Category: Geometry

Kayoed Kangaroo

This challenge considers an upset kangaroo making a series of discrete jumps in order to retaliate, and thus involves trigonometry.

Kangaroo chasing Sheep

Red Kangaroos in Australia compete with sheep for food. Sheep is grazing when it bumps into Kangaroo. The kangaroo wishes to learn the sheep some respect by either pummelling it, or pushing it into a nearby pond. Sheep wants to escape Kangaroo, but is slower than Kangaroo. However, Kangaroo is only capable of moving in jumps. Ergo, if Sheep stays far enough away to avoid pummelling, but too close for Kangaroo to jump, it is safe, and thus can avoid retribution indefinitely. Precisely formulated:

  • Kangaroo can jump up to 8 meters
  • Kangaroo cannot jump fewer than 5 meters
  • Kangaroo can jump at most fource per second
  • Kangaroo can pummel the sheep they are within 1 meter
  • Sheep can flee at 10 meters per second
  • Sheep can flee for 1 second before before Kangaroo pursues
  • There is a nearby pond, a perfect circle and 100 meters in radius
  • Sheep does not like swimming or being pummelled;
  • Kangaroo wishes to punish Sheep for perceived slight

Therefore, the challenge is thus: what is the maximum distance from the pond from which Kangaroo can punish Sheep?

Solution

Cantankerous Circle

The average distance to the centre of a circle is two thirds the radius. For simplicity, say the radius is 1; the average is simply 2/3. How is this average defined? We will be exploring continuous extrapolations of averages using calculus, geometry, and more.

\int_a^b \operatorname f x \ \delta x \, \div (b-a)

We consider this to be the arithmetic mean value of f from a to b. This can be generalized to more than one dimension. But there are other types of average than the arithmetic mean. Consider the following other types of average, specifically applied to a circle:

\iint_R\ln{\operatorname f x}\ \delta A = \ln\text{GM}\cdot A \\[32pt] \iint_R{(\operatorname f x)}^2\ \delta A = \text{QM}^2\cdot A \\[32pt] \iint_R {1\over\operatorname f x}\ \delta A=\frac A{\text{HM}} \\[32pt] \operatorname f n < \text{median}\ \forall\ n \in R_1, \\[4pt] \operatorname f n > \text{median}\ \forall\ n \in R_2, \\[8pt] \iint_{R_1}\delta A = \iint_{R_2}\delta A,\quad R = R_1 \cup R_2

Since, at the origin, some of these means are discontinuous, ignore the boundary. The challenge is to determine the value of these averages for the distance from the centre.

SOLUTION

Scrupulist Square

This challenge is spans all the way from graph theory, geometry, and calculus—to algebra and combinatorics.

Example set of points for N = 15

Consider a square with diagonal 1. In it, N points are randomly arranged. A graph is created from these points, such that the probability any 2 points touch is equal to 1 minus their distance. Choose a point at random on this graph. The challenge is to calculate a probability regarding this point: what is the probability that a randomly selected node connected to this one is connected to more nodes? Each node has a loop, so if the node was connected to no other nodes the probability of this is 0. For N=1 or 2, this probability is 0. Try to calculate at least one of the following: chance for a specific N > 2, limit of chance as N approaches infinity, and most difficult, generalized probability in terms of N.

SOLUTION

Seriocomic Set

This challenge requires some knowledge of calculus and trigonometry, but mainly algebra.

We begin this challenge with a set of 4 numbers which has the following properties. Can you find them? If you can, good for you. But do you understand the use of this week’s adjective?

a,\,b,\,c,\,d\in\mathbb Z\\[8pt] a\leqslant b\leqslant c\leqslant d\\[8pt] a+b+3=a+d=\cos'{\pi}\\[8pt] \int \frac{d-c}a+1\;\;\delta a\bigg\lt\cos'{(a+b)\pi}\bigg\lt\int\frac{1-a}b-2\;\;\delta b\\[8pt] 0\lt|bc|+1\leqslant a+b+c+d\lt5\\[8pt] bc\notin\{a^2|a\in\mathbb N\}

Note: when calculating integrals, assume the ‘+c’ that you would add at the end is equal to the variable c.

SOLUTION

Radical Rectangles

This challenges covers geometry and algebra.

We have 3 rectangles, X Y and Z. Rectangle X has sides with lengths A and B. Rectangle Y, C and D, and rectangle Z, D and E. Rectangle X has the same perimeter as rectangle Y and the same area as rectangle Z. The total perimeter of the three rectangles is 46 centimetres, and the total area, 39 square centimetres. All sides have integral centimetre lengths. Find the sum A+B+C+D+E.

SOLUTION

Curious Colonies

This science challenge requires an understanding of bacterial growth.

Diagram of the Curious Colonies

You receive an internship at Joana the scientist’s bacteria laboratory. Remember Joana from Pure Water and Spring Forward? She’s studying some exquisite bacteria. No bacteria colonies can grow over others, but they just block each other’s progress. Joana is studying them on a circular dish with diameter 1 metre. Zigzag bacteria colonies’ radii increase by 4 centimetre a minute. Linear bacteria grow at 2 centimetres a minute. Checkerboard bacteria grow at 5 centimetres a minute. Dotty bacteria grow by 1 centimetre every minute, and Inscribed bacteria grow by 3 centimetres a minute. Joana wants to display the results by simultaneously starting one colony of each type on the border of the dish. After some time, all the bacteria colonies will reach an equilibrium. The challenge is to determine where to place the colonies so that they will all end with an equal portion of the dish.

Solution

Salamandriform Sea

Geometry, calculus, and angle theorems are used in this challenge.

You are going on a vacation to an island in a sea. It’s a long journey. First, you have to drive to the sea. You park your car, take off your motorboat which you carried on top of it, and embark onto the sea. Then, you take your boat up to the island, park it, and walk up to your cottage. The sea and island are both ellipses, Whose foci are co-linear and point to your starting point. The sea and island both have their major axes twice the length of their minor axes, and you are a minor sea axis away from the closest point to the sea. Driving is 3 times faster than boating, which is thrice as fast as walking. The sea has only one island, and is 8 ninths water. What is the quickest route to your cottage?

Loopy Lines

You will need to use geometry and algebra to solve this challenge.

Graph of Loopy Lines for this challenge.

We begin this challenge with 2 points. These points are at (1, 1) and (-1, 2). You may know that it only takes 2 points to define a line. You may have guessed that that is what we will use these points for. A second line is drawn, and it passes through the point (3, -2). A third line is then drawn, and it passes through points (2, 3) and (4, 0). A circle is drawn from the 3 intersection points. The radius of the circle is 5 units. The challenge is to find it’s centre.

solution

Sophisticated Streetcar

Algebra, trigonometry, and calculus are used in today’s challenge.

You are taking a streetcar through a city. Your destination is in the middle of a large park, and you are late. You can jump of the streetcar at any time while it is on the side of the park. Your destination is 1 kilometre from the road. The streetcar moves at 15 m/s, and you move at 5 when you run, which you will to try to be on time. You calculate when you will need to jump off the streetcar to arrive as soon as possible, and jump of exactly then. For how long do you run?

Solution