Math Is Fun Forum

One Friday evening Sam goes to his favorite restaurant. He gives the doorman a one dollar tip, then spends half of the remaining money, having a great time, of course. On his way out he gives the doorman another one dollar tip. But once out, feeling he could do with more fun, he enters the restaurant next door where he gives the new doorman $ 1 then spends the half of the remaining money, and on his way out does not forget to present the doorman with $ 1. Now, believe it or not, he repeats the whole thing in the third and then the fourth restaurant. Leaving the later, (after giving $ 1 to the doorman of course), Sam has the curiosity to inspect the content of his pockets, realizing that he has exactly $ 1 left. How much did he have initially?

without a calculator prove

How many solutions does the equation

have on the set of positive integers?

At what time instant after 8 o'clock do the short and long hands of a clock enclose the equal angles with the horizontal?

find all real x,y such that

If

and

and

prove

Solve the following equation on the set of integers:

prove

is an integer