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Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$. What is the geometric intuition behind this inequality?
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1answer
32 views

Geometric inequality

Given Triangle ABC, let D be mid point of BC and let E and F be two point on sides AB and AC respectively such that angle EDF is right angle (90°). Prove that BE + CF > EF. I was able to do it by ...
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1answer
53 views

Help in solving this geometric inequality

I have the following geometry problem with me: "The altitudes through the vertices A,B,C of an acute angles triangle meet the opposite sides at D,E and F respectively, and AB > AC. The line EF meets ...
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1answer
53 views

Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

For $a_1, \ldots , a_n \in \mathbb{R}, a_1 < a_2 < \cdots <a_n$ and $a_i \ne 0$, show that $\dfrac{n}{a_1 - a_0} + \dfrac{n - 1}{a_2 - a_1} + \cdots + \dfrac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^...
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0answers
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3
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1answer
111 views

For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$

In the interior of $\triangle ABC$ we take the arbitrary point $P$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\...
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1answer
82 views

Maximizing $\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$ for $x$, $y$, $z$ the angles of a non-acute triangle

$x$, $y$, $z$ are the angles of a triangle, one of which is not less than $\frac{\pi}{2}$. Find the maximum value of the expression $$\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$$ I ...
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1answer
43 views

A triangle has sides x, x+4, and 3x−5. What is the possible range of x?

A triangle has sides $x, x+4,$ $and$ $3x−5$. What is the possible range of x? The answer to this is not: $2x-9$ $<$ $x$ $<$ $4x-1$ Which is gotten from: --> $(3x-5)-(x+4)<x<3x-5+x+4$ ...
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1answer
187 views

A nice IMO 1983 inequality from a stronger inequality

If you are interested in IMO $1983$ please see: $$3[a^2b(a-b)+b^2c(b-c)+c^2a(c-a)]\geqq b(a+b-c)(a-c)(c-b),$$ where $a,b,c$ are three side-lengths of a triangle. If $c≠{\rm mid}\{a,b,c\}$, the ...
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1answer
50 views

Prove that in every triangle the inequality $a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $ takes place

Prove that in every triangle the inequality $$a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $$ takes place, with the usual notations ($a,b,c$ lengths of sides, $r_a, r_b, r_c$ radii of coresponding ...
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1answer
79 views

Proving the inequality $\angle A+\angle COP < 90^\circ$ in $\triangle ABC$

In an acute angled $\triangle ABC$, $AP \perp BC$and $O$ is its circumcenter. If $\angle C \ge \angle B + 30^\circ$, then prove that $$\angle A + \angle COP < 90^\circ$$ My Attempt: Extending the ...
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1answer
50 views

I cannot reproduce a geometric result algebraically

Take a parabola $u(x)=ax^2 + bx + c$, where $a<0$, and draw a secant on an upward-sloping portion of the parabola from $(w-h, u(w-h))$ to $(w+h, u(w+h))$, for some $w \leq -\frac b{2a} - |h|$. Now ...
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1answer
27 views

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $ I feel like the products between different variables (i.e. not x^2, y^2, z^2) give this inequality the $ \gt $ sign and I don't ...
3
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1answer
46 views

Equality in triangle obtuse

$m_1, m_2, m_3 $ are sides-lengths of a triangle such that $m_1\sqrt{m_1}+m_2\sqrt{m_2}=m_3\sqrt{m_3}$. Prove that this triangle is an obtuse-angled triangle. I don't have idea make run this ...
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1answer
399 views

Patterns in inequalities of triangle involving angles.

I was reading this page and wondered as why, inequalities for $\cos A$ (with argument $A$) become the same inequality for $\sin\frac{A}{2}$ (with argument $\frac{A}{2}$), similarly for $\tan$ and $\...
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3answers
381 views

Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest. I tried using Ptolemy's theorem but don't know how to make ...
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1answer
38 views

Inequality triangle Radon substitutions

I have this inequality: $$\sum \frac {a^3}{p-a}\geq 8(2R-r)^2$$ I have tried using Radon substitutions and I get this: $$\sum \frac{(y+x)^3}{x}\geq 8(2R-r)^2$$ I know from Holder that : $$\sum \frac{(...
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2answers
59 views

Is there a right triangle with angles $A$, $B$, $C$ such that $A^2+B^2=C^2$?

A right angle triangle with vertices $A,B,C$ ($C$ is the right angle), and the sides opposite to the vertices are $a,b,c$, respectively. We know that this triangle (and any right angle triangle) has ...
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1answer
147 views

Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Let $a,b,c\in \Bbb R^+$ such that $a+b+c=abc$. Prove that $$\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$$ Idea 1.From $a+b+c=abc\Leftrightarrow \frac{1}{ab}+...
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0answers
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Generalizing Heron's Formula for Cyclic $n$-gons

Consider the following extension of Heron's Formula. For a cyclic $n$-gon $C$ with side lengths $x_1, x_2, \dots, x_n$ and semi-perimeter $P = \frac{1}{2} \left( x_1 + x_2 + \dots + x_n\right)$ define:...
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2answers
112 views

Write in a concise form a set in 4-dimensional space.

I have the following subset of the 4-dimensional space $(x_1,x_2,x_3,y)$: \begin{align} \mathcal I=&\left\{x_1=0\, , \, y\in[0, 1]\, , \, 0 \leq x_2\leq y\, , \, 0 \leq x_3\leq y\right\}\cup\\ &...
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1answer
101 views

A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

Let $\triangle ABC$ have sides $BC=a$, $CA=b$, and $AB=c$. Let $m_a$, $m_b$, $m_c$ be the medians to $BC$, $CA$, and $AB$, respectively. Prove that $$(ab+bc+ca)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{...
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2answers
51 views

A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$? [duplicate]

Let $a_1,\dots,a_d,b_1,\dots,b_d$ be positive real numbers. Then $$ \prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$$ and equality holds if and only if $\frac{a_i}{a_j}=\frac{...
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3answers
69 views

geometry inequality

$M$ is a point in $\triangle ABC$. $AM$ intersect with $BC$ at $A_1$. $BM$ intersect with $AC$ at $B_1$. $CM$ intersect with $AB$ at $C_1$. Proof that: $$AA_1 \times BB_1 \times CC_1 \geq 27 (MA_1 \...
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2answers
72 views

Proof using AM-GM inequality

The questions has two parts: Prove (i) $ xy^{3} \leq \frac{1}{4}x^{4} + \frac{3}{4}y^{4} $ and (ii) $ xy^{3} + x^{3}y \leq x^{4} + y^{4}$. Now then, I went about putting both sides of $\sqrt{xy}...
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2answers
44 views

Find at least one solution to matrix inequality

I have the following problem posed: find at least one vector $\mathbf{x}$ such that $$ A\mathbf{x} + \mathbf{b} \geqslant \mathbf{0} $$ for a given matrix $A$ and vector $\mathbf{b}$. Nothing is ...
3
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3answers
224 views

Difficult inequality with three real variables

For any real $e, t, \sigma$ such that \begin{aligned} \label{s} 0&<e<1\,,\\ 0&<t<\pi\,,\qquad\qquad\qquad(1)\\ -\pi/2&\leqslant\sigma\leqslant\pi/2 \end{aligned} the ...
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2answers
71 views

Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [closed]

A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
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5answers
658 views

Minimizing $\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$

What is the minimum value of $$\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$$ I tried getting the first derivative, but I can't solve the equation when I put $y = 0$. Methods without using calculus ...
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2answers
87 views

INMO 1998 question

This was question 5 of 1998 INMO. Suppose $a,b,c$ are three real numbers such that the quadratic equation $$ x^2 - (a +b +c )x + (ab +bc +ca) = 0 $$has roots of the form $\alpha + i \beta$ where $\...
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1answer
65 views

Complex number inequalities

I got the following question below : I tried and got this as the region , is this correct ?
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0answers
38 views

lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.

Let $ABC$ be an acute triangle. The goal is to prove: \begin{align*} &(a)\;\;\;\pi(2R-r) < aA+ bB + cC < 4(2R-r)\\[2pt] &\qquad\qquad\text{[where in the above, $A,B,C$ denote the ...
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1answer
70 views

Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$

If $a,b,c$ are three sides of a triangle then I need to prove that $$ {\frac {a}{a+b}}+{\frac {b}{b+c}}+{\frac {c}{c+a}}\geq \frac{3}{2}\, \left( { \frac {a}{a+b}}+{\frac {b}{b+c}} \right) \left( {\...
2
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1answer
89 views

Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$ My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is ...
3
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2answers
66 views

Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$

I am dealing with the test of the OBM (Brasilian Math Olimpyad), University level, 2017, phase 2. As I've said at others topics (questions 1 and 2, this last yet open, here), I hope someone can help ...
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5answers
110 views

Finding the values of $x$ and $y$ for which $x^2 + y^2$ is a minimum [closed]

Three squares are shown in the diagram. The largest has side $AB$ of length $1$. The others have side $AC$ of length $x$, and side $DE$ of length $y$. As $D$ moves along $AB$, the values of $x$ and $...
2
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1answer
31 views

$A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$

Let $ABCD$ a square of length $ 2$. If we consider $A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$ and $B_1, B_2\in [BC]$ s.t. $BB_1 < BB_2< 1$ and $C_1, C_2\in [CD]$ $D_1, D_2\in [DA]$ with ...
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1answer
87 views

Inequality about Schur

Let a,b,c be the sides of triangle such that $a+b+c=1$. Prove that $$5(ab+bc+ca)\geq18abc+a+b+c$$ I tried to prove: $$5(ab+bc+ca)\geq18abc+a+b+c$$$$10(ab+ac+bc)\geq36abc+2(a+b+c)$$$$a(5b+5c-2-12bc)+b(...
4
votes
1answer
221 views

inequality in geometry

Given triangle $ABC$ has $BC=a$, $CA=b$, $AB=c$ and $M$ is a point of the triangle plane. Prove that: \begin{align*} \cos \dfrac{A}{2} \cdot MA+\cos \dfrac{B}{2} \cdot MB+\cos \dfrac{C}{2} \cdot ...
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0answers
90 views

Known proofs of isodiametric inequality

Isodiametric inequality is the following statement: Suppose that $A\subset\mathbb R^n$ is a compact set of diameter at most 2. Then the volume of $A$ is less than or equal to the volume of a unit ...
1
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2answers
290 views

Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$

Prove that if $a,b,c$ are the lengths of the edges of a given triangle, then the following inequality holds: $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + ...
2
votes
3answers
67 views

Prove inequality inside a triangle

"Let $ABC$ be a triangle with centroid $G$. A line $PQ$ is drawed in the triangle such that it passes through $G$ and intersects the sides $AB$ and $AC$ in $P$ and $Q$ respectively." Prove that: $$\...
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6answers
127 views

Proving $\frac{1}{\sin(A/2)}+\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\ge 6$, where $A$, $B$, $C$ are angles of a triangle [duplicate]

If $A$, $B$, and $C$ are the angles of a triangle, then $$\frac{1}{\sin \left(\frac{A}{2}\right)}+\frac{1}{\sin\left(\frac{B}{2}\right)}+\frac{1}{\sin\left(\frac{C}{2}\right)}\ge 6$$ I have used ...
0
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1answer
50 views

inequality proving for intersection

AB and CD are line segments, $AB=CD=1$, intersecting in point O, $\enspace$ $AB\cap CD =O$, $\angle AOC=60^{\circ}$. Prove that $AC+BD\geq1$. $\enspace$ What I tried: $AO+BO=1$, $\enspace$ $CO+DO=1$ ...
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3answers
60 views

Draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$

I need to draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$, where $x_1 \in [0, \frac{1}{2}]$, $x_2 \in [\frac{1}{2}, 1]...
0
votes
1answer
91 views

Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.

Problem For any $\triangle ABC$, let $a,b,c$ be the length of the side respectively. Show that $$(2b+2c-2a)^3(a+b+c)\geq18a^2bc.$$
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2answers
66 views

Minimal area of a right triangle with inradius $1$

I got this question, solved it, then forgot how I solved it. What is the minimal area of a right triangle with inradius $1$? My attempt: $r=\frac{a+b-c}2$, so $a+b=c+2$ $a^2+b^2=c^2$ This gives ...
2
votes
2answers
77 views

A geometric inequality for a triangle ABC

I have to prove that: $ \frac {a^2}{w_a^2} + \frac {b^2}{ w_b^2} +\frac {c^2}{ w_c^2} \ge 4,$ for the sides $a,b,c$ of the triangle $ABC,$ $ w_a, w_b, w_c $ the angle bisectors and $s$ its ...
3
votes
2answers
102 views

Proving a weird inequality

Recently I saw this interesting inequality problem from Singapore Math Olympiad 2016 (Open Section, Special Round), but I am not sure how to start, here it goes: Real numbers $a,b,c$ are such that $...
2
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1answer
54 views

A geometrical inequality with a sphere

A bee flied 4 meters (in total) and back to the original spot. Prove its path can be inscribed in a sphere with radius 1m. I am bad at $3D$ geometry, so I tried to reduce it to a plane, but I still ...