Old problem

by kwin, May 16, 2025, 3:44 AM

Let $a, b, c > 0$ . Prove that:
$(a^2+b^2)(b^2+c^2)(c^2+a^2)(ab+bc+ca)^2 \ge 8(abc)^2(a^2+b^2+c^2)^2$

Interesting inequalities

by sqing, May 15, 2025, 2:51 PM

Let $a,b,c \geq 0$ and $abc+2(ab+bc+ca) =32.$ Show that
$ka+b+c\geq 8\sqrt k-2k$ Where $0<k\leq 4.$
$ka+b+c\geq 8$ Where $k\geq 4.$
$a+b+c\geq 6$ $2a+b+c\geq 8\sqrt 2-4$

This post has been edited 1 time. Last edited by sqing, Yesterday at 3:15 PM

inequalities proposed

inequalities

4-vars inequality

by xytunghoanh, May 15, 2025, 2:10 PM

For $a,b,c,d \ge 0$ and $a\ge c$ , $b \ge d$ . Prove that
$a+b+c+d+ac+bd+8 \ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{cd}+\sqrt{da}+\sqrt{ac}+\sqrt{bd})$ .

This post has been edited 1 time. Last edited by xytunghoanh, Yesterday at 2:16 PM
Reason: add condition

inequalities

Cycle in a graph with a minimal number of chords

by GeorgeRP, May 14, 2025, 7:51 AM

In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.

This post has been edited 1 time. Last edited by GeorgeRP, Yesterday at 3:57 AM

combinatorics

graph theory

Team Selection Test

Interesting inequalities

by sqing, May 10, 2025, 1:29 PM

Let $a,b,c\geq 0 , (a+k )(b+c)=k+1.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2k-3+2\sqrt{k+1}}{3k-1}$ Where $k\geq \frac{2}{3}.$
Let $a,b,c\geq 0 , (a+1)(b+c)=2.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq 2\sqrt{2}-1$ Let $a,b,c\geq 0 , (a+3)(b+c)=4.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{7}{4}$ Let $a,b,c\geq 0 , (3a+2)(b+c)= 5.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2(2\sqrt{15}-5)}{3}$

inequalities proposed

inequalities

Inspired by KhuongTrang

by sqing, Jan 21, 2024, 1:29 PM

Let $a,b,c\ge 0$ and $a+b+c=3.$ Prove that
$\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{6}{abc+5}$ $\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{7}{abc+6}$ $\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{21}{17(abc+1)}$ $\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{42}{17(abc+2)}$

inequalities proposed

algebra

inequalities

Sum of bad integers to the power of 2019

by mofumofu, Mar 11, 2019, 11:37 AM

Given coprime positive integers $p,q>1$ , call all positive integers that cannot be written as $px+qy$ (where $x,y$ are non-negative integers) bad, and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$ . Prove that there exists a positive integer $n$ , such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$ .

number theory

China TST

Floor function and coprime

by mofumofu, Jan 9, 2018, 11:41 AM

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$ , such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$ .

number theory

floor function

function

Collinearity with orthocenter

by liberator, Jan 4, 2016, 9:38 PM

Let $ABC$ be an acute triangle with orthocenter $H$ , and let $W$ be a point on the side $BC$ , lying strictly between $B$ and $C$ . The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$ , respectively. Denote by $\omega_1$ is the circumcircle of $BWN$ , and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$ . Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$ , and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$ . Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand

geometry

IMO

RMM 2013 Problem 3

by dr_Civot, Mar 2, 2013, 10:43 AM

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ . The lines $AB$ and $CD$ meet at $P$ , the lines $AD$ and $BC$ meet at $Q$ , and the diagonals $AC$ and $BD$ meet at $R$ . Let $M$ be the midpoint of the segment $PQ$ , and let $K$ be the common point of the segment $MR$ and the circle $\omega$ . Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

geometry

circumcircle

geometric transformation

homothety

reflection

Submit

hi guys $~~~$
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by Morrigan_Black, Jan 28, 2022, 1:05 PM
still waiting for the mathlinks camp lol
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by adityaguharoy, Jun 20, 2016, 4:11 AM
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by MinionLA, Jun 9, 2016, 11:43 PM
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118 shouts

An Introduction to the Theory of Mathematics

by kwin, May 16, 2025, 3:44 AM

by sqing, May 15, 2025, 2:51 PM

by xytunghoanh, May 15, 2025, 2:10 PM

by GeorgeRP, May 14, 2025, 7:51 AM

by sqing, May 10, 2025, 1:29 PM

by sqing, Jan 21, 2024, 1:29 PM

by mofumofu, Mar 11, 2019, 11:37 AM

by mofumofu, Jan 9, 2018, 11:41 AM

by liberator, Jan 4, 2016, 9:38 PM

by dr_Civot, Mar 2, 2013, 10:43 AM