There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.
Prove the mass of all the weights is the same, if it is known that:
a) the mass of each weight in grams is an integer;
b) the mass of each weight in grams is a rational number;
c) the mass of each weight could be any real (not negative) number.
Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?
A number set \(M\) contains \(2003\) distinct positive numbers, such that for any three distinct elements \(a, b, c\) in \(M\), the number \(a^2 + bc\) is rational. Prove that we can choose a natural number \(n\) such that for any \(a\) in \(M\) the number \(a\sqrt{n}\) is rational.
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?
a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)
Is “I see what I eat” the same thing as “I eat what I see”?
To make it not so confusing let’s change the wording to make it more “mathematical”
“I see what I eat”=“If I eat it then I see it”
“I eat what I see”= “If I see it then I eat it”
Was the March Hare right? Is “I like what I get” the same thing as “I get what I like”?
Do you remember the example from the previous maths circle?
“Take any two non-equal numbers \(a\) and \(b\), then we can write; \(a^2 - 2ab + b^2 = b^2 - 2ab + a^2\).
Using the formula \((x-y)^2 = x^2 - 2xy + y^2\), we complete the squares and rewrite the equality as \((a-b)^2 = (b-a)^2\).
As we take a square root from the both sides of the equality, we get \(a-b = b-a\). Finally, adding to both sides \(a+b\) we get \(a-b + (a+b) = b-a + (a+ b)\). It simplifies to \(2a = 2b\), or \(a=b\). Therefore, All NON-EQUAL NUMBERS ARE EQUAL! (This is gibberish, isn’t it?)”
Do you remember what the mistake was? In fact we have mixed up two things. It is indeed true “if \(x=y\), then \(x^2 = y^2\)”. But is not always true “if \(x^2 = y^2\), then \(x=y\).” For example, consider \(2^2 = (-2)^2\), but \(2 \neq (-2)!\) Therefore, from \((a-b)^2 = (b-a)^2\) we cannot conclude \(a-b = b-a\).
Was the Dormouse right? Is “I breathe when I sleep” the same thing as “I sleep when I breathe”?