Number Theory

by TUAN2k8, Apr 18, 2025, 3:25 PM

Find all positve integers m such that $m+1 | 3^m+1$
fun nt
number theory
L

Inspired by 1984 IMO Problem 1

by sqing, Apr 18, 2025, 3:22 PM

Let $ a,b,c\geq 0 $ and $ a+b+c+a^2+b^2+c^2-abc=1. $ Prove that
$$ab+bc+ca- abc\le3\sqrt{3}-5$$$$ab+bc+ca-2abc\le18\sqrt{3}-31$$
This post has been edited 1 time. Last edited by sqing, 15 minutes ago
inequalities proposed
algebra
L

Inspired by 1984 IMO Problem 1

by sqing, Apr 18, 2025, 3:12 PM

Let $ a,b,c\geq 0 $ and $ a+b+c+a^2+b^2+c^2=1. $ Prove that
$$ab+bc+ca-2abc\le\dfrac{9}{2}-\dfrac{17}{6}\sqrt{\dfrac{7}{3}}$$$$ab+bc+ca-4abc\le\dfrac{13}{2}-\dfrac{25}{6}\sqrt{\dfrac{7}{3}}$$
This post has been edited 1 time. Last edited by sqing, 8 minutes ago
inequalities
inequalities proposed
L

Interesting inequality

by sqing, Apr 18, 2025, 2:57 PM

Let $ a,b,c\geq 0 $ and $ a+b+c=3. $ Prove that$$ab(1-c)+bc(1-a)+ca(1-b)+\frac{9}{4}abc \leq\frac{9}{4} $$
inequalities proposed
inequalities
L

Sum of divisors

by DinDean, Apr 18, 2025, 2:47 PM

Does there exist $M>0$, such that $\forall m>M$, there exists an integer $n$ satisfying $\sigma(n)=m$?
$\sigma(n)=$ the sum of all positive divisors of $n$.
number theory
L

Existence of perfect squares

by egxa, Apr 18, 2025, 9:48 AM

Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \]is a perfect square.
number theory
Perfect Squares
L

3D russian combo

by egxa, Apr 18, 2025, 9:41 AM

A natural number \(N\) is given. A cube with side length \(2N + 1\) is made up of \((2N + 1)^3\) unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a \(2 \times 2 \times 2\) cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?
combinatorics
L

A problem with non-negative a,b,c

by KhuongTrang, Mar 4, 2025, 3:50 PM

Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
inequalities
L

Circles tangent to AD and AB intersect on AC

by gghx, Aug 3, 2024, 2:33 AM

In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.
geometry
L

Constructing sequences

by SMOJ, Mar 31, 2020, 7:09 AM

Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.
combinatorics
L

The oldest, shortest words — "yes" and "no" — are those which require the most thought.

avatar

adityaguharoy
Archives
- April 2021
Shifting to wordpress (scoutmathematics.wordpress.com)
+ February 2021
Moving to wordpress
+ March 2020
Wish everyone safety from Covid-19
+ March 2018
Convex and concave functions in Real numbers -- Basic 1
+ February 2018
Relaxing the to-be satisfied conditions for Squeezing Monotone sequences
1. Algebra and sigma algebra on a set
+ January 2018
Broken Stick Problem with 3 pieces forming a triangle
Ramsey problem: 3 people in 6 pairwise knowing or strangers
Happy New Year 2018 wishes
+ December 2017
Christmas is coming -- Advance wishes
a^x> = 1+x for all real numbers means a=e
+ November 2017
For invertible matrix A if B is (A^2)^-1 then AB=BA
Kobayashi's theorem(on prime factors of elements of set)
A Theorem of "prime form"
+ September 2017
Fish gallery
Search function of AoPS
A puzzle around 17 and four primes
Diophantus identity
Candido's identity
At least one of G or G' is connected for simple G on finite
If G is self complementary on n vertices then 4|n(n-1)
No continuous bijection between (a,b) and [a,b]
Abel's partial summation identity for real sequences
AB invertible then A has right inverse B has left inverse
Gauss Jordan Elimination method on matrices
A & B nonzero n x n matrices, AB=0 means A,B non invertible
For matrices A,B in M_n(R) AB=I_n if and only if BA=I_n
+ August 2017
The Twelvefold Way
Calls to find gcd by Euclid is max for consecutive Fibonacci
+ July 2017
Constructing a function differentiable only at one point (0)
A and B are bijective implies A^2 and B^2 are bijective sets
Locally finite sets
Korselt Criterion
Interesting Reverse Relation under square between 12,&,21
Finding an interesting number
gcd(a^n,b^n)=(gcd(a,b))^n
+ June 2017
Zsigmondy's theorem
A brilliant creation of Lonesan sir
Lonesan sir's work
Explaining what a congruent to b modulo c means for c not 0
BOTTEMA Maple Software
Apery's constant
If x+a/x+b is rational, x irrational & a,b rational then a=b
Zeckendorf theorem
Pythagoras
a,b,c form acute triangle iff a^2,b^2,c^2 form triangle
a^n,b^n,c^n are always sides of triangle & a<=b<=c then b=c
Chess Puzzles (Mate in 6)
Defining ordered pair and studying from and the definition
Fiber (in mathematics) has two different meanings
A simple bijective function from R to (0,1)
Exists no continuous function f:R to R mapping QtoR/Q&R/QtoQ
Sophie Germain identity (factorization)
Cauchy-Bunyakovsky-Schwartz inequality for integrals
w^3=1,w nonreal & a,b,c real & aw^2+bw+c=0 implies a=b=c
sum_cyc((sin A)/(sin B),(A)/(B),(tan A)/(tan B))A,B,C acute
Bertrand's postulate
The Pythagorean Tree
The Rotating Barnstar.
+ May 2017
Cauchy Condensation Test theorem for convergence of series
The diophantine equation (x^2-n^2y^2=1) in integers
Non existence of function f(xy)=x+y for every real x,y
22/7 > π
+ April 2017
Non existence of function f(x+y)=xy for every x,y in real
Primes checked modulo 6
Weirestrass function (continuous but not differentiable)
259*39*(any number)=numbernumber..... (interesting)
The inequality motto (By BQ)
Function continuous at only the points x_1,x_2,.....,x_n
Solving the equation n(a^2)=b^2 ; n is not a perfect square
Number of Real solution in x of eqn:a^x+b^x=c^x(0<a<b<c)
P(P(x))=x implies P(x)=x
Easy floor function inequality
Hermite's identity
Ex Falso Quodlibet
Koch Curve
Sierpinski triangle
Combinatorial Arguments(O(A)<O(B) implies N(A)<N(B))
Trigonometry equation(sine^(odd)(x)+cosine^(odd)(x)=1)
Volterra's function(diff..derivative not Riemann integrable)
A nice Bolzano Weirestrass convergence ~ Example 1
VJIMC 2017, Category I, Problem 1 : Functional Equation
2017 HMMT ~ Algebra/NT #3: Functional Equation
Sqing inequality
Differentiability implies Continuity of Real Valued Function
Magma (algebra)
+ March 2017
2013 PUMAC team 10
Finding a limit
Fishing with nemo and dory
Applying ~ Real numbers NOT bijective to Rational numbers
A combinatorial counting (B2-2004)
f(x) has degree 2 and f(nonreal)=nonreal then f has degree 1
Polynomial as difference of two monotone polynomials
+ February 2017
Polynomial Approximation to Continuous functions ~Example 1
Cosine , sine , tangent of rational multiples of &pi;
+ January 2017
Stronger Pigeonhole Principle
Willy's Lemma
Content of a polynomial -- introduced
+ December 2016
Rules of Propositional Calculus
FRT FRS of degree 2
+ April 2016
Sequence
Shouts
Submit

  • hi guys $~~~$

    by Yiyj1, Apr 9, 2025, 7:25 AM

  • You will be remembered

    by giangtruong13, Feb 26, 2025, 3:38 PM

  • 2025 shout!

    by just_a_math_girl, Jan 12, 2025, 7:25 AM

  • 2024 shout

    by bachkieu, Aug 22, 2024, 12:52 AM

  • helooooooooooo

    by owenccc, Sep 27, 2023, 12:59 AM

  • hullo :<

    by gracemoon124, Jul 14, 2023, 2:33 AM

  • hello $ $

    by LeoLionTank, Feb 17, 2023, 9:02 PM

  • Halo thear

    by HoRI_DA_GRe8, Oct 18, 2022, 7:44 AM

  • hi!!
    just found this and I can't wait to read more!
    so happy to have found this blog!

    by Morrigan_Black, Jan 28, 2022, 1:05 PM

  • still waiting for the mathlinks camp lol

    by CinarArslan, Jan 9, 2022, 1:55 PM

  • hello :D

    by CyclicISLscelesTrapezoid, Nov 29, 2021, 6:36 PM

  • Right below the shout box it says how many it has.

    by pith0n, May 11, 2021, 5:08 AM

  • Oh really the blog has 100 posts! I never counted the number of posts here. If I get some free time I will create a new page on my wordpress website and there I will post all the contents of this blog. So, make sure that you check the wordpress site.

    by adityaguharoy, Mar 21, 2021, 2:58 PM

  • Nice blog!

    by DCode10, Mar 10, 2021, 7:00 PM

  • Hi adityaguharoy! Nice blog!

    by masadca, Feb 4, 2021, 9:19 PM

118 shouts
Tags
number theory
algebra
calculus
Inequality
function
real analysis
Real Analysis 1
real numbers
combinatorics
continuity
geometry
polynomial
Wikipedia
inequalities
linear algebra
prime numbers
rational numbers
Sequence
Vectors and Matrices
Convergence
functional equation
gallery
identity
Irrational numbers
Lemma
mathematics
Matrices
algorithm
Calculus 1
countable sets
definition
differentiability
easy
equation
Example
images
Integral
interesting
Links
probability
set theory
trigonometry
uncountable sets
Vectors
analysis
bijection
bijective function
complex numbers
continuous function
convergence and divergence
counting
differentiation
Diophantine equation
Fibonacci sequence
fishes
Fractals
GCD
Geometric Inequalities
graph theory
Greatest Integer Function
interesting number
inverse of matrices
logic
lonesan
modulo
non-existence
numbers
pi
Pictures
puzzles
pythagoras
Recreation
Sequence and Series
sequence definition
series
Solution
solve
Theorem
triangle inequality
tribute
12-21
1968
2018
22dividedby7
259 X 39
acute angled triangle
announcement
AoPS
Apery s constant
article
Attachment
barnstar
Bertrand s postulate
Bolzano Weirestrass
BOTTEMA
bounds
bq
Candido s identity
Category I
Cauchy condensation
Celebration
chess
chess-puzzles
collection
combinatorial-number theory
Community
complement graph
complex-geometry
computer
Computer Programming
computer-programming
concave functions
Congruency
connected graph
construction
content of a polynomial
continuous
Convex Functions
convex-concave
Coronavirus
Cos
cosine rule
Covid-19
cube-root of 1
definitions
degree 2
Determinants
differentiable
Digits
Diophantus identity
divergence
Euclidean algorithm
euclidean geometry
Euler s number
ex falso quodlibet
factorization
false
Fiber
Floor
foundational mathematics
FRS degree 2
FRT
Function Construction
functions
Gauss Jordan Elimination
google
graph
greatest common divisor
greetings
Happy New Year
Hermite s identity
history
HMMT
infinity
Integers
integrable
integral-calculus
integration
irrational
isomorphic graphs
isomorphism in graph
kobayashi
Koch curve
Koch snowflake
Korselt
Korselt criterion
limit
link
Locally finite set
magma
Maple
mathematical theory
mathematicians
matrix
Measure theory
Memory
merry christmas
method
modular arithmetic
modulo 6
motto
notation
number
number of outcomes
Number of Real Number solution
number puzzles
Order
ordered pair
pascal s triangle
pattern
PDF
pigeonhole principle
polynomial approximation
positive real numbers
precautions
predicate claculus
predicate logic
prime
Prime number
project Euler
propositional calculus
propositional logic
Putnam
pythagorean tree
Quadratic
Ramsey
Ramsey Theory
rational
Real number equations
reverse under square
riemann integral
Safety
search
self complementary graphs
Sets
Sierpenski
Sierpinski Triangle
Sierpnski
sin
slogan
snowflake
software
song
squaring
Stone-Weirestrass
stronger PhP
Tan
tends
terminology
Tradition
Triangle
trigonometric inequalities
truth
twelvefold way
unity
VJIMC
Volterra s function
Weirestrass
willy s lemma
xzlbq
zeckendorf theorem
Zsigmondy
About Owner
  • Posts: 4655
  • Joined: Apr 29, 2014
Blog Stats
  • Blog created: Apr 26, 2016
  • Total entries: 101
  • Total visits: 26044
  • Total comments: 61
Search Blog
a