If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is $ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$

Given a triangle $ABC$ with $BC=a$, $CA=b$, $AB=c$, $\angle BAC = \alpha$, $\angle CBA = \beta$, $\angle ACB = \gamma$. Prove that $$ a \sin(\beta-\gamma) + b \sin(\gamma-\alpha) +c\sin(\alpha-\beta) = 0.$$

Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that \[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\] and find all cases of equality.

Let $\epsilon$ be a primitive seventh unit root. Which integers occur in $|\alpha|^2$ in form, where $\alpha$ is an element of the seventh circular field $\mathbb Q(\epsilon)$?

Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB = 3$, $BC = 2$, $CD = 6$, and $DA = 8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that \[\frac{DX}{BD} = \frac{1}{4} \quad \text{and} \quad \frac{BY}{BD} = \frac{11}{36}.\] Let $E$ be the intersection of intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF \cdot XG$? $\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91 - 12\sqrt{3}}{4}\qquad\textbf{(D) }\frac{67 - 10\sqrt{2}}{3}\qquad\textbf{(E) }18$

(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have? $\text{(A)}\ 32 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 36\qquad \text{(E)}\ 38$

Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point.

Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.

Determine all pairs of positive real numbers $(a, b)$ for which there exists a function $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation $ f(f(x))=af(x)- bx $

Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations \[\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.\]

Prove that it is possible to choose different real numbers $a_1, a_2, . . . , a_{10}$ that the equation $$(x - a_1)(x -a_2).... (x -a_{10}) = (x + a_1)(x + a_2) ...(x + a_{10})$$ will have exactly $5$ different real roots.

Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$, Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of $n$, what is the minimum number of colors Elmo could have used? [i]Michael Ren[/i]

Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

Show that the equation $x^2+y^2+z^2-xy-yz-zx=3$ has an infinity solutions over nonnegative integers.

Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

Find the number of five-digit positive integers, $n$, that satisfy the following conditions: (a) the number $n$ is divisible by $5$, (b) the first and last digits of $n$ are equal, and (c) the sum of the digits of $n$ is divisible by $5$.

In the country of Drilandia, which has at least three cities, there are bidirectional roads connecting some pairs of cities such that any city can be reached from any other. Two cities are called [i]close[/i] if one can reach the other by using at most two intermediary cities. The mayor, Drilago, fortified the road system by building a direct road between each pair of close cities that were not already connected. Prove that after the expansion, there exists a journey that starts and ends at the same city, where each city except the first is visited exactly once, and the first city is visited twice (once at the beginning and once at the end).

An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?

Let $n$ be a natural number $n>3$. Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

At a math contest which has $ 5 $ problems, each candidate has solved $ 3 $ problems. Among these candidates, for any group of $ 5 $ candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.

Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .