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\begin{center}
{\large A selection of problems}\\
published in 
Crux Mathematicorum, \\ 
American Mathematical Monthly, and Mathematics Magazine\\
Iliya Bluskov
\end{center}
\begin{enumerate}
\item (CM) 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 $\lim_{n \rightarrow \infty} a_n$.

\item (CM) 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 $S$.

\item (CM) 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 $n$.

\item (CM) 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.

\item (CM) 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 $n$.

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

\item (CM) The equation $x^3+ax^2+(a^2-6)x+(8-a^2)=0$ has only positive roots.
Find all possible values of $a$.

\item (CM) 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
$|A_i \cap A_j|\leq 2$ for all $i\neq j$. What is the smallest $n$ possible ?

\item (CM) (with S. Kapralov) Find all sequences $a_1\leq a_2 \leq ... \leq a_n$ of 
positive integers
such that
\[\begin{array}{lcr}
a_1+a_2+...+a_n&=&26\\
a_1^2+a_2^2+...+a_n^2&=&62\\
a_1^3+a_2^3+...+a_n^3&=&164\\
\end{array}\]

\item (CM) 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 
$\lim_{n \rightarrow \infty} b_n =\infty$.

\item (CM) Let $B_1, B_2,...,B_b$ be $k$-element subsets of $\{1,2,...,n\}$ such that
$|B_i \cap B_j|\leq 1$ for all $i\neq j$. Show that 
\[b\leq \left \lfloor \frac{n}{k}  \left \lfloor \frac{n-1}{k-1} \right \rfloor 
\right \rfloor .\]

\item (CM) (with G. MacGillivray) Find a closed form expression for the $n$ by $n$ 
determinant
\[  \left| \begin{array}{rrrrcrrr}
n & -1 & -1 & -1 & \ldots & -1 & -1 & -1\\
-1 & 3 & -1 & 0 & \ldots & 0 & 0 & 0\\
-1 & -1 & 3 & -1 & \ldots & 0 & 0 & 0\\
-1 & 0 & -1 & 3 & \ldots & 0 & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
-1 & 0 & 0 & 0 & \ldots & 3 & -1 & 0\\
-1 & 0 & 0 & 0 & \ldots & -1 & 3 & -1\\
-1 & 0 & 0 & 0 & \ldots & 0 & -1 & 3\\
\end{array} \right|.\] 

\item (CM) Find 364 five-element subsets $A_1,A_2,...,A_{364}$ of a $17$-element
set such that\\ 
$|A_i \cap A_j|\leq 3$ for all $1\leq i<j\leq 364$.

\item (CM) Prove that for every positive integer $n$
\[\left\lceil {\frac {n^2+3n+1}{n^2+2n}}\left\lceil {\frac 
{n^2+3n}{n^2+2n-1}}... 
\left\lceil {\frac {n^2+2n+2}{n^2+n+1}}\right\rceil... \right\rceil\right\rceil 
=n+1.\]

\item (CM) 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 $n=22$.

\item (AMM) Find 
\[\lim_{n \rightarrow \infty}\frac{1}{n}\left\lceil \frac 
{3n^2+2n}{3n^2}\left\lceil \frac {3n^2+2n-1}{3n^2-1}... 
\left\lceil {\frac {3n^2+1}{3n^2-2n+1}}\right\rceil... \right\rceil\right\rceil 
. 
\]

\item (MM) Let $k_1,k_2,...,k_n$ be integers such that $k_i>2$, 
$i=1,2,...,n$, and let
\[N=\sum_{i=1}^n {k_i \choose 2}.\]
Prove that
\[\sum_{1\leq i<j\leq n}{k_i \choose 2}{k_j \choose 2}+3\sum_{i=1}^n {k_i +1 
\choose 4}={N \choose 2}.\]

\item (AMM) Let $N$ be a positive even integer. A placement of queens
on an $N \times N$ chessboard is a set of $N$ squares on the board such that none of these
squares lies on either long diagonal, and no two of these squares lie in a single row,
column, or diagonal (that is, the queens are non-attacking). A cover of an $N \times N$
board is a set of $N - 2$ disjoint placements. (Thus on a $4 \times 4$ board, the placements
$\{(1, 3), (2, 1), (3, 4), (4, 2)\}$ and $\{(1, 2), (2, 4), (3, 1), (4, 3)\}$ form a cover.)

(a) Show that there exists a cover of the $N \times N$ board if $N + 1$ is prime.

(b) Give an example of an even $N$ for which $N + 1$ is not prime and for which there
is no cover.

(c) Give an example of an even $N$, and a cover, for which $N + 1$ is not prime.

\item (MM) Prove that
\[\sum_{m=0}^n\sum_{r=0}^{n-m}\frac{(-2)^r}{m!r!} {2n-m-r \choose n}=\sum_{r=0}^{n}\frac{(-1)^r}{r!}{2n-r\choose n}\, .\]

\item (CM) Let $B=[b_{ij}]$ be an $n\times k$ matrix with entries in the set 
of residues modulo $v$, such that 
%$b_{ij}\neq b_{is}$ if $j\neq s$. 
the $k$ entries in each row of $B$ are pairwise distinct.
Form the $n\, \times \, [k(k-1)]$ matrix of 
differences $D=[d_{ip}]$, where $d_{ip}=b_{ij}-b_{is} \pmod v$, $j\neq s$, 
$1\leq j,s\leq k$ and $1\leq i\leq n$. Let $O_q$, $q=1,2,...,v-1$ be the number 
of occurrences of 
the residue $q$ in the matrix $D$. Show that the sum \[ \sum_{q=1}^{v-1}O_q \] 
does not depend on $v$.


\end{enumerate}

\end{document}
