Questions tagged [problem-solving]

Question 1

I want to deepen my probability knowledge. Started solving One Thousand Exercises in Probability, but apparently it turned out to be tough for me. Sometimes it is ...

Question 2

The game of nim is played with two players againts each other ,by removing 1 or many stones from only one pile in each turn from n piles each pile with $n_1,...,n_k$ and a player cannot skip a turn. ...

Question 3

I've been struggling through this and created a bunch of options where 12 work such as alternating colour 1 and 2 around the edges and having colour 3 in the middle, but I have a suspicion that there ...

Question 4

I recently came across this nice inequality, which looks simple but elegant. Here’s my short proof — and I’d love to see alternative approaches, preferably using only classical inequalities (Cauchy–...

Question 5

Consider a rectangle $ABCD$ with $BC = 2AB$. Let $L$ be the midpoint of side $AD$. From $L$, draw a perpendicular to diagonal $AC$ that intersects: $AC$ at point $K$ $BC$ at point $F$ Let $M$ be the ...

Question 6

A problem from Le Gall's Measure Theory, Probability and Stochastic Processes (Chapter 9, Exercise 9.11(4)), which I'm not really sure what it is asking: Let $(Y_n)$ be a sequence of i.i.d. real ...

Question 7

Let $ABCD$ be a square. Let $Dx$ be a ray from vertex $D$ that intersects side $BC$ internally at point $E$. Draw $BH$ perpendicular to ray $Dx$, where $BH$ intersects $Dx$ at point $F$ and intersects ...

Question 8

Given a square $ABCD$ with $E$ an interior point on side $CD$ (not at the endpoints). Construction: From vertex $D$, draw ray $Dx$ perpendicular to $AE$, intersecting side $BC$ at point $H$ From ...

Question 9

My question is about a very erratic quotient space. I encountered this space in some topology exercise. The space $X$ is described in the following: Let $\mathbb R^2=\{(x,y):x,y\in \mathbb R\} $ be ...

Question 10

Given triangle $ABC$ with altitude $AD$ (where $D \in BC$). At point $A$, construct a perpendicular to $AC$, and on the half-plane that does not contain $B$, take point $E$ such that $AE = AD$ and $AE ...

Question 11

I wish to discuss about the following question from general topology, involving set of ordinals: Problem: Let $X=[0,\Omega)$ be the set of all ordinals strictly smaller than the first uncountable ...

Question 12

We say that a sequence $x_1, x_2,\ldots, x_n$ is increasing if $x_i ≤ x_{i+1}$ for all $1 ≤ i < n$. How many ways are there to fill an 8 x 8 table with numbers 1, 2, 3, and 4 such that: • The ...

Question 13

Here is the problem statement again: If $a, b, c, d, e, f > 0$ prove that $$\frac{ab}{a+b} +\frac{cd}{c+d} + \frac{ef}{e+f} \leq \frac{(a+c+e)(b+d+f)}{a+b+c+d+e+f}$$ The solution given in my book ...

Question 14

For two teams, A and B, they play a best of 5 series with the probability of team A winning one game is $p$ and each game is independent. What is the probability that team A will win the matchup? I ...

Question 15

Given $ABCD$ is a isosceles trapezium. $EF$ is parallel to $DC$ and $AB$. If $AB=25$ and $CD=20$, and $DF=\frac{3}{5}BD$, what the length of $EF$? Let $O$ is intersection of $DB$ and $AC$. I think to ...

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Questions tagged [problem-solving]

Probability Practice book

The Intuition behind the nim game and the XOR?

Maximum number of non-crossing edges in a 7-node network, such that nodes can be three-colored without same-colored nodes being connected

Prove that for all real $a,b,c$ holds $a^2 + b^2 + c^2 + 2 + (abc)^2 \ge 2(ab + bc + ca).$

Perpendicular segments from special points in a rectangle with ratio $BC = 2AB$

$Y_n$ is Cauchy distributed if $S_n/n \sim Y_1$ and $Y_i$ are symmetric [duplicate]

Isosceles trapezoid and special triangles in a square with perpendicular construction

Collinearity of three points in a square with perpendicular and parallel constructions

Investigating the topological properties of a quotient space of $\mathbb R^2$, given by identification of irrational lines through $0$.

Parallelogram formed by altitude, perpendicular construction, and angle bisector in a triangle

Investigating Countability Axioms for the Space of Ordinals $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal.

Iranian combinatorics olympiad 2024 problem 3

If $a, b, c, d, e, f > 0$ prove that $\frac{ab}{a+b} +\frac{cd}{c+d} + \frac{ef}{e+f} \leq \frac{(a+c+e)(b+d+f)}{a+b+c+d+e+f}$ [duplicate]

Probability win a 5 game series [duplicate]

Length of $EF$ in isosceles trapezium.

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