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.
It is known that \(\cos \alpha^{\circ} = 1/3\). Is \(\alpha\) a rational number?
Let \(f (x)\) be a polynomial of degree \(n\) with roots \(\alpha_1, \dots , \alpha_n\). We define the polygon \(M\) as the convex hull of the points \(\alpha_1, \dots , \alpha_n\) on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon \(M\).
a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of \(36^{\circ}\) at the vertex are incommensurable.
b) Invent a geometric proof of the irrationality of \(\sqrt{2}\).
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.
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).
The following words/sounds are given: look, yar, yell, lean, lease. Determine what will happen if the sounds that make up these words are pronounced in reverse order.