A sharp one with 3 var (2)

by mihaig, May 26, 2025, 3:15 AM

Let $a,b,c\geq0$ satisfying
$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$ Prove
$a+b+c+\sqrt{abc}\geq4.$

Interesting inequality

by sqing, May 26, 2025, 3:14 AM

Let $(a+b)^2+(a-b)^2=1.$ Prove that
$0\geq (a+b-1)(a-b+1)\geq -\frac{3}{2}-\sqrt 2$ $-\frac{9}{2}+2\sqrt 2\geq (a+b-2)(a-b+2)\geq -\frac{9}{2}-2\sqrt 2$

This post has been edited 2 times. Last edited by sqing, 2 hours ago

inequalities proposed

inequalities

Interesting inequality

by sqing, May 26, 2025, 2:43 AM

Let $a,b\geq 0, 2a+2b+ab=5.$ Prove that
$a+b^3+a^3b+\frac{101}{8}ab\leq\frac{125}{8}$

inequalities proposed

inequalities

Inspired by RMO 2006

by sqing, May 24, 2025, 3:24 PM

Let $a,b >0 .$ Prove that
$\frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1}$ Where $k\geq 0.03$
$\frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3$

This post has been edited 1 time. Last edited by sqing, Saturday at 3:34 PM

inequalities proposed

algebra

inequalities

Number of functions satisfying sum inequality

by CyclicISLscelesTrapezoid, Jul 9, 2023, 5:44 AM

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$ . A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called nice if there exists an index $k$ such that $\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.$ Prove that the number of nice functions is at least $n^n$ .

function

combinatorics

Math Problem of the week#3

by Keith50, May 9, 2021, 6:53 AM

Math Problem of the week#3 wrote:

$\textbf{\color{blue}{AMC 12 2000/25:}}$ Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) $[asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$
$\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

Solution

f(1)f(2)...f(n) has at most n prime factors

by MarkBcc168, Jul 15, 2020, 2:22 AM

Let $f(x) = 3x^2 + 1$ . Prove that for any given positive integer $n$ , the product
$f(1)\cdot f(2)\cdot\dots\cdot f(n)$ has at most $n$ distinct prime divisors.

Proposed by Géza Kós

This post has been edited 1 time. Last edited by MarkBcc168, Jul 16, 2020, 1:53 PM

number theory

cyberspace

Inequality em981

by oldbeginner, Sep 22, 2016, 8:49 AM

Let $a, b, c>0, a+b+c=3$ . Prove that
$\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}$

inequalities

algebra

IMO Shortlist 2010 - Problem G7

by Amir Hossein, Jul 17, 2011, 2:44 AM

Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$ , and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too.

Proposed by Géza Kós, Hungary

$[asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; } T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H[i],G[j]),al[i][j],fontsize(8));[/asy]$

geometry

geometric transformation

homothety

incenter

IMO Shortlist

IMO 2009, Problem 2

by orl, Jul 15, 2009, 2:27 PM

Let $ABC$ be a triangle with circumcentre $O$ . The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$ . respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$ . Suppose that the line $PQ$ is tangent to the circle $\Gamma$ . Prove that $OP = OQ.$

Proposed by Sergei Berlov, Russia

This post has been edited 2 times. Last edited by orl, Jul 15, 2009, 11:04 PM

Simple triangle geometry [a fixed point]

by darij grinberg, May 18, 2004, 8:25 PM

Three distinct points $A$ , $B$ , and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$ . Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$ . Suppose $\Gamma$ meets the segment $PB$ at $Q$ . Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$ .

Attachments:

This post has been edited 4 times. Last edited by djmathman, May 27, 2018, 3:46 PM
Reason: edited wording according to https://anhngq.files.wordpress.com/2010/07/imo-2003-shortlist.pdf

geometry

IMO Shortlist

Fixed point