Continuity of function and line segment of integer length

For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$ .
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$ .

Determine the equality case.

5 replies

Nuran2010

Apr 29, 2025

sqing

an hour ago

Inequality with 4 variables

bel.jad5 1

N an hour ago by sqing

Source: Own

Let $a$ , $b$ , $c$ $d$ positive real numbers. Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}$

1 reply

1 viewing

bel.jad5

Sep 5, 2018

sqing

an hour ago

Chain of floors

Assassino9931 0

an hour ago

Source: Vojtech Jarnik IMC 2025, Category I, P2

Determine all real numbers $x>1$ such that
$\left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots$ for any positive integer $n$ .

0 replies

Assassino9931

an hour ago

0 replies

a^n + b is divisible by p but not by p^2

Assassino9931 0

an hour ago

Source: Vojtech Jarnik IMC 2025, Category I, P1

Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$ , there is a prime number $p$ (possibly depending on $a,b,n$ ) such that $a^n + b$ is divisible by $p$ , but not divisible by $p^2$ .

0 replies

Assassino9931

an hour ago

0 replies

Do not try to bash on beautiful geometry

ItzsleepyXD 8

N 2 hours ago by Ianis

Source: Own , Mock Thailand Mathematic Olympiad P9

Let $ABC$ be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$

8 replies

ItzsleepyXD

Wednesday at 9:30 AM

Ianis

2 hours ago

Sum of points' powers

Suntafayato 3

N 2 hours ago by Ianis

Given 2 circles $\omega_1, \omega_2$ , find the locus of all points $P$ such that $\mathcal{P}ow(P, \omega_1) + \mathcal{P}ow(P, \omega_2) = 0$ (i.e: sum of powers of point $P$ with respect to the two circles $\omega_1, \omega_2$ is zero).

3 replies

Suntafayato

Mar 24, 2020

Ianis

2 hours ago

PAMO 2017 Shortlst: Sum of maxima of adjacent pairs in permutation

DylanN 1

N 3 hours ago by MelonGirl

Source: 2017 Pan-African Shortlist - I4

Find the maximum and minimum of the expression
$\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),$ where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$ .

1 reply

DylanN

May 5, 2019

MelonGirl

3 hours ago

Bijective quartic modulo p

DottedCaculator 12

N 3 hours ago by MathLuis

Source: ELMO 2024/6

For a prime $p$ , let $\mathbb{F}_p$ denote the integers modulo $p$ , and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$ . Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$ , there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$ . (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)

Aprameya Tripathy

12 replies

DottedCaculator

Jun 21, 2024

MathLuis

3 hours ago

No More than √㏑x㏑㏑x Digits

EthanWYX2009 3

N 4 hours ago by MathisWow

Source: 2024 April 谜之竞赛-3

Let $f(x)\in\mathbb Z[x]$ have positive integer leading coefficient. Show that there exists infinte positive integer $x,$ such that the number of digit that doesn'r equal to $9$ is no more than $\mathcal O(\sqrt{\ln x\ln\ln x}).$

Created by Chunji Wang, Zhenyu Dong

3 replies

EthanWYX2009

Mar 24, 2025

MathisWow

4 hours ago

IMO ShortList 2001, number theory problem 3

orl 10

N 4 hours ago by OronSH

Source: IMO ShortList 2001, number theory problem 3

Let $a_1 = 11^{11}, \, a_2 = 12^{12}, \, a_3 = 13^{13}$ , and $a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.$ Determine $a_{14^{14}}$ .

10 replies

orl

Sep 30, 2004

OronSH

4 hours ago

Continuity of function and line segment of integer length

egxa 2

N Apr 23, 2025 by NO_SQUARES

Source: All Russian 2025 11.8

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of $f$ . It is known that the graph of $f$ contains exactly $N$ chords, one of which has length 2025. Find the minimum possible value of $N$ .

2 replies