Number Theory

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

Find all positve integers m such that $m+1 | 3^m+1$

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

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

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

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

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

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

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

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

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 $R_m=R_0$ .