ALGEBRA INEQUALITY

by Tony_stark0094, Apr 23, 2025, 12:17 AM

$a,b,c > 0$ Prove that $\frac{a^2+bc}{b+c} + \frac{b^2+ac}{a+c} + \frac {c^2 + ab}{a+b} \geq a+b+c$

Checking a summand property for integers sufficiently large.

by DinDean, Apr 22, 2025, 5:21 PM

For any fixed integer $m\geqslant 2$ , prove that there exists a positive integer $f(m)$ , such that for any integer $n\geqslant f(m)$ , $n$ can be expressed by a sum of positive integers $a_i$ 's as
$n=a_1+a_2+\dots+a_m,$ where $a_1\mid a_2$ , $a_2\mid a_3$ , $\dots$ , $a_{m-1}\mid a_m$ and $1\leqslant a_1<a_2<\dots<a_m$ .

This post has been edited 1 time. Last edited by DinDean, an hour ago
Reason: I forgot one condition for a_i's.

number theory

combinatorics

Iran second round 2025-q1

by mohsen, Apr 19, 2025, 10:21 AM

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

number theory

Woaah a lot of external tangents

by egxa, Apr 18, 2025, 5:14 PM

A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle $\Omega$ . Circles $\omega_a, \omega_b, \omega_c, \omega_d$ are inscribed in triangles $DAB, ABC, BCD, CDA$ , respectively. Common external tangents are drawn between $\omega_a$ and $\omega_b$ , $\omega_b$ and $\omega_c$ , $\omega_c$ and $\omega_d$ , and $\omega_d$ and $\omega_a$ , not containing any sides of quadrilateral $ABCD$ . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle $\Gamma$ . Prove that the lines joining the centers of $\omega_a$ and $\omega_c$ , $\omega_b$ and $\omega_d$ , and the centers of $\Omega$ and $\Gamma$ all intersect at one point.

geometry

inscribed

Dear Sqing: So Many Inequalities...

by hashtagmath, Oct 30, 2024, 5:52 AM

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you

inequalities

Sqing

9x9 Board

by mathlover314, May 6, 2023, 9:41 PM

There is a $9x9$ board with a number written in each cell. Every two neighbour rows sum up to at least $20$ , and every two neighbour columns sum up to at most $16$ . Find the sum of all numbers on the board.

combinatorics

Bunnies hopping around in circles

by popcorn1, Dec 12, 2022, 5:47 PM

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$ . The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$ . She hops along $\gamma$ from $A_1$ to $A_2$ , then from $A_2$ to $A_3$ , until she reaches $A_{2022}$ , after which she hops back to $A_1$ . When hopping from $P$ to $Q$ , she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$ ; if $\overline{PQ}$ is a diameter of $\gamma$ , she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points.

Kevin Cong

This post has been edited 5 times. Last edited by v_Enhance, Dec 19, 2022, 4:04 AM

combinatorics

USA TST

integer functional equation

by ABCDE, Jul 7, 2016, 7:52 PM

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that $f(x-f(y))=f(f(x))-f(y)-1$ holds for all $x,y\in\mathbb{Z}$ .

IMO Shortlist

functional equation

IMO Shortlist 2013, Number Theory #1

by lyukhson, Jul 10, 2014, 6:08 AM

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
$m^2 + f(n) \mid mf(m) +n$
for all positive integers $m$ and $n$ .

Estonian Math Competitions 2005/2006

by STARS, Jul 30, 2008, 1:17 AM

A $9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

combinatorics unsolved

combinatorics

Submit

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math_exploration

by Tony_stark0094, Apr 23, 2025, 12:17 AM

by DinDean, Apr 22, 2025, 5:21 PM

by mohsen, Apr 19, 2025, 10:21 AM

by egxa, Apr 18, 2025, 5:14 PM

by hashtagmath, Oct 30, 2024, 5:52 AM

by mathlover314, May 6, 2023, 9:41 PM

by popcorn1, Dec 12, 2022, 5:47 PM

by ABCDE, Jul 7, 2016, 7:52 PM

by lyukhson, Jul 10, 2014, 6:08 AM

by STARS, Jul 30, 2008, 1:17 AM