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Prove that for every natural number n $>$ 1 the equality: [$n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:$\\$

$\\$Alex is 1 year older than V,$\\$Beatrice is 2 years older than W,$\\$Victor is 3 years older than X,$\\$Gregory is 4 years older than Y.$\\$

Who is older and by how much: Deborah or Z?

Anna’s garden is a grid of $n \times m$ squares. She wants to have trees in some of these squares, but she wants the total number of trees in each column and in each row to be an odd number (not necessarily the same, they just all need to be odd). Show that it is possible only if $m$ and $n$ are both even or both odd and calculate in how many different ways she can place the trees in the grid.

An $8 \times 8$ chessboard has 30 diagonals total (15 in each direction). Is it possible to place several chess pieces on this chessboard in such a way that the total number of pieces on each diagonal would be odd?

Out of $7$ integer numbers, the sum of any $6$ is a multiple of $5$. Show that every one of these numbers is a multiple of $5$.

Several films were nominated for the “Best Math Movie“ award. Each of the 10 judges secretly picked the top movie of their choice. It is known that out of any 4 judges, at least 2 voted for the same film. Prove that there exists a film that was picked by at least 4 judges.

Can you decorate an $8 \times 8$ cake with chocolate roses in such a way that any $ 2 \times 2$ piece would have exactly 2 roses on it, and any $3 \times 1$ piece would have exactly one rose? Either draw such a cake or explain why this is not possible.

A $3 \times 3$ magic square is a square with different number from $1$ to $9$ in each of its $9$ cells. The numbers in each row, column and diagonal sum up to $15$. Show that there is a number $5$ in the centre of the square.

Francesca, Isabella and Lorenzo played chess together. Each child played $10$ rounds. $\\$

a) What was the total number of rounds? $\\$ b) Is it possible that Lorenzo played more rounds with Isabella than with Francesca?

There are $36$ warrior tomcats standing in a $6 \times 6$ square formation. Each cat has several daggers strapped to his belt. Is it possible that the total number of daggers in each row is more than $50$ and the total number of daggers in each column is less than $50$?

A group of Martians and a group of Venusians got together for an important talk. At the start of the meeting, each Martian shook hands with 6 different Venusians, and each Venusian shook hands with 8 different Martians. It is known that 24 Martians took part in the meeting. How large was the delegation for Venus?

There are $5$ directors of $5$ banks sitting at the round table. Some of these banks have a negative balance (they owe more money that they have) and some have a positive balance (they have more money that they owe). It is known that for any 3 directors sitting next to each otehr, their 3 banks together have a positive balance. Does it mean that the $5$ banks together have a positive balance?

There are some cannons in every fortress on the Cannon Island. The star marks the Grand Fortress, the capital, and the 10 circles mark 10 smaller fortresses. The total number of cannons located in all the fortresses along the east-west road is known to be $130$. The total number of cannons along each of the other 3 roads is $80$. Also it is known that there is a total of $280$ cannons in all the fortresses. How many cannons are in the capital?

Orcs and goblins, 40 creatures altogether, are standing in a rectangular formation of $4$ rows and $10$ columns. Is it possible that the total number of orcs in each row is $7$, while the number of orcs in each column is the same?

The distance between Athos and Aramis, galloping along one road, is 20 leagues. In an hour Athos covers 4 leagues, and Aramis – 5 leagues.

What will the distance between them be in an hour?

Is it possible to fill a $5 \times 5$ table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?

30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?

In each square of a rectangular table of size $M \times K$, a number is written. The sum of the numbers in each row and in each column, is 1. Prove that M = K.

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.