Školjka

%V0 In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, $$|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .$$

%V0 Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: $$bf(2x) = f(x), \quad 0 \leq x \leq 1/2$$ $$f(x) = b+(1-b)f(2x-1), 1/2 \leq x \leq 1$$ where $\displaystyle b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$

%V0 Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

%V0 Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that $$\max_{1 \leq i<j \leq n} P_iP_j > \frac{\sqrt 3}{2}(n -1) \min_{1 \leq i<j \leq n} P_iP_j$$

%V0 Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

%V0 Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$