Problems

A rectangle of size $199\times 991$ is drawn on squared paper. How many squares intersect the diagonal of the rectangle?

The intelligence agency of the Galactic Empire intercepted the following coded message from the enemy planet Medusa: $ABCDE+BADC=ACDED$.

It is known that different numbers are represented by different letters, and that the same numbers are represented by the same letters. Two robots attempted to decode this message and each one got a different answer. Is this possible, or should one of the robots be melted down as scrap metal?

Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?

The board has the form of a cross, which is obtained if corner boxes of a square board of $4\times 4$ are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?

Prove that: $$a_1{a}_2{a}_3\cdots a_{n-1}{a}_n\times {10}^3\equiv a_{n-1}{a}_n\times {10}^3(\mathrm{mod}4),$$ where $n$ is a natural number and $a_i$ for $i=1,2,\dots ,n$ are the digits of some number.

In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?

You are given 10 different positive numbers. In which order should they be named $a_1,a_2,\dots ,a_{10}$ such that the sum $a_1+2a_2+3a_3+\cdots +10a_{10}$ is at its maximum?

Prove that, if $b=a-1$, then $$(a+b)(a^2+b^2)(a^4+b^4)\cdots (a^{32}+b^{32})=a^{64}-b^{64}.$$

Carpenters were sawing some logs. They made 10 cuts and this produced 16 pieces of wood. How many logs did they saw?

Snow White cut out a big square of cotton fabric and placed it in a chest. The first gnome came, took out the square of fabric from the chest, cut it into four squares, put these back in the chest and left. Later the second gnome came and took out one of the squares and then cut it into four pieces and placed all of these in the chest. Then came the third gnome. He also took out one of the squares and cut it into four squares and put them all back in the chest. The rest of the gnomes also did the same thing. How many squares of fabric were in the chest after the seventh gnome left?