# CTK Insights

• ## Pages

31 Dec

### Review of Mathematical Plums by R. Honsberger

TweetReview of Mathematical Plums by Ross Honsberger. Mathematical Plums is the fourth book in the Dolciani Mathematical Expositions series

31 Dec

### Calculus Proof of the Pythagorean Theorem

TweetCalculus Proof of the Pythagorean Theorem. Begin with a right triangle drawn in the first quadrant. The legs are variables xand y and the hypotenuse is a fixed positive value c, where the verte xof the angle whose sides contain xand c is the origin

31 Dec

### Filling Pool with Fluids

TweetCreatures from a nearby dimension penetrate our reality and want to fill a large swimming pool with their lime-scented nutritional fluid. From Hose A pours a green slime that would, by itself, take 30 minutes to fill the pool. From Hose B surges a crimson slime that, by itself, would take 20 minutes to fill […]

31 Dec

### Homothety between In- and Excircles

TweetA circle is inscribed in triangle ABC. MN is the diameter perpendicular to the side AC. Let L be the intersection of BM with AC. Prove that AN = CL

31 Dec

### Minus One But What a Difference

TweetProve that the equation x^n + y^n = x^(n-1), n greater than 2 has infinitely many solutions in positive integers

31 Dec

### Dissection of Triangle into Rhombus by Hubert Shutrick

TweetIn an acute triangle ABC, the median AM is longer than side AB. Prove that you can cut the triangle into three parts out of which you can construct a rhombus. From 2010 All-Russian Olympiad, grade 9

31 Dec

### Diophantine Quadratic Equation in Three Variables

Tweetxx+ yy + zz = 59^n is solvable for all positive integers n

31 Dec

### Helly's Theorem

TweetHelly's theorem, amended history, a proof in the plane, references to generalizations

31 Dec

### First Applications of Helly's Theorem

TweetGiven s (s > 0) points in the plane such that every three of them are contained in a disk of radius 1. Prove that all s points are contained in a disk of radius 1