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Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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sum of projected area and a generalization of Oppenheim's inequality

From this post: In $\mathbb R^4$, let $U$ be a 2D plane, let $\pi_1$ be the projection from $U$ onto $xy$-plane and $\pi_2$ be the projection from $U$ onto $zw$-plane, then $\det\pi_1+\det\pi_2\le1$, ...
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In a triangle if $P\left(\frac{1}{a-b+x}+\frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)=\frac{1}{2}$ then $x=\pm\frac{3}{4}, \pm\frac{3}{5}$ or $0$?

Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let $$ f(x) = P\...
Nilotpal Sinha's user avatar
2 votes
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Inequality in an acute-angled triangle

Let $ABC$ be an acute-angled triangle and $M$ a point inside it. We denote by $C_1$, $C_2$, $C_3$ the centers of the Nine-Point circles corresponding to the triangles $BMC$, $CMA$, and $AMB$, ...
math.enthusiast9's user avatar
1 vote
1 answer
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Longest diagonal in a convex quadrilateral [closed]

Is is true that, in any convex quadrilateral $\mathcal{Q}$, the longest distance between any two points of $\mathcal{Q}$ is attained when we join two non-adjacent vertices of $\mathcal{Q}$? I suppose ...
Jamai-Con's user avatar
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In the plane there are given vectors $a, b, c, d, $, the sum of which is equal to $0$. Prove the inequality!

In the plane there are given vectors $a, b, c, d, $, the sum of which is equal to $0$. Prove the inequality $$\mid a \mid+ \mid b \mid + \mid c \mid + \mid d \mid \geq \mid a+d \mid + \mid b+d \mid + ...
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Geometric inequalities using sides, radius and semi-perimeter

Prove that: $$\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}S}+1 \ge 2\cdot\frac{4R+r}{\sqrt{3}p}$$ where $a, b, c$ are the sides of the triangle, $S$ is the area, $R$ is the circumradius, $r$ is the inradius and ...
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3 votes
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Sum of sides of a quadrilateral and sum of its diagonals

A proposition says that the sum of the sides of a quadrilateral is greater than the sum of its diagonals. The proof for this proposition goes as follows:- In quadrilateral $\mathrm{ABCD,}$ we have in $...
Sumant's user avatar
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Magnitude of sum of vectors $\le2$

Let $V_1V_2\cdots V_{2n}$ be a convex inscribed polygon of the unit circle. Let $\mathbf x=\sum\limits_{k=1}^n\overrightarrow{V_{2k-1}V_{2k}}$, prove that $|\mathbf x|\le2$. I started by setting ...
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Prove or disprove the inequality if $a,b,c>0$, $a \geq b+c$.

Prove or disprove the inequality $$a^2b+a^2c+b^2a+b^2c+c^2a+c^2b \geq 7abc$$ if $$a,b,c>0, a \geq b+c.$$ I thought to use this evaluation: $$a^2b+b^2c+c^2a \geq 3abc.$$ So we have: $$a^2b+a^2c+b^2a+...
 Alice Malinova's user avatar
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Prove or disprove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1$.

Prove or disprove that the inequality $$\dfrac{1}{\sqrt{1+x}}+\dfrac{1}{\sqrt{1+y}}+\dfrac{1}{\sqrt{1+z}} \geq 1$$ is valid if $x,y,z$ are positive numbers and $$xyz=1.$$ My solution is: Let $$x=\...
 Alice Malinova's user avatar
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Prove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1.$

Is given that $x,y,z$ are positive numbers and $xyz=1$, prove that $$\dfrac{\dfrac{1}{x}}{\sqrt{z^2+1}}+\dfrac{\dfrac{1}{y}}{\sqrt{x^2+1}}+\dfrac{\dfrac{1}{z}} {\sqrt{y^2+1}}>\sqrt{2}.$$ What have ...
 Alice Malinova's user avatar
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Proving a simple vector inequality [duplicate]

Can you prove that, if $\vec{x}, \vec{y}$ are real vectors and $\vec{x}$'s elements are nonnegative, then $$ \sum_i x_i \sum_j x_j y_j^2 \geq \left( \sum_i x_i y_i \right)^2 $$ I thought it followed ...
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Feasibility - When is a set defined by linear inequalities empty (in $\mathbb{R}^n$, with $n \geq 4$)?

Consider the set $\mathcal{S} \subseteq \mathbb{R}^n$, defined by the following $m > n \geq 4 $ linear inequalities: $$\begin{cases} a_{1,1}x_1 + a_{1,2}x_2 + \ldots + a_{1,n} x_n \leq b_1\\ ~\...
the_candyman's user avatar
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Norm inequality for Banach Lattices

Let $X$ be a Banach lattice, $\varepsilon>0$ and $x,y\in S_X$. If $\||x|\pm y\|\le1+\varepsilon$, then is it true that $\||x|+|y|\|\le1+f(\varepsilon)$? Where $f$ is some real function satisfying $\...
Stefano Ciaci's user avatar
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1 answer
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Showing $(4H-2L)e^{\frac{H+L}{k}}-2(H-L)(e^{H/k}+e^{\frac{H+2L}{k}})-L(e^{2H/k}+e^{2L/k})\geq 0$ for $k>0$, $H>0>L$, and $H+L>0$

Let $k>0$, $H>0>L$, and $H+L>0$. Prove that $$(4H-2L)e^{\frac{H+L}{k}}-2(H-L)(e^{H/k}+e^{\frac{H+2L}{k}})-L(e^{2H/k}+e^{2L/k})\geq 0.$$ I'm trying to prove the statement in the title ...
econra2017's user avatar
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Prove that $x\sin A+y\sin B+z\sin C\leqslant \frac{\sqrt{\left( x^2+k \right) \left( y^2+k \right) \left( z^2+k \right)}}{k}$

In triangle $ABC$, let $x,y,z,k>0$, prove that $$x\sin A+y\sin B+z\sin C\leqslant \frac{\sqrt{\left( x^2+k \right) \left( y^2+k \right) \left( z^2+k \right)}}{k}$$ where $k$ satisfies $\frac{x^2}{x^...
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$ab+bc+ca \le 4{\sqrt 3}\Delta$ for a triangle with sides $a$, $b$ and $c$ with area $\Delta$ [duplicate]

If $a$, $b$ and $c$ are the sides of a triangle with area $\Delta$, prove that $ab + bc + ca \le 4\sqrt3\Delta$ and prove that the equality holds iff the triangle is equilateral. I tried to approch ...
archbishop's user avatar
7 votes
4 answers
219 views

$\cos(a)+\cos(b)-\cos(a+b)\geq 1$

I am trying to prove that $$\cos(a)+\cos(b)-\cos(a+b)\geq 1$$ For $a,b \geq 0$ and $0\leq a+b\leq 180^°$ I have checked in Wolfram Alpha that the inequality is true, but I am not able to prove it. The ...
Juan Moreno's user avatar
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5 votes
1 answer
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Convex polygons - For $n>3$, $\sum_{i=1}^{n} \cos\theta_i < \cos\left(\sum_{i=1}^{n} {\theta_i}\right)$

It seems that for any convex polygon $P$ with $n>3$ sides and $n$ interior angles $\theta_i$, $$\sum_{i=1}^{n} \sin\theta_i > \sin\left(\sum_{i=1}^{n} {\theta_i}\right)$$ $$\sum_{i=1}^{n} \cos\...
Juan Moreno's user avatar
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How to prove inequalities in geometry

Outline: In a triangle $\triangle ABC$ let $\overline{AB}$ be the longest of the three sides. Let $G$ be the centroid of $\triangle ABC$ and $M$ the midpoint of $\overline{AB}$. Furthermore, let a ...
restpegel's user avatar
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$\displaystyle \frac{h_a}{l_a} + \frac{h_b}{l_b} + \frac{h_c}{l_c} \leq \frac{\sqrt{3(p^2 + r^2 - 8Rr)}}{2R} + \sqrt{3\frac{2R - r}{2R}}$

For a triangle with standard conventions, prove the inequality in the title, that is: $$\frac{h_a}{l_a} + \frac{h_b}{l_b} + \frac{h_c}{l_c} \leq \frac{\sqrt{3(p^2 + r^2 - 8Rr)}}{2R} + \sqrt{3\left(1 - ...
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1 answer
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Geometric-arithmetic mean inequality applied to eigenvalues

I've applied the arithmetic-geometric mean inequality to the eigenvalues of a positive definite matrix $X$, so $det(X)^{1/n}≤tr(x)/n$. Now I would like to show when equality holds. I already found out ...
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Distance from vector vs distance from projection

Let $x,y,z,w \in \mathbb{R}^n$. Assume $x^\top w \geq 0$, $y^\top w \geq 0$, $z^\top w = 0$ and $\|x\| = \|y\| = \|z\| = 1$, for a general norm $\|\cdot \|$. Moreover, define $p(x,w)$ to be the ...
durdi's user avatar
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2 votes
1 answer
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Euclidean angle vs euclidean distance between two vectors

Let $x,y,z \in \mathbb{R}^n$ and define the euclidean angle between two vectors as $$ a(x,y) := \arccos\left(\frac{x^\top y}{\|x\|_2\|y\|_2}\right). $$ Assuming $\|x\| = \|y\| = \|z\| = 1$ for a ...
durdi's user avatar
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Another proof of Euler's inequality via the half-angle formulas

The Euler's inequality is an immediate consequence of Euler's identity in a triangle, $$OI^2=R^2−2Rr.$$ An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
Emmanuel José García's user avatar
2 votes
1 answer
134 views

Showing $\sum_{cyc}\tan\frac\alpha2\tan\frac\beta2\geq4$ for a cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with sides $a$, $b$, $c$ and $d$. Denote $s$ the semiperimeter and let $\angle{DAB}=\alpha$, $\angle{ABC}=\beta$, $\angle{BCD}=\gamma$ and $\angle{CDA}=\delta$. ...
Emmanuel José García's user avatar
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1 answer
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Isoperimetric Inequalities for Infinite Regular Graphs

Say we have the two-dimensional regular graph $G=(\mathbb{Z}^2,S)$ with generator $S$ such that for every vertex we have $\operatorname{deg}(v)=\lvert S\rvert=\text{const.}$ for all $v\in G$. If we ...
Caesar.tcl's user avatar
3 votes
3 answers
121 views

Why is the triangle inequality equivalent to $a^4+b^4+c^4\leq 2(a^2b^2+b^2c^2+c^2a^2)$?

Consider the existential problem of a triangle with side lengths $a,b,c\geq0$. Such a triangle exists if and only if the three triangle inequalities $$a+b\geq c,\quad b+c\geq a\quad\text{and}\quad c+a\...
Ramen Nii-chan's user avatar
4 votes
0 answers
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A Pedoe-like Geometric Inequality

Problem Given two triangles $ABC$ and $A'B'C'$ where $a,b,c$ and $a',b',c'$ are the corresponding sides and $F, F'$ denotes the areas of the two triangles. Prove: \begin{equation} a^2a'^2+b^2b'^2+c^2c'...
E. Huang's user avatar
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1 answer
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Geometric inequality in regular pentagon

Let $ABCDE$ a regular pentagon inscribed in a circle of center $O$. Let $P$ an interior point of the pentagon from which we consider parallel line segments to all the sides of the pentagon. We know ...
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1 vote
1 answer
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Question about a geometric inequality

Question: Studying some geometric inequalities about arbitrary points, I thought of the following conjecture: Define triangle $ABC$ and let $M$ be an arbitrary point inside triangle $ABC$. Let $MD \...
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Sum of distances from interior point to vertices is greater than double the sum of distances to edges

Let $\triangle ABC$, $M\in Int(\triangle ABC)$. Let $MD\perp BC$, $ME\perp AC$ and $MF\perp AB$, $D\in BC$, $E\in AC$, $F\in AB$. Prove that $$MA+MB+MC\geq 2\cdot(MD+ME+MF).$$ My only idea is to use ...
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0 answers
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Geometric inequality with variable point in a triangle and circumcircles' radii

Problem statement Let $\triangle ABC$ a triangle and $M$ a point inside it. Let $\mathcal{C_c}$ be the circumcircle of $\triangle MAB$ and $\mathcal{C_a}$ and $\mathcal{C_b}$ similarly defined. Note $...
MathStackExchange's user avatar
1 vote
1 answer
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How to prove this inequality involving $\tanh$?

Let $a>0, b >0$ and let $0<\alpha<\theta<\pi$. Prove that $$\frac{1}{\tanh \left(\frac{1}{\frac{\sin (\alpha )}{a \sin (\theta )}+\frac{\sin (\theta -\alpha )}{b \sin (\theta )}}\...
ccriscitiello's user avatar
3 votes
2 answers
336 views

How to show an inequality in an inner product space?

Let $V$ be a real inner product space with inner product $\langle\cdot\,,\cdot\rangle$. For $u,v,w\in V$, how to show the following inequality $$\langle u,v\rangle \langle u,w\rangle\:\leq\: \frac{1}{...
Prof.Hijibiji's user avatar
3 votes
3 answers
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Determining all $(a,b)$ on the unit circle such that $2x+3y+1\le a(x+2)+b(y+3)$ for all $(x,y)$ in the unit disk

In the middle of another problem, I came up with the following inequality which needed to be solve for $(a, b)$ : $$2x+3y+1\le a(x+2)+b(y+3)$$ for all $(x, y)\in\mathbb{R}^2$ with $x^2+y^2\le1.$ Here ...
Bumblebee's user avatar
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4 votes
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Prove that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$ if $a^2+b^2+c^2+abc=4$.

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2+abc=4$. How do you prove that $\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$? My Approach: I ...
Omair Siddique's user avatar
0 votes
1 answer
158 views

In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Problem: Given a,b,c are length of triangle. Prove that: $$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$$ Happy Vietnamese Women's ...
Sickness's user avatar
3 votes
2 answers
255 views

Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression

Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression: a) $$P=\frac{4a}{b+c-a} + \frac{9b}{c+a-b} + \frac{16c}{a+b-c}$$ b) $$P=\frac{a^3}{2a+bc} + \...
Trường Hưng Nguyễn's user avatar
3 votes
1 answer
1k views

Largest possible side of a triangle when $\cos(3P)+ \cos(3Q)+ \cos(3R) = 1$.

In triangle $PQR$, $\cos(3P)+ \cos(3Q)+ \cos(3R) = 1$. Two sides of the triangle have lengths of $15 cm$ and $18 cm$. If the length of the third side of the triangle PQR is $\sqrt{m}$ cm, then the ...
Ganit's user avatar
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4 votes
1 answer
228 views

For an acute angled triangle $ABC,$ if $p=\frac{\sqrt3+\sin A+\sin B+\sin C}{2\sin A\sin B\sin C}$, find the range of $p$

$$ p=\frac{\sqrt3+\sin A+\sin B+\sin C}{2\sin A\sin B\sin C}$$ $\displaystyle \sin A+\sin B+\sin C=4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$ and $\displaystyle \sin A\sin B\sin C=8\cos\frac{A}{2}...
Tatai's user avatar
  • 755
4 votes
2 answers
94 views

Bounds of an Expression

if $(a,b,c)>0$ and $abc(a+b+c)=3$ Then what can you say about the bounds of $(a+b)(b+c)(c+a)$ ? Hint: My Approach 1.assumed $a\geq b \geq c$ 2.Tried to think geometrically 3.used the Hadwiger-...
StackpackedKar's user avatar
1 vote
3 answers
146 views

Show that $\frac{P_a}{a^2}+\frac{P_b}{b^2}+\frac{P_c}{c^2}\ge\frac{3}{4R}$. When is the equality reached?

Show that $\dfrac{P_a}{a^2}+\dfrac{P_b}{b^2}+\dfrac{P_c}{c^2}\ge\dfrac{3}{4R}$. When is the equality reached? We're dealing with an acute triangle $ABC$ where $h_a,h_b$ and $h_c$ are altitudes. $P_a,...
NikolDimitrova's user avatar
0 votes
1 answer
43 views

Let $h_{1},h_{2},h_{3}$ be the altitudes and $m_{1},m_{2},m_{3}$ be the medians of the triangle ABC.

Show that:$$\frac{h_1}{m_1}+\frac{h_2}{m_2}+\frac{h_3}{m_3}\leq3$$ So, I was wondering if we could prevent all the hefty geometry and solve this using Chebyshev's or the Rearrangement inequality. ...
Parth Shresth's user avatar
5 votes
3 answers
286 views

Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $

Show that for positive reals $x,y,z$ the following inequality holds and that the constant cannot be improved $$ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \...
vand's user avatar
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2 votes
1 answer
143 views

Does this class of triangles satisfy a certain geometric inequality?

If $\Delta{}ABC$ is a triangle, call the segment perpendicular to $AB$ and containing $C$ the altitude segment at $C$. In brief my question is the following: is it true that for all $0<\theta<\...
Sam van der Poel's user avatar
1 vote
4 answers
123 views

Given $a,b,c$ are sides of a triangle, Prove that :- $\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$

Given $a,b,c$ are sides of a triangle, Prove that :- $$\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$$ What I Tried:- I was able to solve the left hand side inequality. From RMS-AM ...
Anonymous's user avatar
  • 4,254
7 votes
4 answers
344 views

Prove that if $a,b,c$ are sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$

Prove that, if $a,b,c$ are the sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$. What I Tried:- Nothing special really came in ...
Anonymous's user avatar
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0 answers
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Systems of linear homogenous inequalities: getting started

I have a number of questions of varying difficulties related to satisfying largish (as large as possible) systems of linear inequalities. I gather these aren't easy, so I'd be happy to get numerical ...
David's user avatar
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0 votes
1 answer
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Proving a conditional algebraic inequality

Let $x,y \in (0,1)$, and suppose that $$ x^2-2x+y^2<0. $$ How to prove that $$ -x^3-xy^2+4y^2 \ge 0. $$ holds? The motivation comes from a certain geometric problem (a bit long to describe here).
Asaf Shachar's user avatar
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