numbers
Minkowski's Theorem
The Drake Equation
The Multinomial Theorem
The Binomial Theorem
The Hodge Conjecture
Penrose Tilings
P vs NP Problem
The Menger Sponge
The Koch Snowflake
Newton's Third Law of Motion
Newton's Second Law of Motion
Newton's First Law of Motion
Bernoulli Numbers
P-adic Analysis
The Mandelbrot Set
Cardinality
Bayes' Theorem
Parabolas
Angle Trisection
The Angle Bisector Theorem
The Riemann Hypothesis
The Collatz Conjecture
Bertrand's Box Paradox
Coin Rotation Paradox
The Birthday Problem
Prime Numbers and Multiples of 24
Simon's Favorite Factoring Trick
Fermat's Factoring Method
Conic Sections
The Fibonacci Sequence
Problem of the Day
The Triangle Inequality
Heron's Formula
The Goldbach Conjecture
Finding the Circumference of the Earth
The Banach-Tarski Paradox
Problem of the Day
Fermat's Last Theorem
Problem of the Day
The Pythagorean Theorem
Problem of the Day
Problem of the Day
The Golden Ratio
Finding the Digits of Pi
The Collatz Conjecture
The Four Color Theorem
Euler's Identity
Infinity
Fractals
Famous Mathematical Conjectures
How to Prepare for the AMC 10
How to Calculate the Digits of Pi (Without Computers)
How to Calculate the Digits of Pi with Python
Free Books: AMC 8, AMC 10/12, Mathematical Formulas
How to Prepare for the National Science Bowl
How to Prepare for MATHCOUNTS
How to Prepare for the AMC 8
Welcome!
Shouts
Submit

  • Check out the Pascal's Law post. I included a cartoon from the xkcd serial webcomic.

    by aoum, Today at 1:04 AM

  • If you leave a comment on one of my posts—especially older ones—I might not see it right away.

    by aoum, May 2, 2025, 11:55 PM

  • 100 posts!

    by aoum, Apr 21, 2025, 9:11 PM

  • Very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very cool (The maximum of the factorial machine is 7228！

    by Coin1, Apr 21, 2025, 4:44 AM

  • cool blog and good content but it looks eerily similar to chatgpt

    by SirAppel, Apr 17, 2025, 1:28 AM

  • 1,000 views!

    by aoum, Apr 17, 2025, 12:25 AM

  • Excellent blog. Contribute?

    by zhenghua, Apr 10, 2025, 1:27 AM

  • Are you asking to contribute or to be notified whenever a post is published?

    by aoum, Apr 10, 2025, 12:20 AM

  • nice blog! love the dedication c:
    can i have contrib to be notified whenever you post?

    by akliu, Apr 10, 2025, 12:08 AM

  • WOAH I JUST CAME HERE, CSS IS CRAZY

    by HacheB2031, Apr 8, 2025, 5:05 AM

  • Thanks! I'm happy to hear that! How is the new CSS? If you don't like it, I can go back.

    by aoum, Apr 8, 2025, 12:42 AM

  • This is such a cool blog! Just a suggestion, but I feel like it would look a bit better if the entries were wider. They're really skinny right now, which makes the posts seem a lot longer.

    by Catcumber, Apr 4, 2025, 11:16 PM

  • The first few posts for April are out!

    by aoum, Apr 1, 2025, 11:51 PM

  • Sure! I understand that it would be quite a bit to take in.

    by aoum, Apr 1, 2025, 11:08 PM

  • No, but it is a lot to take in. Also, could you do the Gamma Function next?

    by HacheB2031, Apr 1, 2025, 3:04 AM

60 shouts
Contributors
aoum • HacheB2031 • sadas123
Tags
Problem of the Day
Fractals
geometry
combinatorics
Millennium Prize Problems
poll
Riemann Hypothesis
calculus
Collatz Conjecture
Factorials
graph theory
infinity
pi
Sir Issac Newton
AMC
Bernoulli numbers
Chudnovsky Algorithm
Exponents
Gauss-Legendre Algorithm
Goldbach Conjecture
Koch snowflake
MAA
Mandelbrot Set
Mastering AMC 1012
MATHCOUNTS
Matroids
Nilakantha Series
number theory
P vs NP Problem
P-adic Analysis
paradoxes
Polynomials
primes
probability
Ramsey Theory
1d
2D
3d
4d
algebra
Algorithmic Applications
AMC 10
AMC 8
angle bisector theorem
Angle trisection
Applications in Various Fields
Arc Sine Formula
Archimedes Method
Banach-Tarski Paradox
Basel Problem
Basic Reproduction Number
Bayes Theorem
Bell Curve
Bertrand s Box Paradox
binomial theorem
Birthday Attack
Birthday Problem
buffon s needle
Cantor s Infinite Sets
cardinality
catalan numbers
Center of a Triangle
Chicken McNugget Theorem
Circumference
Coin Rotation Paradox
computer science
conditional probability
conic sections
Conjectures
Cryptography
Cyclic Numbers
Cyclic Sieving Phenomenon
Different Sizes of Infinity
Diophantine Equations
Diophantinve Approximation
Dirichlets Approximation
Diseases
Double Factorials
Drake Equation
epidemiology
euclidean geometry
Euler Characteristic
Euler s Formula for Polyhedra
Euler s Identity
Euler s totient function
Euler-Lagrange Equation
Fermat s Factoring Method
fermat s last theorem
Fibonacci sequence
finite
First Dimenstion
four color theorem
Fourth dimension
Fractals and Chaos Theory
free books
Gamma function
Golden Ratio
Graham s Number
Graph Minor Theorem
gravity
Greedoids
Gregory-Liebniz Series
Hailstone Problem
Heron s Formula
Higher Dimensions
Hilbert s Hotel
Hilberts Hotel
Hodge Conjecture
ideal gas law
Inclusion-exclusion
infinite
Irrational numbers
Kruskals Tree Theorem
Laplace s Equation
Law of Force and Acceleration
legendre s theorem
Leibniz Formula
logarithms
logic
Lucas-Lehmer Numbers
Mastering AMC 8
Matrices
Medoids
Menger Sponge
Mersenne numbers
Minkowskis Theorem
modular arithmetic
Multinomial Theorem
Multiples of 24
National Science Bowl
Newton s First Law of Motion
Newton s Second Law of Motion
Newton s Third Law of Motion
normal distribution
Parabolas
Paradox
Pascal s Law
pascal s triangle
Penrose Tilings
physical chemistry
pie
pigeonhole principle
platonic solids
Price s Equation
prime numbers
Ptolemys Theorem
Pythagorean Theorem
Python
Ramsey s Theorem
recursion
Reproduction Rate of Diseases
Riemann Zeta Function
Second Dimension
Sequences
Sequences of Binomial Type
Sets
Sierpinski Triangle
Sierpiski Carpet
Sierpiski Triangle
Simon s Factoring Trick
Squaring the Circle
statistics
Sums of Like Powers
Taylor series
The Birthday Problem
The Book of Formulas
The HalesJewett Theorem
The Law of Action and Reaction
The Law of Inertia
The Lost Boarding Pass Problem
thermodynamics
Third Dimension
Topological Insights
triangle inequality
trigonometry
twin prime conjecture
Umbral Calculus
Van der Waerdens Theorem
venn diagram
Wallis Product
Zeno s Paradoxes
About Owner
  • Posts: 0
  • Joined: Nov 2, 2024
Blog Stats
  • Blog created: Mar 1, 2025
  • Total entries: 119
  • Total visits: 1431
  • Total comments: 40
Search Blog
a