BMAD

I disagree with your conclusion in the second case.

I forgot, say the smaller chord is 5 cm in length and the bigger chord 7 cm in length

How many circles exist with the following properties... 1. Contains two chords that are 2 cm from each other 2. The longer chord only gets as close as 1 cm from the center. 3. The shorter chord only gets as close as 3 cm from the center. versus How many circles exist with the following properties... 1. Contains two chords that only gets as close as 2 cm from each other. 2. The longer chord only gets as close as 1 cm from the center. 3. The shorter chord only gets as close as 3 cm from the center.

Yes. Mating is random so anyone may end up with anyone but if they are related by more than 10% no offspring we be produced. Also, we are assuming that any mating that occurs with someone not too related will always result in pregnancy.

I have the same questions that DejMar raises, plus some others. Does every male mate mate exactly once every day with a random female? - yes, everyone mates at most once a day. Do the original ten citizens live for exactly 4 days? yes. they were all miraculously born on the same day and none of them related. How is percentage relationship reckoned? Common parents account for 25%, common grandparents account for 12.5%, common great grandparents 6.25% and so on. Does "sustain" require only that after 7 days at least one citizen is living? Yes. Will this society at least survive a week.

Solution https://www.simonsfoundation.org/multimedia/mathematical-impressions-multimedia/the-bicycle-pulling-puzzle/

Suppose there are nine points located at (0,0) ; (1,0) ; (2,0) ; (0,1) ; (1,1) ; (2,1) ; (0,2) ; (1,2); (2,2) Your opponent lets you move first. You are to remove the fewest amount of points necessary to prevent your opponent from forming a square. On their turn, they will be adding two points at a location of their choosing to build a square (if they can). What is the least amount of points, and which ones are they, that you should remove.

I believe you replace the ? with digits.

You are brought to a crime scene. You are told that a thief just made off with a bag full of diamonds, escaping on a bicycle. You come across the following pair of bicycle tracks in the snow, no doubt made by the fleeing thief. But which way did the thief go?

If you pull straight back on the lower pedal of your bicycle with a string going perpendicular to the ground, will the bike move forward or backward initially? Why?

@CaptainEd Forgive if my English is not clear this happens to me a lot... But unless my intuition is wrong... If the center of the circle is southwest of the midpoint between the two values (1,1.5) then the arc curves away from the origin. If the center of the circle is northeast of the midpoint then the arc curves towards the origin while intersecting the two points. Then it should follow that there is a point, somewhere on that lone where the arc curves towards and away from the origin while going through the point at the same time (making the axis tangent I believe) or it neither curves towards or away from the origin. The last case, I argue does not meet the conditions of the question but maybe I over thought this one. @Buddyboy3000 I used an approach similar to your similar triangles in conjunction with Captain Ed's formula

Does the area approach anything as A goes to infinity? Also, I still question the domain of this line. I don't believe a value is feasible for all real numbers

after testing your center point of (1.5,1), I have found that it doesn't work in producing a consistent radius. You are using your formula incorrectly i think.

I am curious by how you found your line and how you know the line works? Is there another possible line containing solutions? Though I don't disagree with your assertion that there are infinite circles I do not believe the domain of your line is correct.

Suppose that there is an arc from a circle going through the point (0,3) and (2,0). How many different circles (different radius, different centers) could make this arc? Without using calculus or trig, find the area under the arc between the x and y axis?

This is what I had in mind

Assume that a dorito's shape is a perfect isosceles triangle with the ratio of its longest sides to shortest being 2:1. If you were to make three linear cuts through the chip, what is the most amount of sides of a part of the chip that can be formed?

Am I understanding your claim correctly? Are you stating that it is impossible for a ball to give the appearance to a bystander of traveling straight up and down over the course of 8 seconds?

A baseball player hit a ball high into the air. It took a perfectly linear path and came straight back down through the point where the man made contact after 8 seconds. Of course the ball only appeared to travel Linearly to us bystanders. Since the earth was spinning throughout the 8 second flight of the ball, describe the true flight the ball took during this period assuming earth's core is (0,0,0), make any other assumptions you need except leave the height of the ball unknown.

how about...

Say there is a new society of ten unrelated people. Each time a person is born into their society, that person immediately becomes an adult while another random same-gender adult dies at the other's birth. These individuals live for four days, providing them four opportunities to mate before expiring at the end of the fourth day. Each male randomly mates with another female. In order to conceive one must mate with another who is less than 10% related to themselves. What is the probability this population will sustain itself for a week?

What you say is true if and only if time 2,3 & 4 is greater than zero. at zero there are two solutions, 80mph, 60mph for the fly meaning that the fly never left either car until the collision. But since it is given that the fly starts with the car, the fly must be going 80mph.

purely a guess at the approach but