[b]p1.[/b] You have $10$ bags of $2006$ coins each and a scale which can show exact weight. You know that nine bags contain honest coins which weigh $10$ grams each. The remaining bag contains false coins which weigh $11$ grams each. All bags look alike, so you don’t know which bag contains false coins. How to determine the bag with false coins by one weighing? (This means that you can put some coins on the scale, read the result and make a decision based on the result).


[b]p2.[/b] Is it true that if the lengths of all three altitudes of a triangle are less than $1$ centimeter, then its area is less than $100$ square centimeters?


[b]p3.[/b] Find the smallest possible whole number with the following property: its half is a square of some whole number, its third is a cube of some whole number and its fifth is the fifth power of some whole numer.


[b]p4.[/b] Martian bus tickets have six-digit numbers, so that all tickets are numbered from $000001$ to $999999$. Martians think that the ticket is lucky if the sum of the first three digits is equal to the sum of the last three digits. Prove that the total sum of all 6-digit numbers which appear on the lucky tickets is divisible by $13$.


PS. You should use hide for answers.