3 var inequalities

by sqing, Apr 23, 2025, 1:13 PM

Let $a,b> 0$ and $a+b\leq 2ab .$ Prove that
$\frac{ a + b }{ a^2(1+ b^2)} \leq\frac{1 }{\sqrt 2}-\frac{1 }{2}$ $\frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \sqrt 2-1$ $\frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq\frac{\sqrt5 }{2}$

inequalities proposed

inequalities

algebra

\frac{1}{5-2a}

by Havu, Apr 23, 2025, 9:56 AM

Let $a,b,c \ge \frac{1}{2}$ and $a^2+b^2+c^2=3$ . Find minimum:
$P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.$

inequalities

Equal angles with midpoint of $AH$

by Stuttgarden, Mar 31, 2025, 1:10 PM

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$ , satisfying $AB<AC$ . The tangent line at $A$ to the circumcicle of $ABC$ intersects $BC$ in $T$ . Let $X$ be the midpoint of $AH$ . Prove that $\angle ATX=\angle OTB$ .

geometry

Spain

1:1 correspondance + Graph Theory

by jaydenkaka, Oct 24, 2024, 9:43 AM

Lets define Graph G as a graph with "n" vortexes and no edges. Define C(G) as a number of cycles that starts from a point, visit all points exactly once, and comes back to the point that they started (The paths made can't cross each other). Define R(G) as a number of routes that starts from a point, visit all points exactly once, and finishes at another point. (The paths made cannot cross each other.) Show that R(G)≥n*C(G). Also, show the reason why is it inequality instead of an equation, and show the equrilibrum conditions.

(See attachment for example)

Attachments:

This post has been edited 1 time. Last edited by jaydenkaka, Oct 24, 2024, 9:44 AM

graph theory

inequalities

ISL 2023 C2

by OronSH, Jul 17, 2024, 12:22 PM

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:

every term in the sequence is less than or equal to $2^{2023}$ , and
there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$ ) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which $s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.$

This post has been edited 1 time. Last edited by OronSH, Jul 17, 2024, 12:28 PM

combinatorics

Flipping L's

by MarkBcc168, Jul 17, 2024, 12:15 PM

Let $m$ and $n$ be positive integers greater than $1$ . In each unit square of an $m\times n$ grid lies a coin with its tail side up. A move consists of the following steps.

select a $2\times 2$ square in the grid;
flip the coins in the top-left and bottom-right unit squares;
flip the coin in either the top-right or bottom-left unit square.

Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.

Thanasin Nampaisarn, Thailand

This post has been edited 1 time. Last edited by MarkBcc168, Jul 24, 2024, 4:00 PM
Reason: author

IMO Shortlist

combinatorics

AZE IMO TST

2024 IMO P1

by EthanWYX2009, Jul 16, 2024, 1:13 PM

Determine all real numbers $\alpha$ such that, for every positive integer $n,$ the integer
$\lfloor\alpha\rfloor +\lfloor 2\alpha\rfloor +\cdots +\lfloor n\alpha\rfloor$ is a multiple of $n.$ (Note that $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$ For example, $\lfloor -\pi\rfloor =-4$ and $\lfloor 2\rfloor= \lfloor 2.9\rfloor =2.$ )

Proposed by Santiago Rodríguez, Colombia

This post has been edited 2 times. Last edited by EthanWYX2009, Jul 19, 2024, 5:33 AM
Reason: change to original text

algebra

IMO

<DPA+ <AQD =< QIP wanted, incircle circumcircle related

by parmenides51, Sep 22, 2020, 11:38 PM

Let $I$ be the incentre of acute-angled triangle $ABC$ . Let the incircle meet $BC, CA$ , and $AB$ at $D, E$ , and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$ , such that $F$ lies between $E$ and $P$ . Prove that $\angle DPA + \angle AQD =\angle QIP$ .

(Slovakia)

This post has been edited 1 time. Last edited by parmenides51, Sep 22, 2020, 11:41 PM

Domain of (a, b) satisfying inequality with fraction

by Kunihiko_Chikaya, Feb 26, 2014, 7:47 AM

For real constants $a,\ b$ , define a function $f(x)=\frac{ax+b}{x^2+x+1}.$

Draw the domain of the points $(a,\ b)$ such that the inequality :

$f(x) \leq f(x)^3-2f(x)^2+2$

holds for all real numbers $x$ .

Max and min of Sum of d_k^2

by Kunihiko_Chikaya, Feb 27, 2012, 6:41 PM

Given $n$ points $P_k(x_k,\ y_k)\ (k=1,\ 2,\ 3,\ \cdots,\ n)$ on the $xy$ -plane.
Let $a=\sum_{k=1}^n x_k^2,\ b=\sum_{k=1}^n y_k^2,\ c=\sum_{k=1}^{n} x_ky_k$ . Denote by $d_k$ the distance between $P_k$ and the line $l : x\cos \theta +y\sin \theta =0$ . Let $L=\sum_{k=1}^n d_k^2$ .

Answer the following questions:

(1) Express $L$ in terms of $a,\ b,\ c,\ \theta$ .

(2) When $\theta$ moves in the range of $0\leq \theta <\pi$ , express the maximum and minimum value of $L$ in terms of $a,\ b,\ c$ .

trigonometry

algebra proposed

algebra

Submit

I still exist as well.
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ORZ CODER
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[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
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I see you're making changes to the CSS.
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And I took part of it from CaptainFlint and then added a ton of modifications.
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fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
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see this PM
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I can't get the cursor to work.
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If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
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Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
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No. I don't have IDLE.
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Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
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I don't know how to change it....
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Do you know how the change the into thing above my avatar a different color? It's too hard to see.
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by sonone, Mar 31, 2020, 4:56 PM