Find all functions such that for all real numbers and ,
Proposed by novus677

Suppose die is laid down on an infinite grid of squares. The die starts perfectly aligned in one of the square of the grid with the facing up and the facing forward. The die can be "rolled" along an edge such that is moves exactly one unit up, down, left, or right and rotates correspondingly in that axis.

Determine, with proof, all possible squares relative to the starting square for which it is possible to roll the die to that square and have it in the same orientation ( facing up and facing forward).

(This problem may be hard to visualize/understand. If you don't understand the question, please ask and I can clarify.)

How many ways are there to color the regions of a three-set Venn Diagram with colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.

Non ai way btw pls

Five persons wearing badges with numbers are seated on chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)

How many ways are there to divide a square into three rectangles, all of whose sides are integers? Assume that two configurations which are obtained by either a rotation and/or a reflection are considered the same.



Answer Confirmation

Given triangle . Points and at segment and respectively such that and . The ray is extended to point such that . It's given , compute .

What's the difference between writing and ?

What happens when you rotate a line of slope by degrees? All I've got is that if , then the new line should still have slope since . :/

On a second thought, is it even possible to rotate such a line?

I remember reading about this because "five points determine a conic" and "three points are sufficient to determine a circle". So there are two special points on all circles (smth like that).

A genie has tied and blindfolded you, and has placed four coins in front of you on a square plate's vertices - the coins do not start out all heads or all tails.

The genie makes a deal with you: When the coins are all heads or all tails, he will release you, but until then, you are bound and blindfolded. You may turn any number of coins you like, and say "ALAKAZAM" to indicate you are ready for the genie to inspect the coins. The genie will inspect the coins, and will either free you (if the coins are right), or will rotate the table degrees - you won't know which.

After he rotates the table, you may flip coins again, and shout "ALAKAZAM" again. You may repeat this as many times as you wish.

You realize that there's a way to escape - and this way to escape requires you shouting "ALAKAZAM" at most times. Find the minimal value of , and describe the strategy.

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let be the probability that at least two of the three had been sitting next to each other. If is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Consider the real number, written in decimal notation,

where, starting from the third decimal place after the comma, each digit is equal to the remainder when the sum of the two previous digits is divided by 10. We can state that:

(a) is an integer

(b) is an integer

(c) is an integer

(d) is an algebraic irrational number

(e) is a transcendental (non-algebraic) irrational number