All Questions
Tagged with geometric-probability contest-math
9
questions
5
votes
4
answers
263
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Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square
Question: Suppose that points $P_1$, $P_2$, and $P_3$ are chosen uniformly at random on the sides of a square $T$. Compute the probability that $$\frac{[\triangle P_1 P_2 P_3]}{[T]}>\frac{1}{4}$$ ...
- 690
1
vote
1
answer
59
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Viewing sides of a hexagon in a circle
I need help with this problem:
Take a circle with radius r, and place a regular hexagon of side length 2 so that the circle and hexagon are concentric. The probability of picking a point on the ...
- 572
1
vote
0
answers
118
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Further thoughts on 1992-Putnam-Archive's A6 [duplicate]
The original question is here:
"A–6 Four points are chosen at random on the surface of a
sphere. What is the probability that the center of the
sphere lies inside the tetrahedron whose vertices ...
- 11
0
votes
1
answer
506
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The probability of the dart hitting the smaller ring is?
A dart is randomly thrown at a circular board on which two concentric rings of radii $R$ and $2R$ having the same width (width much less than $R$) are marked. The probability of the dart hitting the ...
- 2,912
17
votes
3
answers
980
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A random sphere containing the center of the unit cube
Inspired by a Putnam problem, I came up with the following question:
A point in randomly chosen in the unit cube, a sphere is then created using the random point as the center such that the sphere ...
- 645
1
vote
1
answer
531
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What is the probability if we throw dart towards a large square but it should hit only the inner part of small square $FEHG$ inscribed in it?
Let $ABCD$ be a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are $x$ and $y$, where $\frac{x}{y}$ = $3$. A dart is thrown towards the square ...
- 1,188
5
votes
1
answer
100
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Two points of a square $K$ determine a diagonal of another square that is contained in $K$
Let $K:=[0,1]^2$ be a square on $\mathbb{R}^{2}$. We select 2 random points $A$, $B$ $\in [0,1]^{2}$ in this square. What is the probability that the square whose diagonal is the line segment $AB$, is ...
- 139
0
votes
2
answers
146
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Do the lengths of all three segments necessarily have the same distribution?
Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have ...
user316074
-8
votes
2
answers
468
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Geometry Probability Question
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Hi everyone I found this interesting question; help is appreciated! :)...
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