Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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15 views

Lipschitz maps cannot increase the volume of a Borel set by a factor greater than $k^n$.

Let $(M,\,g,\,d,\,vol_g)$ be a Riemannian manifold with metric $g$, geodesic distance $d$ and volume form measure $vol_g = \sqrt{\det(g_{ij})}\cdot m$ ($m$ = Lebesgue measure) and a Lipschitz map $f:K\...
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2answers
86 views

Minimize area of n-gon circumscribed around unit circle

Given regular unit circle and a n-gon, circumscribed around this unit circle. I need to minimize area of n-gon. Also i need to find the limit of n-gon area with $n\rightarrow \infty$. Intuitively i ...
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0answers
95 views

BMO1 2010-11 Q6

I have attempted this question but I believe there is a better way of doing this. Please leave some comment on how I should improve my answer/ alternative ways of doing it. Thanks! Question: Let $a, ...
3
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3answers
49 views

Prove that $AD\cdot BD \cdot CD \leq \dfrac{32}{27}$ where $ABC$ is a triangle of circumradius 1 and $D\in (BC)$.

Let triangle $ABC$ of circumradius $1$ and $D$ a point on side $(BC)$. Prove that $$AD\cdot BD\cdot CD\leq \dfrac{32}{27}.$$ My idea. By letting $\alpha = \dfrac{BD}{BC}$ (of course $0<\alpha &...
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0answers
26 views

Is a bound on a variable useful if the bound is a function of itself?

This is an odd question, but say I have a bound given by $$ \frac{1}{a} - \frac{1}{b} \geq c$$ If we rearrange, we have $$ b \geq a(1 + bc) $$ But, is this useful? I know the word useful is kind ...
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1answer
24 views

Is this “constrained” infimum zero?

Let $s>0$ be a fixed real number. Does $$ \inf_{x,y>0,\, xy \ge \frac{1}{4},\,x-y=s}(\sqrt{x}-\sqrt{y})^2\big( (\sqrt{x}+\sqrt{y})^2-2\big)=0\,\,\,?$$ Note that by the AM-GM inequality, we ...
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1answer
40 views

How to find the sides of the triangle where $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$ [closed]

$\Delta ABC$ is such that $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$. How to find the sides of the triangle? Is there a simple way?
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1answer
67 views

Does this inequality hold with some constant factor $c>0$?

Does there exist a real number $c>0$ such that $$ (x-1)^2+(y-1)^2-2(\sqrt{xy}-1)^2\ge c\big( (x-\sqrt{xy})^2+(y-\sqrt{xy})^2 \big) \tag{*}$$ holds for every positive real numbers $x,y$ such that $...
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1answer
56 views

How to analyze the equation $(x-y)^2=2\big( (x+y)-2\sqrt{xy} \big)$?

Suppose that $x,y$ are positive real numbers and that $$ (x-y)^2=2\big( (x+y)-2\sqrt{xy} \big). \tag{*}$$ Then Mathematica claims that one of the following $3$ options holds: $$1. \, \, \, x=y.$$ $...
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2answers
48 views

Inequality $a^ab^bc^c \geq (a+b-c)^a(b+c-a)^b(c+a-b)^c$

I found a problem where now I essentially have to show the following inequality $$a^ab^bc^c \geq (a+b-c)^a(b+c-a)^b(c+a-b)^c$$ where $a,b,c$ are the sides of a triangle. I have tried a lot of ...
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2answers
67 views

If the side lengths of a triangle increase and the third side is fixed, the opposite angle decreases

Suppose that I have an Euclidean triangle in the plane with side lengths $a,b,c$. Denote the angle opposite $c$ by $\theta$. I am trying to prove, that if we keep $c$ fixed, and increase $a,b$, ...
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36 views

Parallelepiped inequality and probabilistic proof

Let $(v_i)_{i=1}^n$ be linearly independent unit vectors of $\mathbb{R}^n$ and $V=\{\sum_{i=1}^n \alpha_iv_i:0 \le \alpha_i\le 1\}$. Given $v=\sum_{i=1}^nx_iv_i \in V$ prove that there exists a ...
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2answers
150 views

Minimize $|a-1|^3+|b-1|^3$ with constant product $ab=s$

Let $0<s$, and define $$ F(s):=\min_{a,b \in \mathbb{R}^+,ab=s} \left(|a-1|^3+|b-1|^3\right). $$ I would like to find proofs for the claim $$ F(s)=\begin{cases} 1 - 3 s - 2s^{3/2}=F\big(a(s),b(s)\...
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3answers
142 views

Proving a complicated looking inequality in a simple way

This is again a search for alternative proofs: Let $0 <s \le 1$, and suppose that $0 <a,b $ satisfy $$ ab=s,a+b=1+\sqrt{s}. \tag{1}$$ I have a proof for the assertion $$ 2(1-\sqrt s)^3 \le |...
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5answers
68 views

If $a,b,c$ are the sides of a triangle, then $\dfrac{a}{b+c-a}+\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}$ is:

If $a,b,c$ are the sides of a triangle, then $\dfrac{a}{b+c-a}+\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}$ is: $A)$ $\le3$ , $B)$ $\ge3$, $(C)$ $\ge2$, $(D)$ $\le2$ My attempt is as follows:- $$\dfrac{1}{2}\...
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2answers
52 views

Let $a,b,c$ be side lengths of a triangle, $a+b+c=1$. Prove that $P=a^3+b^3+c^3+3abc<\frac{1}{4}$.

Let $a,b,c$ be side lengths of a triangle such that $a+b+c=1$. Prove that $$a^3+b^3+c^3+3abc<\frac{1}{4}\,.$$ I solved this question. However, I'd like to know if there's a neater solution that ...
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1answer
39 views

Proofs: Relationships in Triangles

Question: In the diagram ABC is a triangle in which AB = AC and BC = 1. D is the point on BC such that $\angle BAD=\theta$ and $\angle CAD=2\theta$, $BD=x$ and $CD=1-x$. i) Use the sine rule in each ...
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0answers
40 views

Prove that : $\prod_{cyc}(2a^2+2b^2-c^2)\prod_{cyc}(a+b-c)\leq a^2b^2c^2(a+b+c)^3$ [duplicate]

Let $a,b,c$ be the sides for given triangle $ABC$ , then the following inequality true : $$\prod_{cyc}(2a^2+2b^2-c^2)\prod_{cyc}(a+b-c)\leq a^2b^2c^2(a+b+c)^3$$ Actually Michael Rozenberg gave a ...
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2answers
104 views

Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$

Let $m_{a},m_{b},m_{c}$ be the lengths of the medians and $a,b,c$ be the lengths of the sides of a given triangle , Prove the inequality : $$m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$$ Where : $s : \...
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2answers
113 views

Inequality involving the angle bisectors of a triangle

Let $l_a,l_b,l_c$ denote the lengths of angle bisectors of a triangle with sides $a,b,c$ and semiperimeter $s$. I am looking for the best constant $K>0$ such that $$l_a^2+l_b^2+l_c^2> K s^2.$$ ...
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1answer
57 views

Law of Cosines and Heron's Formula in inequalities

I had the following question: Suppose $a$, $b$, and $c$ are non-zero real numbers, and $x$, $y$, and $z$ satisfy the equations $$ bx + ay = c, cx + az = b, cy + bz = a. $$ Prove that $-1 < x,...
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2answers
206 views

For acute $\triangle ABC$, prove $(\cos A+\cos B)^2+(\cos A+\cos C)^2+(\cos B+\cos C)^2\leq3$

Prove that, in an acute $\triangle ABC$, $$(\cos A+\cos B)^2+(\cos A+\cos C)^2+(\cos B+\cos C)^2\leq3$$ I tried this, but I can't to this. I used $AM\geq GM$ and got $$3\geq\cos(A-B)+\cos(A-C)+\...
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1answer
50 views

Geometric inequality in triangle with inradius

I am trying to examine if the inequality $ab+bc+ca \ge 12Rr+ \frac{a^2+b^2+c^2}{3}$ holds for a triangle. Since $ab+ca+ca \ge 4r(5R-r)$ (Bottema, p. 53) it is enough to prove $12r(2R-r) \ge a^2+b^2+c^...
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1answer
40 views

the sum of the reciprocals of the lengths of the interior angle bisectors of a triangle and the sum of the reciprocals of the lengths of the sides

I have been struggling with this problem: Prove that the sum of the reciprocals of the lengths of the interior angle bisectors of a triangle is greater than the sum of the reciprocals of the ...
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1answer
76 views

The maximum for $xy \sin \alpha + yz \sin \beta +zx \sin \gamma$.

Question: Deduce the maximum of $xy \sin \alpha + yz \sin \beta +zx \sin \gamma$ if $x,y,z$ are real numbers that satisfy $x^2+3y^2+4z^2=6$ with $0<\alpha,\beta,\gamma<\pi$ such that $\alpha+\...
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2answers
85 views

A geometric inequality for acute triangle $\sum \sqrt \frac{b^2+c^2-a^2}{a^2+2bc} \leq \sqrt3$

I am trying to prove $\sum \sqrt \frac{b^2+c^2-a^2}{a^2+2bc} \leq \sqrt3$ for an acute triangle. There is: $$\sum \sqrt \frac{b^2+c^2-a^2}{a^2+2bc} \leq \sqrt{3\sum\frac{b^2+c^2-a^2}{a^2+2bc}}$$ So ...
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3answers
81 views

Triangle's circumcircle inequality: Is it suitable for high schoolers?

I want to give this problem to high schooler students and I want to check if it is suitable for high school level. Let $A$, $B$, $C$ be the sides of a triangle and $R$ be the triangle's ...
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1answer
40 views

Distance between mid point of two sides of a quadrilateral

Given any quadrilateral ABCD. Let X be the midpoint of side AB and Y be the midpoint of side CD. How can I prove that XY is not greater than max{AC, BD} ? Intuitively I see it is true in all cases, ...
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2answers
64 views

Prove that at least one area is less then one quarter of area of ABC

Question- $P,Q,R$ are points on the sides $BC,CA,AB$ of $\Delta ABC$. Prove that the area of at least one of the triangles $AQR, BRP, CPQ$ is less than or equal to one quarter of the area of $\Delta ...
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1answer
63 views

In ABC find points X,Y,Z such that AXYZ is rhombus

Question - In ABC find points X,Y,Z on AB,BC,CA such that AXYZ is rhombus and area of AXYZ <= 1/2 AREA OF ABC My try - I know it very easy but I am not getting ...I take midpoints of sides ...
3
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1answer
122 views

Hard inequality problem (if $(a_2-a_1)^2 + (a_3-a_2)^2 + \ldots + (a_{2n}-a_{2n-1})^2 = 1$ …)

I have a hard problem : If $$(a_2-a_1)^2 + (a_3-a_2)^2 + \ldots + (a_{2n}-a_{2n-1})^2 = 1$$ where $a_1,a_2...,a_{2n} \in \mathbb{R}$ What is the maximum of $$(a_{n+1}+a_{n+2}+…+a_{2n})−(a_1+...
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2answers
55 views

Trouble with sum of inverse integers vs. inverse of sum of integers [duplicate]

I'm having trouble trying to show the following: $$ \frac{1}{r_{1}}+\dots+\frac{1}{r_{v}} \ge \frac{v^{2}}{r_{1}+\dots+r_{v}} $$ where $r_{i}>1$ for all i and $r_{i}$ are not necessarily equal. ...
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2answers
78 views

Prove that $m_a\geq \dfrac{b^2+c^2}{4R}$

Let triangle ABC, $m_a$ the lenght of the median from A, $b,c$ the lenghts of the segments AC and AB respectively and R the circumradius. Prove that: $m_a\geq \dfrac{b^2+c^2}{4R}$. I found this in a ...
3
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2answers
92 views

Prove $\prod\limits_{k=1}^N (1+0.8\cdot r_{k})-1\leq \left( \prod\limits_{k=1}^{N}(1+r_{k}) -1 \right)\cdot0.8$

Today I was asked to make a compound interest calculation in two different ways. And from this real life application, arose an interesting inequality that I was able to verify empirically but not ...
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1answer
41 views

Circular arrangement of numbers with constraint on 3 consecutive triples of them gives a unique solution

Around a circle write 1000 numbers, such that each three consecutive numbers A, B, C (B is between A and C) satisfy $A^2+C^2 \leq B-(1/8)$ . Find the maximum and the minimum value for the sum of the ...
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1answer
43 views

Solving Inequality Relation:

I have this inequality that I deduce from the problem: $$ \frac{3}{a+b} < \frac{1}{2} \left(\frac{1}{a} + \frac{1}{b}\right) $$ For the sake of problem: assuming $a,b > 0$ and $a \neq b$. ...
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3answers
79 views

For △ABC, prove $\frac a{h_a} + \frac b{h_b} + \frac c{h_c} \ge 2 (\tan\frac{\alpha}2+ \tan\frac{\beta}2 + \tan\frac{\gamma}2)$

Given $\triangle ABC$, (using the main parameters and notation), prove that $$ \frac{a}{h_a} + \frac{b}{h_b} + \frac{c}{h_c} \ge 2 \cdot \left(\tan\frac{\alpha}{2} + \tan\frac{\beta}{2} + \tan\frac{\...
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0answers
75 views

For convex cyclic hexagon $ABCDEF$, show $AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \geq 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$ [duplicate]

Given a convex hexagon $ABCDEF$ inscribed in the circle, prove that $$AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \;\geq\; 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$$ ("$AC$" means the ...
1
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2answers
75 views

A point inside a triangle

I am given a triangle $\triangle ABC$ with side lengths $a,b,c$ and a point $P$ inside it. $R_A=PA$, $R_C=PC$, $R_C=PC$ the distances from point $P$ to the sides $BC, AC, AB$ are $d_a, d_b, d_c$ ...
3
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3answers
60 views

Inequality Solution Correctness

Let $a,b,c \in \Re $ and $ 0 < a < 1 , 0 < b < 1 , 0 < c < 1 $ & $ \sum_{cyc} a = 2$ Prove that $$ \prod_{cyc}\frac{a}{1-a} \ge 8$$ My solution $$ \prod_{cyc}\frac{a}{1-a} \...
2
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1answer
75 views

Given $a, b, c > 0$ such that $a^2 + b^2 + c^2 + abc = 4$, prove that $\sum_{cyc}\frac{b}{\sqrt{(c^2 + 2)(a^2 + 2)}} \ge 1$.

Given positives $a, b, c$ such that $a^2 + b^2 + c^2 + abc = 4$, prove that $$\large \frac{a}{\sqrt{(b^2 + 2)(c^2 + 2)}} + \frac{b}{\sqrt{(c^2 + 2)(a^2 + 2)}} + \frac{c}{\sqrt{(a^2 + 2)(b^2 + 2)}} \ge ...
2
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1answer
72 views

Prove that for all acute triangles $\triangle ABC$, $r_a + r_b + r_c \ge m_a + m_b + m_c$. [duplicate]

Let $r_b$ and $m_b$ respectively be the exradius of the excircle opposite $B$ and the median drawn from the midpoint of side $CA$ of acute triangles $\triangle ABC$. Prove that $$\large r_a + r_b + ...
3
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1answer
84 views

Proof of inequality in triangle

I am back after a long time, with this question In a $\Delta ABC; r_1+ r = r_2 + r_3, \angle ABC > \dfrac{\pi}{3}$. Then prove that $b < 3a$. Here $r_1$ is exradius of excircle formed by ...
10
votes
4answers
660 views

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Let $ABCD$ be a regular tetrahedron with center $O.$ Consider two points $M,N,$ such that $\overrightarrow{NO}=-3\overrightarrow{MO}.$ Prove or disprove that $$NA+NB+NC+ND\geq MA+MB+MC+MD$$ I ...
0
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2answers
85 views

In triangle $ABC$, find maximum value of $\sin A \cos B + \sin B \cos C + \sin C \cos A$

In triangle $ABC$, find maximum value of $$\sin A \cos B + \sin B \cos C + \sin C \cos A$$ We could make $\cos C = - \cos(A+B)$ and $\sin C = \sin(A+B)$. But then we have a rather awkward ...
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1answer
39 views

Given that R is interior to triangle BAD, prove $BR+DR<BA+AD$

I need help with this proof for my math class!
2
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4answers
100 views

prove that : $a^{2}+b^{2}+c^{2}+4abc<\frac12$

Let $a,b,c$ be a sides of triangle Such that : $a+b+c=1$ Then prove that : $a^{2}+b^{2}+c^{2}+4abc<\frac{1}{2}$ My effort : Since $a+b+c=1$ $\implies$ $2S=sr=bc\sin A=\frac{abc}{2R}$ Also :...
1
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0answers
127 views

Prove that $\frac{1}{q}+\frac{1}{p}>\frac{1}{r}$ when you know that:

Let $ABC$ be a triangle. Two points $P$ and $Q$ are on sides $AB$ and $AC$, respectively. The point $R$ is the intersection of lines $BQ$ and $CP$. The distances from the points $P$, $Q$, and $R$ to ...
10
votes
4answers
699 views

A bound for $\sqrt\frac{b+c-a}a+\sqrt\frac{c+a-b}b+\sqrt\frac{a+b-c}c$ in a triangle

Assume that $ABC$ is a triangle with $a\geq b\geq c$, where the angle $A$ has a fixed value. We denote by $\Sigma$ the sum $$\sqrt\frac{b+c-a}a+\sqrt\frac{c+a-b}b+\sqrt\frac{a+b-c}c.$$ Then the only ...
1
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2answers
47 views

If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$ [duplicate]

If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$. Also show that equality occurs in the above inequality when $a=b=c$ ...

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