Let $M$ be the (tridiagonal) $10\times10$ matrix $$M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}$$Show that $M$ has exactly nine positive real eigenvalues (counted with multiplicities).

On the radius $OA$ of a certain circle, as on the diameter, a circle is constructed. A ray is drawn from the center $O$, intersecting the larger and smaller circles at points $B$ and $C$, respectively. Show that the lengths of arcs $AB$ and $AC$ are equal.

What is the remainder when \[\sum_{k=0}^{100}10^k\] is divided by $9$?

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]

Let $n$ be a fixed positive integer. To any choice of real numbers satisfying \[0\le x_{i}\le 1,\quad i=1,2,\ldots, n,\] there corresponds the sum \[\sum_{1\le i<j\le n}|x_{i}-x_{j}|.\] Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$.

Let $n,k$ be integers greater than $1$, $n<2^k$. Prove that there exist $2k$ integers none of which are divisible by $n$, such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by $n$.

Suppose $x_0, x_1, \ldots , x_n$ are integers and $x_0 > x_1 > \cdots > x_n.$ Prove that at least one of the numbers $|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|,$ where \[F(x) = x^n + a_1x^{n-1} + \cdots+ a_n, \quad a_i \in \mathbb R, \quad i = 1, \ldots , n,\] is greater than $\frac{n!}{2^n}.$

For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

Let $G$ be a group and $H$ a proper subgroup. If there exist three group homomorphisms $f,g,h:G\rightarrow G$ such that $f(xy)=g(x)h(y)$ for all $x,y\in G\setminus H$, prove that: [list=a] [*] $g=h$. [*] If $G$ is noncommutative and $H=Z(G)$, then $f=g=h$.

At Olympic High School, $\frac25$ of the freshmen and $\frac45$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? $\text{(A)}$ There are five times as many sophomores as freshmen. $\text{(B)}$ There are twice as many sophomores as freshmen. $\text{(C)}$ There are as many freshmen as sophomores. $\text{(D)}$ There are twice as many freshmen as sophomores. $\text{(E)}$ There are five times as many freshmen as sophomores.

A permutation of the integers \(2020, 2021,...,2118, 2119\) is a list \(a_1,a_2,a_3,...,a_{100}\) where each one of the numbers appears exactly once. For each permutation we define the partial sums. $s_1=a_1$ $s_2=a_1+a_2$ $s_3=a_1+a_2+a_3$ $...$ $s_{100}=a_1+a_2+...+a_{100}$ How many of these permutations satisfy that none of the numbers \(s_1,...,s_{100}\) is divisible by $3$?

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

Let $ ABC$ be an acute-angled triangle. $ C_{1}$ and $ C_{2}$ are two circles of diameters $ AB$ and $ AC$, respectively. $ C_{2}$ and $ AB$ intersect again at $ F$, and $ C_{1}$ and $ AC$ intersect again at $ E$. Also, $ BE$ meets $ C_{2}$ at $ P$ and $ CF$ meets $ C_{1}$ at $ Q$. Prove that $ AP=AQ$.

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

Prove that there are infinitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.

Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.

A string of letters is called $good$ if it contains a continuous substring $IMONST$ in it. For example, the string $NSIMONSTIM$ is $good$, but the string $IMONNNST$ is not. Find the number of good strings consisting of $12$ letters from $I$, $M$, $O$, $N$, $S$, $T$ only.

A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.

A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that: 1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$. 2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\] a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have: \[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\] b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.

Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation. [i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality $ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.

For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\] Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.

Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy (i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$ (ii) $f(1) = 1,$ (iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$. Find, with proof, the smallest constant $\, c \,$ such that \[ f(x) \leq cx \] for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.

Suppose that \[x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009.\] Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\displaystyle\sum_{n=1}^{2008}x_n$.