3 var inquality

by sqing, Apr 16, 2025, 2:49 AM

Let $a,b,c> 0$ and $4(a+b) +3c-ab \geq10$ . Prove that
$a^2+b^2+c^2+kabc\geq k+3$ Where $0\leq k \leq 1.$
$a^2+b^2+c^2+abc\geq 4$

This post has been edited 1 time. Last edited by sqing, 3 hours ago

inequalities proposed

inequalities

The old one is gone.

by EeEeRUT, Apr 16, 2025, 1:37 AM

An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$ .

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots,$ there are infinitely many positive integers $n$ with $a_n = b_n$ .

This post has been edited 2 times. Last edited by EeEeRUT, 4 hours ago

algebra

A Projection Theorem

by buratinogigle, Apr 16, 2025, 1:30 AM

In triangle $ABC$ , prove that
$a = b\cos C + c\cos B.$

geometry

Ant wanna come to A

by Rohit-2006, Apr 15, 2025, 6:47 PM

An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$ . Find probability. For example $ABCDCDEDEA$ is valid but $ABABCDEABA$ is not valid.

*Too many edits, my brain had gone to a trip

Attachments:

This post has been edited 4 times. Last edited by Rohit-2006, 3 hours ago

combinatorics

Interesting inequalities

by sqing, Apr 15, 2025, 8:32 AM

Let $a,b$ be reals such that $a^2+ab+b^2 =1$ . Prove that
$\frac{8}{ 5 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq 1$ $\frac{9}{ 5 }\geq\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq 1$ $\frac{27}{ 14 }\geq \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq 1$ Let $a,b$ be reals such that $a^2+ab+b^2 =3$ . Prove that
$\frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$ $\frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$ $\frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$

This post has been edited 3 times. Last edited by sqing, Yesterday at 8:53 AM

inequalities proposed

inequalities

Turbo's en route to visit each cell of the board

by Lukaluce, Apr 14, 2025, 11:01 AM

Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$ , the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey

This post has been edited 1 time. Last edited by Lukaluce, Monday at 11:54 AM

Perhaps a classic with parameter

by mihaig, Jan 7, 2025, 6:01 PM

Find the largest positive constant $r$ such that
$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$ for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$

inequalities

parameterization

Pls solve this FE

by ItzsleepyXD, Nov 26, 2023, 8:14 AM

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $f(x^2f(x+y))=f(xyf(x))+xf(x)^2$ for all real numbers $x$ and $y$ .

functional equation

Functional equation in R

BMO Shortlist 2021 A5

by Lukaluce, May 8, 2022, 5:01 PM

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$f(xf(x + y)) = yf(x) + 1$ holds for all $x, y \in \mathbb{R}^{+}$ .

Proposed by Nikola Velov, North Macedonia

This post has been edited 2 times. Last edited by Lukaluce, May 10, 2022, 2:31 PM

Connected graph with k edges

by orl, Nov 11, 2005, 6:25 PM

Suppose $\,G\,$ is a connected graph with $\,k\,$ edges. Prove that it is possible to label the edges $1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.

Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $\,u,v\,$ belongs to at most one edge. The graph $G$ is connected if for each pair of distinct vertices $\,x,y\,$ there is some sequence of vertices $\,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $\,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $\,G$ .