Is it possible to arrange \(10\) numbers in a row such that:
The sum of any \(4\) consecutive numbers is positive.
The sum of any \(5\) consecutive numbers is positive.
The sum of any \(7\) consecutive numbers is negative?
In how many different ways can you place \(12\) chips in the squares of a \(4\times 4\) board so that
there is at most one chip in each square, and
every row and every column contains exactly three chips?
On a TV screen the number \(1\) appears. Every minute that passes by, the number that is currently on the screen increases by the sum of its digits. For example: if at some point the number \(12\) appears on the screen, the next number will be \(12+(1+2)=15.\) Will the number \(123456\) ever appear on the screen?
A book club with 37 members is reading the following books: “Brave New World", “Dracula" and “Flatland". Each member chose one of the books, though some people chose more than one book. We know that:
23 people chose “Brave New World";
18 people chose “Dracula";
26 people chose “Flatland";
7 people chose all three books.
How many people chose at least two books?
Jan wants to paint a map with \(95\) countries on it, where only one colour can be used for each country. He has \(33\) different colours to paint with and he must use each of yellow, blue, green, purple and red at least once. How many ways of painting the map are there?
On the first day Robinson Crusoe tied the goat with a single piece of rope by putting one peg into the ground. What shape did the goat graze?
On a distant planet called Hexaris, there live two alien species: the Blipnors and the Quantoodles.
The chief alien writes on a board: “There are \(100\) aliens on this planet. Of these, \(24\) are Blipnors and \(32\) are Quantoodles.”
At first this seems confusing — the numbers do not seem to add up! Then you remember that the aliens use a different base for their numeral system.
What base are they using?
Take the numbers \(0,1,2,\dots,3^k-1\), where \(k\) is a whole number.
Show that you can pick \(2^k\) of these numbers so that, among the numbers you picked, no number is the average of two other chosen numbers.
What is the smallest number of weights that allows us to weigh any whole number of grams of gold from \(1\) to \(100\) on a two-pan balance? The weights may be placed only on the left pan.
Three complex number \(a,b,c\) are called affinely independent if whenever \(t,s,u\) are real numbers such that \(ta+sb+uc = 0\) and \(t+s+u=0\), we have that \(t=s=u=0\). Show that three complex numbers \(a,b,c\) are affinely independent if and only if they are not collinear.