Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$

Find the number of distinct non-empty subsequences of the binary string \[01001111010110.\] Note: A subsequence of a string $S$ is any string which can be formed by deleting some characters from $S$ while keeping the order of the remaining characters. For example, ``ab'' and ``ccm'' are a subsequences of ``ccamb'', but ``abc'' is not. [i]Team #8[/i]

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$: \[ a_i | a_{i+1}\]

There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.

The vertices of a regular $45$-gon are painted into three colors so that the number of vertices of each color is the same. Prove that three vertices of each color can be selected so that three triangles formed by the chosen vertices of the same color are all equal.

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that $$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$.

In a tetrahedron $ ABCD $ the edges $AD$, $ BD $, $ CD $ are equal. $ ABC $ Non-collinear points are chosen in the plane. $ A_1$, $B_1$, $C_1 $ The lines $DA_1$, $DB_1$, $DC_1 $ intersect the surface of the sphere circumscribed about the tetrahedron at points $ A_2$, $B_2$, $C_2 $, different from the point $ D $. Prove that the points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie on the surface of a certain sphere.

For each positive integer $n$, denote by $O(n)$ its greatest odd divisor. Given any positive integers $x_1 = a$ and $x_2 = b$, construct an innite sequence of positive integers as follows: $x_n = O(x_{n-1} + x_{n-2})$, where $n = 3,4,...$ (a) Prove that starting from some place, all terms of the sequence are equal to the same integer. (b) Express this integer in terms of $a$ and $b$.

In elections to the City Duma, each voter, if he goes to the polls, casts a vote for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the sociological service of the mayor's office is considered good if it correctly predicts the number of votes cast for at least one of the candidates, and bad otherwise. Prove that for any forecast, voters can turn out to vote in such a way that this forecast turns out to be bad.

Find all possible values of the number $$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$ where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.

For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate $$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$ for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.

[b]a)[/b] Give example of two irrational numbers $ a,b $ having the property that $ a^3,b^3,a+b $ are all rational. [b]b)[/b] Prove that if $ x,y $ are two nonnegative real numbers having the property that $ x^3,y^3,x+y $ are rational, then $ x $ and $ y $ are both rational. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$ respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$ such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.

Consider the sequence of real numbers $a_n$ satisfying the recurrence $$a_na_{n+2}-a_{n+1}^2-(n+1)a_na_{n+1}=0.$$ Given that $a_1=1$ and $a_2=2018$, compute $$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$

Find $x$ and $y$ ($a$ and $b$ parameters): $$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$

A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$

Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively. Prove that if $h$ is the distance from $A$ to line $BC$, then \[ \frac{R_B+R_C}{R} \le \frac{BC}{h}. \]

One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel $50$ miles per hour while heading to our final vacation destination..." Hannah teases her husband, "You $\textit{would}$ drive that $\textit{slowly}\text{!}$" Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of $50$ miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route. If our faster return rate is an integer number of miles per hour, and our average speed for the $\textit{whole round trip}$ is $\textit{also}$ an integer number of miles per hour, what must be our speed during the return trip?" Michael pipes up, "How about $4950$ miles per hour?!" Wendy smiles, "For the sake of your $\textit{other}$ children, please don't let $\textit{Michael}$ drive." Jerry adds, "How about we assume that we never $\textit{ever}$ drive more than $100$ miles per hour. Michael and Wendy, let Josh and Alexis try this one." Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?

Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

If for every real $x$, $ax^2 + bx+c \geq 0$, where $a,b,c$ are reals such that $a<b$, what is the smallest value of $\dfrac{a+b+c}{b-a}$? $ \textbf{(A)}\ \dfrac{5}{\sqrt 3} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \dfrac{\sqrt 5}2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \dfrac{\sqrt 7}2 $

Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$? [asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("$E$",(1,0),SE); label("$F$",(25,10),NW); label("$A$", (0,0), SW); label("$B$", (0,10), NW); label("$C$", (26,10), NE); label("$D$", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy] [i]Proposed by Nathan Xiong[/i]

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?