Compute the minimum value of $$\frac{x^4+2x^3+3x^2+2x+10}{x^2+x+1}$$ where $x$ can be any real number.

Determine all the injective functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, such that for each pair of integers $(m, n)$ the following conditions hold: $a)$ $f(mn) = f(m)f(n)$ $b)$ $f(m^2 + n^2) \mid  f(m^2) + f(n^2).$

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $ABC$ be a triangle. Points $D, E, F$ are on segments $BC$, $CA$, $AB$, respectively. Suppose that $AF = 10$, $F B = 10$, $BD = 12$, $DC = 17$, $CE = 11$, and $EA = 10$. Suppose that the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect again at $X$. Find the circumradius of $\vartriangle EXF$.

Petya and Kolya play the following game: they take turns changing one of the coefficients $a$ or $b$ of the quadratic trinomial $f = x^2 + ax + b$: Petya is on $1$, Kolya is on $1$ or $3$. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds $a$ and $b$ regardless of Petya's game? [hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?[/hide]

[b]a)[/b] Prove that if $G$ is $2$-connected, then it has a cycle with the length at least $\min\{n(G),2\delta(G)\}$. (10 points) [b]b)[/b] Prove that every $2k$-regular graph with $4k+1$ vertices has a hamiltonian cycle. (10 points)

Find the number of positive and negative squares in the canonical form of the quadratic form $\sum_{i<j}(x_i-x_j)^2$ in $n$ variables. The same for the form $\sum_{i<j}x_i x_j$.

What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times ... \times 50$ is divisible by $10^n$?

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$

The base of a pyramid $ABCDV$ is a rectangle $ABCD$ with the sides $AB=a$ and $BC=b$, and all lateral edges of the pyramid have length $c$. Find the area of the intersection of the pyramid with a plane that contains the diagonal $BD$ and is parallel to $VA$.

Justify the statement that $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\dots}}}}}.$$

The sequence $ (a_n)$ is defined by $ a_1\equal{}a_2\equal{}a_3\equal{}1$ and $ a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2$ for all $ n \ge 3.$ Prove that $ a_n$ is a positive integer for all $ n \ge 1$.

Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}\plus{}\sqrt{5})^{2000}.$

For postive integers $k,n$, let $$f_k(n)=\sum_{m\mid n,m>0}m^k$$ Find all pairs of positive integer $(a,b)$ such that $f_a(n)\mid f_b(n)$ for every positive integer $n$.

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

Denote $\textrm{SL}_2 (\mathbb{Z})$ and $\textrm{SL}_3 (\mathbb{Z})$ the sets of matrices with $2$ rows and $2$ columns, respectively with $3$ rows and $3$ columns, with integer entries and their determinant equal to $1$. $\textbf{(a)}$ Let $N$ be a positive integer and let $g$ be a matrix with $3$ rows and $3$ columns, with rational entries. Suppose that for each positive divisor $M$ of $N$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $N$ and a matrix $\gamma_M \in \textrm{SL}_3 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . \] Moreover, if $q_1 = 1$, prove that $\det (g) = N$ and $g$ has the following shape: \[ g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), \] where $a_{ij}$ are all integers, $i, j \in \{ 1, 2, 3 \} .$ $\textbf{(b)}$ Provide an example of a matrix $g$ with $2$ rows and $2$ columns which satisfies the following properties: $\bullet$ For each positive divisor $M$ of $6$ there exists a rational number $q_M$, a positive divisor $f (M)$ of $6$ and a matrix $\gamma_M \in \textrm{SL}_2 (\mathbb{Z})$ such that \[ g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) \] and $q_1 = 1$. $\bullet$ $g$ does not have its determinant equal to $6$ and is not of the shape \[ g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), \] where $a_{ij}$ are all positive integers, $i, j \in \{ 2, 3 \}$. [i](Radu Toma)[/i]

Let there's a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ Find all functions $ f$ that satisfies: a) $ f(2u)\equal{}f(u\plus{}v)f(v\minus{}u)\plus{}f(u\minus{}v)f(\minus{}u\minus{}v)$ b) $ f(u)\geq0$

There are four circles. The chord$ AB$ is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to $AB$ is equal to $1$. The second, third and fourth circles are located inside the larger segment and touch the chord $AB$. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality: \[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]

Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$ $\textbf{(A)} ~47 \qquad\textbf{(B)} ~94 \qquad\textbf{(C)} ~227 \qquad\textbf{(D)} ~471 \qquad\textbf{(E)} ~542$

Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? [list] [*] some rotation around a point of line $\ell$ [*] some translation in the direction parallel to line $\ell$ [*] the reflection across line $\ell$ [*] some reflection across a line perpendicular to line $\ell$ [/list] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Marine biologists are studying a new species of shellfish whose first generation consists of $100$ shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5\mid N$ always holds), they divide themselves into $N/5$ groups of $5$ shellfish each. Each group collectively produces $15$ offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n^{\mathrm{th}}$ generation, its value is $1/3^n$. Find the maximum possible total value of the pearls in the colony. [i]Proposed by: Csongor Beke, Cambridge[/i]

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?