Find the complex numbers $ z$ for which the series \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\] converges and find its sum.

In $\triangle ABC$ let $D$ and $E$ be points on $AB$ and $AC$ respectively. The circumcircle of $\triangle CDE$ meets $AB$ again at $F$, and the circumcircle of $\triangle ACD$ meets $BC$ again at $G$. Show that if the circumcircles of $DFG$ and $ADE$ meet at $H$, then the three lines $AG$, $BE$, and $DH$ concur. Proposed by [i]Oron Wang[/i] inspired by [i]Tiger Zhang[/i]

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

[b]a.[/b] There are $5$ identical paper triangles on the table. Each can be moved in any direction parallel to itself (i.e., without rotating it). Is it true that then any one of them can be covered by the $4$ others? [b]b.[/b] There are $5$ identical equilateral paper triangles on the table. Each can be moved in any direction parallel to itself. Prove that any one of them can be covered by the $4$ others in this way.

Determine the least real number $k$ such that the inequality $$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$ holds for all real numbers $a,b,c$. [i]Proposed by Mohammad Jafari[/i]

Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.

On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.

Farmer John is grazing his cows at the origin. There is a river that runs east to west $50$ feet north of the origin. The barn is $100$ feet to the south and $80$ feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.

For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that: (i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers. If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.

Let $a = e^{\frac{4\pi i}5}$ be a nonreal fifth root of unity and $b = e^{\frac{2\pi i}{17}}$ be a nonreal seventeenth root of unity. Compute the value of the product \[(a + b) (a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]

Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas \[a_{n+1} = a_n + b_n,\] \[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\] and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.

Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.

In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle? $ \textbf{(A)} 20 \qquad\textbf{(B)} 25 \qquad\textbf{(C)} 45 \qquad\textbf{(D)} 306 \qquad\textbf{(E)} 351$

Let $f(x) = \frac{4^x}{4^x+2}$ for $x > 0$. Evaluate $\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)$

Let $ABC$ be a triangle with $AB \neq AC$ and with incenter $I$. Let $M$ be the midpoint of $BC$, and let $L$ be the midpoint of the circular arc $BAC$. Lines through $M$ parallel to $BI,CI$ meet $AB,AC$ at $E$ and $F$, respectively, and meet $LB$ and $LC$ at $P$ and $Q$, respectively. Show that $I$ lies on the radical axis of the circumcircles of triangles $EMF$ and $PMQ$. Proposed by [i]Andrew Wu[/i]

Let $n \geq 2$ be a positive integer. Consider the following equation: \[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\] a) For $n=2$, solve the given equation in $\mathbb{R}$. b) Prove that, for any $n \geq 2$, the equation has at most $2$ real solutions.

Find all integers $n$ for which $n(n + 2010)$ is a perfect square.

Let $a, b, c, d, e$ be real numbers satisfying the following conditions. \[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.

Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

There's infinity of the following blocks on the table:$1*1 , 1*2 , 1*3 ,.., 1*n$. We have a $n*n$ table and Ali chooses some of these blocks so that the sum of their area is at least $n^2$. Then , Amir tries to cover the $n*n$ table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the $n*n$ table with those blocks chosen by Ali. Let $k$ be the maximum coverable area independent of Ali's choice. Prove that: $$n^2 - \lceil \frac{n^2}{4} \rceil \leq k \leq n^2 - \lfloor \frac{n^2}{8} \rfloor$$ *Note : the blocks can be placed only vertically or horizontally.

$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles. (I. Voronovich)

Find distinct points $A, B, C,$ and $D$ in the plane such that the length of the segment $AB$ is an even integer, and the lengths of the segments $AC, AD, BC, BD,$ and $CD$ are all odd integers. In addition to stating the coordinates of the points and distances between points, please include a brief explanation of how you found the configuration of points and computed the distances.

Given positive integers $a_1, a_2, \ldots, a_{2023}$ such that $$a_k = \sum_{i=1}^{2023} |a_k - a_i|$$ for all $1 \le k \le 2023,$ find the minimum possible value of $a_1+a_2+\ldots+a_{2023}.$

In an acute triangle $AB$ the heights $ BE$ and $CF$ intersect at the orthocenter $H$, and $M$ is the midpoint of $BC$. The line $EF$ intersects the lines $MH$ and $BC$ at the points $P$ and $T$ , respectively. $AP$ intersects the cirumcscribed circle of $\vartriangle ABC$ for second time at the point $Q$ . Prove that $\angle AQT= 90^o$. (Fedir Yudin)