Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
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.
Find the largest number of colours in which you can paint the edges of a cube (each edge with one colour) so that for each pair of colours there are two adjacent edges coloured in these colours. Edges are considered to be adjacent if they have a common vertex.
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
The height of the room is 3 meters. When it was being renovated, it turned out that more paint was needed on each wall than on the floor. Can the area of the floor of this room be more than 10 square meters?
In the entry \({*} + {*} + {*} + {*} + {*} + {*} + {*} + {*} = {*}{*}\) replace the asterisks with different digits so that the equality is correct.
16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.
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.
Valentina added a number (not equal to 0) taken to the power of four and the same number to the power two and reported the result to Peter. Can Peter determine the unique number that Valentina chose?