Find \(m\) and \(n\) knowing the relation \(\binom{n+1}{m+1}: \binom{n+1}{m}:\binom{n+1}{m-1} = 5:5:3\).
\(N\) points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the \(k\) colours. Prove that if \(N > \lfloor k!e\rfloor\), then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.
There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if
a) weights can be placed only on one side of the scales;
b) weights can be placed on both sides of the scales?
Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
In the entry \({*} + {*} + {*} + {*} + {*} + {*} + {*} + {*} = {*}{*}\) replace the asterisks with different digits so that the equality is correct.
A chequered strip of \(1 \times N\) is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.
Which of the players can always win (no matter how their opponent played)?
Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.