BMAD

You understood the question perfectly. I guess this one was too easy

Assume we have an equilateral triangle oriented to where one of its base is perpendicular to a vertical line. Pick a point on one of the edges and draw a parallel line to the base from edge to edge; while also measuring from the point to the top most vertex of the triangle. How often are the two line segments the same length?

his answer is right. i was just asking if it extends. If i marked "answered" then no one would look at this question any more.

Someone claims to have invented a Universal Truth Machine (UTM), a machine that takes a proposition as input, and returns "true", "false", or "undecidable" as output. Example: Input Output 1+3 = 4 true 1+2= 4 false this proposition is false undecidable Devise a true proposition that the UTM will claim to be false, thereby disproving the inventor's claim.

Is this claim for any nxm table or only when n = m ? Does OP ask for every nxm, or a particular nxm? Fair question! The way I wrote the OP, I only asked if one exists. I am seeking to extend the OP to see if the above conjecture(s) extends to any nxm and if not, what nxm does it work for?

Both. This one though is from the former Soviet union puzzle challenges.

Is this claim for any nxm table or only when n = m ?

This is a good attempt but is only equal to.

The fourth knife is a distraction. hold it maybe as the rest balances the water.

Is the distance center to center or edge to edge? I'm assuming edge to edge, which makes a better puzzle, but not certain. I am unsure by what you mean but the bottles are the vertices of the triangle and the edges are slightly longer than the knife length (the knifes are congruent)

Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than 75% squares of the same colour?

You need to visualize three soda bottles with narrow necks and four table knives. Place the three bottles on a flat surface so that each bottle forms the corner of a triangle. The distance between the bases of any two bottles should be slightly more than the length of a knife. Using no more than these materials construct a platform on top of the three bottles. No part of any knife may touch the ground. The platform must be strong enough to support a full glass of water.

20 Numbers are written on the board: 1, 2, ... ,20. Two players are putting signs before the numbers in turn (+ or -), where they desire. The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

thank you bonanova

Why cant a line pass through ABE and the common point of ACD and EDF?

yes!

You need to show that in any situation either #1 is true or #2 is true

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

Given 50 segments on the line. Prove that one of the following statements is valid: 1. Some 8 segments have a common point. 2. Some 8 segments do not intersect each other.

How many sides of the convex polygon can equal its longest diagonal?

11?

so we find four snips of papers in a very particular sequence. I was under the impression that i found them randomly, hmm.

ac = 2 ad + bc = 3 bd = -2 c - a = -1 d - b = 3 d = 3 + b b(3+b) = -2 b2+3b+2 = 0 (b+2)(b+1) = 0, b = -2 and d = 1 or b =-1 and d=2 c - a = -1 c = -1 + a ac = 2 a(-1+a)=2 a2-a-2=0 (a-2)(a+1)=0; a=2 and c =1 or a=-1 and c=-2 b = -2 and d = 1 or b =-1 and d=2 a=2 and c =1 or a=-1 and c=-2 ad = 4 or 2 or -1 or -2 bc = -2 or 4 or -1 or 2 given: ad + bc = 3 ad = 4 when bc = -1 so a = 2, d = 2, b = -1, c = 1 ad= -1 when bc = 4 so a = -1, d = 1, b = -2, c = -2 there are two answers

Another 13?