3 var inequality

by sqing, May 1, 2025, 12:47 AM

Let $a,b,c>0 .$ Prove that
$\left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left(1+\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$ $\left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( 1+\frac{a^2+bc}{b^2+ca}+\frac{b^2+ca }{a^2+bc}\right)$

inequalities proposed

inequalities

another problem

by kjhgyuio, May 1, 2025, 12:46 AM

........

Attachments:

number theory

problem interesting

by Cobedangiu, Apr 30, 2025, 5:06 AM

Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

This post has been edited 1 time. Last edited by Cobedangiu, Yesterday at 5:06 AM

algebra

BMO 2024 SL A3

by MuradSafarli, Apr 27, 2025, 12:42 PM

A3.
Find all triples $(a, b, c)$ of positive real numbers that satisfy the system:
$\begin{aligned} 11bc - 36b - 15c &= abc \\ 12ca - 10c - 28a &= abc \\ 13ab - 21a - 6b &= abc. \end{aligned}$

algebra

BMO 2024 SL A1

by MuradSafarli, Apr 27, 2025, 12:24 PM

A1.

Let $u, v, w$ be positive reals. Prove that there is a cyclic permutation $(x, y, z)$ of $(u, v, w)$ such that the inequality:

$\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}$
holds for all positive real numbers $a, b$ and $c$ .

This post has been edited 1 time. Last edited by MuradSafarli, Apr 27, 2025, 12:26 PM

algebra

inequalities proposed

2^x+3^x = yx^2

by truongphatt2668, Apr 22, 2025, 3:38 PM

Prove that the following equation has infinite integer solutions:
$2^x+3^x = yx^2$

number theory

Yet another domino problem

by juckter, Apr 9, 2019, 11:12 AM

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$ , determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$ . Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

combinatorics

EGMO 2019

direct limit

by iarnab_kundu, Nov 26, 2018, 2:00 PM

We consider a locally small category $\mathcal{C}$ . Let $I$ be a small category, which is called the "diagram" category.
Given a covariant functor $\alpha:I\to\mathcal{C}$ we call it a "diagram" in $\mathcal{C}$ .
Example- We consider $I$ is the category of natural numbers, that is for each natural number $n\in\mathbb{N}$ there is an object in $I$ , and there is a unique morphism $n\to m$ if and only if $n\le m$ . Then a functor $\alpha:I\to\mathcal{C}$ is said to be a direct system in the category.
In Algebra we are interested in not only the diagrams in $\mathcal{C}$ but also in the "maps" between them. Let $\alpha,\beta:I\to\mathcal{C}$ be two diagrams. Then a map $m:\alpha\Rightarrow\beta$ is the data of a morphism $m(i):\alpha(i)\to\beta(i)$ for each $i\in I$ such that the (DIAGRAM BELOW) commutes.
Remark-Notice that this generalizes our notion of maps between directed systems.
Remark2- Notice that a map between $\alpha,\beta$ above is just a natural transformation between $\alpha,\beta$ .
Therefore we are naturally interested in studying the category $\mathcal{C}^I$ , the category of "diagrams in $\mathcal{C}$ ".
Notice that there are "constant diagrams". Given $A\in\mathcal{C}$ we have a diagram $C_A:I\in\mathcal{C}$ such that $C_A(i)=A$ for all $i\in I$ , and for each morphism $\phi:i\to j$ we have $C_A(\phi)=\text{id}_A$ . We note that we have a functor $C:\mathcal{C}\to\mathcal{C}^I$ given by $A\to C_A$ . In fact it is interesting to note that the map $\text{Hom}(A,B)\to\text{Hom}(C_A,C_B)$ is a bijection. Thus it is fully faithful. Now we fix a diagram $\alpha:I\to\mathcal{C}$ . We have a contravariant functor $f_{\alpha}:\mathcal{C}^{op}\to\text{Sets}$ given by $C\mapsto\text{Hom}(C_A,\alpha)$ .
Definition- An element $\mathfrak{A}$ is said to be the "limit" of $\alpha$ if it represents the functor $f_{\alpha}$ , that is $f_{\alpha}$ is isomorphic to $h_{\mathfrak{A}}$ .

category theory

Show that XD and AM meet on Gamma

by MathStudent2002, Jul 19, 2017, 4:32 PM

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$ . The points $D$ , $E$ , $F$ are selected on sides $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$ , $\overline{IE}\perp \overline{AI}$ , and $\overline{IF}\perp \overline{AI}$ . Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$ . Prove that lines $XD$ and $AM$ meet on $\Gamma$ .

Proposed by Evan Chen, Taiwan

This post has been edited 2 times. Last edited by v_Enhance, May 6, 2019, 1:36 PM

Medium geometry with AH diameter circle

by v_Enhance, Jun 28, 2016, 2:25 PM

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ . Denote by $M$ , $N$ the midpoints of $\overline{AH}$ , $\overline{BC}$ . Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$ , and meets line $AN$ at a point $Q \neq A$ . The tangent to $\gamma$ at $G$ meets line $OM$ at $P$ . Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$ .

Proposed by Evan Chen

IMO 2010 Problem 5

by mavropnevma, Jul 8, 2010, 8:15 AM

Each of the six boxes $B_1$ , $B_2$ , $B_3$ , $B_4$ , $B_5$ , $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$ , $1\leq j \leq 5$ , remove one coin from $B_j$ and add two coins to $B_{j+1}$ ;

Type 2) Choose a non-empty box $B_k$ , $1\leq k \leq 4$ , remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$ .

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$ , $B_2$ , $B_3$ , $B_4$ , $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Proposed by Hans Zantema, Netherlands