Help me please

by sealight2107, Apr 23, 2025, 2:40 PM

Let $m,n,p,q$ be positive reals such that $m+n+p+q+\frac{1}{mnpq} = 18$ . Find the minimum and maximum value of $m,n,p,q$

inequalities

Interesting combinatoric problem on rectangles

by jaydenkaka, Apr 23, 2025, 2:22 PM

Define act <Castle> as following:
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.

Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)

Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?

This post has been edited 1 time. Last edited by jaydenkaka, an hour ago

combinatorics

geometry

rectangle

(help urgent) Classic Geo Problem / Angle Chasing?

by orangesyrup, Apr 23, 2025, 1:51 PM

In the given figure, ABC is an isosceles triangle with AB = AC and ∠BAC = 78°. Point D is chosen inside the triangle such that AD=DC. Find the measure of angle X (∠BDC).

ps: see the attachment for figure

Attachments:

geometry

Collect ...

by luutrongphuc, Apr 21, 2025, 12:59 PM

Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that:
$f\left(f(xy)+1\right)=xf\left(x+f(y)\right)$

function

functional equation

Rectangular line segments in russia

by egxa, Apr 18, 2025, 10:00 AM

Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?

combinatorics

A game optimization on a graph

by Assassino9931, Apr 8, 2025, 1:59 PM

Let $X_0, X_1, \dots, X_{n-1}$ be $n \geq 2$ given points in the plane, and let $r > 0$ be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points $X_0, X_1, \dots, X_{n-1}$ , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex $X_i$ a non-negative real number $r_i$ , for $i = 0, 1, \dots, n-1$ , such that $\sum_{i=0}^{n-1} r_i = 1$ . Bob then selects a sequence of distinct vertices $X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k}$ such that $X_{i_j}$ and $X_{i_{j+1}}$ are connected by an edge for every $j = 0, 1, \dots, k-1$ . (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$ .) Bob wins if
$\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;$ otherwise, Alice wins. Depending on $n$ , determine the largest possible value of $r$ for which Bobby has a winning strategy.

hard problem

by Cobedangiu, Apr 2, 2025, 6:11 PM

Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$

inequalities

Problem 1

by SpectralS, Jul 10, 2012, 5:24 PM

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$ , and to the lines $AB$ and $AC$ at $K$ and $L$ , respectively. The lines $LM$ and $BJ$ meet at $F$ , and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$ , and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$ , to the ray $AB$ beyond $B$ , and to the ray $AC$ beyond $C$ .)

Proposed by Evangelos Psychas, Greece

geometry

IMO Shortlist

Factor of P(x)

by Brut3Forc3, Apr 4, 2010, 2:45 AM

If $P(x),Q(x),R(x)$ , and $S(x)$ are all polynomials such that $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),$ prove that $x-1$ is a factor of $P(x)$ .

Composite sum

by rohitsingh0812, Jun 3, 2006, 5:39 AM

Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ be positive integers and let $S = a+b+c+d+e+f$ .
Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$ . Prove that $S$ is composite.

Submit

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by EpicSkills32, Apr 26, 2012, 5:11 PM

536 shouts

Seems

by sealight2107, Apr 23, 2025, 2:40 PM

by jaydenkaka, Apr 23, 2025, 2:22 PM

by orangesyrup, Apr 23, 2025, 1:51 PM

by luutrongphuc, Apr 21, 2025, 12:59 PM

by egxa, Apr 18, 2025, 10:00 AM

by Assassino9931, Apr 8, 2025, 1:59 PM

by Cobedangiu, Apr 2, 2025, 6:11 PM

by SpectralS, Jul 10, 2012, 5:24 PM

by Brut3Forc3, Apr 4, 2010, 2:45 AM

by rohitsingh0812, Jun 3, 2006, 5:39 AM