Problems

Age
Difficulty
Found: 1942

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.

Prove the following formulae are true: \[\begin{aligned} a^{n + 1} - b^{n + 1} &= (a - b) (a^n + a^{n-1}b + \dots + b^n);\\ a^{2n + 1} + b^{2n + 1} &= (a + b) (a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - \dots + b^{2n}). \end{aligned}\]

Let \((1 + \sqrt {2} + \sqrt {3})^n = p_n + q_n \sqrt {2} + r_n \sqrt {3} + s_n \sqrt {6}\) for \(n \geq 0\). Find:

a) \(\lim \limits_ {n \to \infty} {\frac {p_n} {q_n}}\); b) \(\lim \limits_ {n \to \infty} {\frac {p_n} {r_n}}\); c) \(\lim \limits_ {n \to \infty} {\frac {p_n} {s_n}}\);

Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: \[F_T(x,z) = \sum_{n=0}^{\infty}T_n(x)z^n;\quad F_U(x,z) = \sum_{n=0}^{\infty}U_n(X)z^n.\]

Definitions of Chebyshev polynomials can be found in the handbook.

We denote by \(P_{k, l}(n)\) the number of partitions of the number \(n\) into at most \(k\) terms, each of which does not exceed \(l\). Prove the equalities:

a) \(P_{k, l}(n) - P_{k, l-1}(n) = P_{k-1, l}(n-l)\);

b) \(P_{k, l}(n) - P_{k-1, l} (n) = P_{k, l-1}(n-k)\);

c) \(P_{k, l}(n) = P_{l, k} (n)\);

d) \(P_{k, l}(n) = P_{k, l} (kl - n)\).