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.
Is it possible to draw from some point on a plane \(n\) tangents to a polynomial of \(n\)-th power?
Prove that if \((p, q) = 1\) and \(p/q\) is a rational root of the polynomial \(P (x) = a_nx^n + \dots + a_1x + a_0\) with integer coefficients, then
a) \(a_0\) is divisible by \(p\);
b) \(a_n\) is divisible by \(q\).
Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).
Prove that the following polynomial does not have any identical roots: \(P(x) = 1 + x + x^2/2! + \dots + x^n/n!\)
For which \(A\) and \(B\) does the polynomial \(Ax^{n + 1} + Bx^n + 1\) have the number \(x = 1\) at least two times as its root?
Prove that the polynomial \(x^{2n} - nx^{n + 1} + nx^{n - 1} - 1\) for \(n > 1\) has a triple root of \(x = 1\).
It is known that \(\cos \alpha^{\circ} = 1/3\). Is \(\alpha\) a rational number?
Let \(a, b\) be positive integers and \((a, b) = 1\). Prove that the quantity cannot be a real number except in the following cases \((a, b) = (1, 1)\), \((1,3)\), \((3,1)\).