A message is encrypted using numbers where each number corresponds to a different letter of the alphabet. Decipher the following encoded text:
1317247191772413816720713813920257178
A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.
Each side in the triangle \(ABC\) is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division (the points \(A\), \(B\), \(C\) cannot be the vertices of triangles) in which neither side is parallel to either side of the triangle \(ABC\)?
How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?
a) One person had a basement illuminated by three electric bulbs. Switches of these bulbs are located outside the basement, so that having switched on any of the switches, the owner has to go down to the basement to see which lamp switches on. One day he came up with a way to determine for each switch which bulb it switched on, descending into the basement exactly once. What is the method?
b) If he goes down to the basement exactly twice, how many bulbs can he identify the switches for?
Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
Peter thought of a number between 1 to 200. What is the fewest number of questions for which you can guess the number if Peter answers
a) “yes ” or “no”;
b) “yes”, “no” or “I do not know”
for every question?
There are 4 coins. Of the four coins, one is fake (it differs in weight from the real ones, but it is not known if it is heavier or lighter). Find the fake coin using two weighings on scales without weights.
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.
Author: D.E. Shnol
On the island of Truthland, all of the inhabitants may be mistaken, but the younger ones never contradict the elders, and when the older ones contradict the younger ones, they (the elders) are not mistaken. Between the residents A, B and C there was such a conversation:
A: B is the tallest.
B: A is the tallest.
C: I’m taller than B.
Does it follow from this conversation that the younger the person, the taller he or she is (for the three people having this conversation)?