Denote by \(\overline{ab} = 10a +b\) the two-digit number whose first and second digits are \(a\) and \(b\) respectively. Do there exist two \(2\)-digit numbers \(\overline{ab}\) and \(\overline{cd}\) such that \(\overline{ab} \times \overline{cd} = \overline{abcd}\)? (Here \(\overline{abcd}\) is a four-digit number with digits \(a\), \(b\), \(c\) and \(d\), i.e. \(\overline{abcd} = 1000a + 100b +10c +d\).)
Sixty children came to a maths circle at UCL. Among any ten children who came to the circle there are three from the same school. Show that there are 15 children from the same school among all the children who came to the maths circle.
The people in Wonderland are having an election. Every voter writes 10 candidate names on a bulletin and puts it in a ballot box.
There are 11 ballot boxes all together. The March Hare, who is counting the votes, is very surprised to discover that there is at least one bulletin in each ballot box. Moreover, he learned that if he takes one bulletin from each ballot box (11 bulletins all together), then there is always a candidate whose name is written in each of the 11 chosen bulletins. Prove that there is a ballot box, in which all the bulletins contain the name of the same candidate.
Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?
Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?
a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.
b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).
Let the number \(\alpha\) be given by the decimal:
a) \(0.101001000100001000001 \dots\);
b) \(0.123456789101112131415 \dots\).
Will this number be rational?
We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?
Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
One hundred gnomes weighing each 1, 2, 3, ..., 100 pounds, gathered on the left bank of a river. They cannot swim, but on the same shore is a rowing boat with a carrying capacity of 100 pounds. Because of the current, it’s hard to swim back, so each gnome has enough power to row from the right bank to the left one no more than once (it’s enough for any one of the gnomes to row in the boat, the rower does not change during one voyage). Will all gnomes cross to the right bank?