26
votes
3answers
450 views

How does one cut onions in a mathematically efficient way?

Perhaps a math degree and cooking don't go hand in hand, but hopefully they do. I have been thinking about this problem for some time when in the kitchen without making any real progress: How does ...
1
vote
0answers
40 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
1
vote
1answer
115 views

How to solve this trigonometric equation / geometric problem

Is there any way to solve this type of equation exactly for x, where a-h are precalculated constants: $a\cos(g x)+b \sin(g x)+c\cos(h x)+d\sin(hx)+ex+f=0$ Or is my only/best option some sort of ...
0
votes
2answers
34 views

Using plant to find depth of water (triangles)

John and Chris were out in their row boat one day, and Chris spied a water lily. Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to ...
1
vote
2answers
73 views

Can someone help out with this geometry problem plase?

I have a pyramid $P$ that has a square pyramid and each of its 4 triangles is equilateral.I also have cuboid $C$ with height 25 and width 50, something like this: So the volume of $C$ is $25 . 50 . ...
2
votes
3answers
79 views

Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
1
vote
2answers
88 views

Solving quadratic system

If $a,b,c\in \mathbb{R}$ satisfy the system $a^2+ab+b^2=9$; $b^2+bc+c^2=16$;. $c^2+ac+a^2=25$. Find $ab+ac+bc$
2
votes
0answers
113 views

Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees ...
1
vote
2answers
89 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
14
votes
3answers
300 views

On a Putnam's 2009 problem [duplicate]

Find all even natural numbers $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + ...
0
votes
1answer
43 views

Finding the length

How to determine the length of PR? We can determine the angle QSP and the length $SR$ then what to do?
2
votes
3answers
103 views

How many isosceles triangles with total side length $100$ are there?

Let the sum of the three sides of a triangle be $100,$ and all the sides are positive integers length, how many possible isosceles triangles are there?
4
votes
2answers
164 views

A challenge geometrical

Let $ABC$ a right isosceles triangle and $M$ the middle point at the hypotenuse $AC$. Inside the triangle, draw a circle that is tangent to $AB$ at $P$ and to $BC$ at $Q$.The line $MQ$ cuts newly to ...
2
votes
2answers
137 views

How would you interpret this question focusing on problem solving?

The first step of problem solving is to understand what the problem is asking, that is where I am stuck. One of the legs of a right triangle has length 4 cm. Express the length of the altitude ...
4
votes
1answer
75 views

How can we prove that this triangle is Equilateral Triangle?

This is a problem which was sent to me by a friend , but i couldn't solve it , in particular , i don't have ideas for that . I hope you can help by hints or any thing . Here is the problem in the ...
5
votes
3answers
566 views

An equation about a rectangle with given perimeter

I am doing a revision calculator paper and am stuck on an algebra question. There is a picture of a rectangle. One side is $x-2,$ another side is $2x +1.$ The question is. Setup and solve an ...
20
votes
4answers
333 views

Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles

I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have: So, how many ways (pattern) that there ...
86
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
1
vote
3answers
94 views

How to find the best point?

I want to find x in the figure below where it must be as close as possible to 0, but not near to red points (the minimum distance of x to each red point must be at least 1).
1
vote
1answer
682 views

Maximum area for fixed perimeter of a triangle

I'm trying to prove that the triangle of largest area for a given perimeter is equilateral, but I'm having some difficulties. I've done 2 different proofs for a similar problem but for rectangles - ...
4
votes
2answers
610 views

Folding a rectangular paper sheet

You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of ...
5
votes
2answers
85 views

Help with a geometry problem

The problem says: A triangle has its lengths in an arithmetic progression, with difference d. The area of the triangle is t. Find the dimensions. the solution says: the notation can be even better if ...
14
votes
2answers
466 views

The Farmyard problem

Problem: There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could ...
-1
votes
2answers
526 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...