A selection of problems
published in Crux Mathematicorum
and
American Mathematical Monthly
Iliya Bluskov

1. The sequence $a_2,a_3,a_4,...$ of real numbers is such that, for each $n$,$a_n > 1$ and the equation $\lfloor a_n x \rfloor =x$ has exactly $n$ different solutions. Find $\lim_{n \rightarrow \infty} a_n$.
2. Find a finite set $S$ of (at least two) points in the plane such that the perpendicular bisector of the segment joining any two points in $S$ passes through exactly two points of $S$.
3. Find all real numbers $\alpha$ for which the equality $\lfloor \sqrt{n+\alpha } + \sqrt{n}\rfloor = \lfloor \sqrt{4n+1} \rfloor$ holds for every positive integer $n$.
4. Consider two concentric circles with radii $R_1$ and $R$ $(R_1>R)$ and a triangle $ABC$ inscribed in the inner circle. Points $A_1, B_1, C_1$ on the outer circle are determined by extending $BC,CA,AB$ respectively. Prove that
$$\frac{F_1}{R_1^2} \geq \frac{F}{R^2},$$


where $F_1$ and $F_2$ are the areas of triangles $A_1B_1C_1$ and $ABC$ respectively, with equality when $ABC$ is equilateral.

5. The sequence $a_0,a_1,a_2,...$ is defined by $a_0=\frac{4}{3}$ and
$$a_{n+1}=\frac{3(5-7a_n)}{2(10a_n+17)}$$


for $n\geq 0$. Find a formula for $a_n$ in terms of $n$.

6. Let $n\geq 2$ and $b_0 \in [2,2n-1]$ be integers, and consider the recurrence
$$b_{i+1}=\left\{\begin{array}{ll}2b_i-1 & \makebox[.5in]{\r......\\2b_i-2n & \makebox[.5in]{\rm if} b_i> n.\end{array}\right.$$


Let $p=p(b_0,n)$ be the smallest positive integer such that $b_p=b_0$.

(a) Find $p(2,2^k)$ and $p(2,2^k+1)$ for all $k\geq 1$.

(b) Prove that $p(b_0,n)$ divides $p(2,n)$.

7. The equation $x^3+ax^2+(a^2-6)x+(8-a^2)=0$ has only positive roots. Find all possible values of $a$.
8. Pairs of numbers from the set $\{7,8,...,n \}$ are adjoined to each of the 20 different (unordered) triples of numbers from the set $\{1,2,...,6\}$, to obtain twenty $5$-element sets $A_1,A_2,...,A_{20}$. Suppose that $\vert A_i \cap A_j\vert\leq 2$ for all $i\neq j$. What is the smallest $n$ possible ?
9. (with S. Kapralov) Find all sequences $a_1\leq a_2 \leq ... \leq a_n$ of positive integers such that
$$\begin{array}{lcr}a_1+a_2+...+a_n&=&26\\a_1^2+a_2^2+...+a_n^2&=&62\\a_1^3+a_2^3+...+a_n^3&=&164\\\end{array}$$
10. Let $\{b_n\}_{n=1}^ \infty$ be a sequence of positive real numbers which satisfies the condition
$$3b_{n+2}\geq b_{n+1}+2b_n$$


for every $n\geq 1$. Prove that either the sequence converges or $\lim_{n \rightarrow \infty} b_n =\infty$.

11. Let $B_1, B_2,...,B_b$ be $k$-element subsets of $\{1,2,...,n\}$ such that $\vert B_i \cap B_j\vert\leq 1$ for all $i\neq j$. Show that
$$b\leq \left \lfloor \frac{n}{k} \left \lfloor \frac{n-1}{k-1} \right \rfloor\right \rfloor .$$
12. (with G. MacGillivray) Find a closed form expression for the $n$ by $n$ determinant
$$\left\vert \begin{array}{rrrrcrrr}n & -1 & -1 & -1 & \ldots ......1 & 0 & 0 & 0 & \ldots & 0 & -1 & 3\\\end{array} \right\vert.$$
13. Find 364 five-element subsets $A_1,A_2,...,A_{364}$ of a $17$-element set such that

$\vert A_i \cap A_j\vert\leq 3$ for all $1\leq i<j\leq 364$.
14. Prove that for every positive integer $n$
$$\left\lceil {\frac {n^2+3n+1}{n^2+2n}}\left\lceil {\frac{n^......+2n+2}{n^2+n+1}}\right\rceil... \right\rceil\right\rceil=n+1.$$
15. Let $(a_1,a_2,...,a_n)$ be a permutation of the integers from 1 to $n$ with the property that $a_k+a_{k+1}+...+a_{k+s}$ is not divisible by $(n+1)$ for any choice of $k$ and $s$, $k\geq 1$, $0\leq s\leq n-k-1$. Find such a permutation


(a) for $n=12$;

(b) for $n=22$.

16. Find
$$\lim_{n \rightarrow \infty}\frac{1}{n}\left\lceil \frac{3n^......n^2+1}{3n^2-2n+1}}\right\rceil... \right\rceil\right\rceil.$$
 
 

 
2000-08-25