Problems

Is it possible for the mean of some 35 whole numbers to equal $6.35$?

A field that will be used to grow wheat has a rectangular shape. This year, the farmer responsible for this field decided to increase the length of one of the sides by $20\mathrm{\%}$ and decrease the length of another side by $20\mathrm{\%}$. The field remains rectangular. Will the harvest of wheat change this year and, if so, then by how much?

Looking back at her diary, Natasha noticed that in the date 17/02/2008 the sum of the first four numbers are equal to the sum of the last four. When will this coincidence happen for the last time in 2008?

A block of cheese comes in packaging with parallel lines of different colours printed on it. If you cut along the red lines then you will get 5 slices of cheese, if you cut along the yellow lines then there will be 7 slices, and along the green lines you will get 11 slices. How many slices will you get if you cut along the lines of all three colours?

The product of two natural numbers, each of which is not divisible by 10, is equal to 1000. Find the sum of these two numbers.

An old analogue clock speeds up by 9 minutes after 24 hours. If you went to sleep at 22:00 and set the correct time on the clock, then for what time should the alarm be set if you want it to go off at exactly 6:00? Explain your answer.

A digital clock shows the time in hours and minutes (for example, 16:15). While practising his counting, Pinocchio finds the sum of all the numbers on this clock (for example, $1+6+1+5=13$). Find the time at which the sum of these numbers will be at its maximum.

Find $x^3+y^3$ if $x+y=5$ and $x+y+x^{2}y+x{y}^2=24$.

On the $xy$-plane shown below is the graph of the function $y=a{x}^2+c$. At which points does the graph of the function $y=cx+a$ intersect the $x$ and $y$ axes?

Find the largest natural number $n$ which satisfies $n^{200}<5^{300}$.