There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?
Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).
Prove that for any natural number \(a_1> 1\) there exists an increasing sequence of natural numbers \(a_1, a_2, a_3, \dots\), for which \(a_1^2+ a_2^2 +\dots+ a_k^2\) is divisible by \(a_1+ a_2+\dots+ a_k\) for all \(k \geq 1\).
Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).
The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?
In the number \(1234096\dots\) each digit, starting with the 5th digit, is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in a row in this number?