Find roots of the equation $$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$

There are 5 points in a rectangle (including its boundary) with area 1, no three of them are in the same line. Find the minimum number of triangles with the area not more than $\frac 1{4}$, vertex of which are three of the five points.

A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? $\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Given that $x$ and $y$ are real numbers, compute the minimum value of $$x^4+4x^3+8x^2+4xy+6x+4y^2+10.$$

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Let be a $ \beta >1. $ Calculate $ \lim_{n\to\infty} \frac{k(n)}{n} ,$ where $ k(n) $ is the smallest natural number that satisfies the inequality $ (1+n)^k\ge n^k\beta . $ [i]Neculai Hârţan[/i]

Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ\perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}.$ Find $\angle CBA$, in degrees. [i]Proposed by Nathan Ramesh

Let $n \in \mathbb{N}, n \ge 3.$ $a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$ $b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

A class has $n > 1$ boys and $n$ girls. For each arrangement $X$ of the class in a line let $f(X)$ be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with $f(X) = 0$ be $A$, and the number of arrangements with $f(X) = 1$ be $B$. Show that $B = 2A$.

Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.

A convex flat quadrilateral is given. Prove that for the ratio $q$ of the largest to the smallest of all distances, for any two vertices: $q \ge \sqrt2$. [hide=original wording]Gegeben sei ein konvexes ebenes Viereck. Es ist zu beweisen, dass fur den Quotienten q aus dem großten und dem kleinsten aller Abstande zweier beliebiger Eckpunkte voneinander stets gilt: q >= \sqrt2.[/hide]

A rhombus is inscribed in triangle $ ABC$ in such a way that one of its vertices is $ A$ and two of its sides lie along $ AB$ and $ AC$. If $ \overline{AC} \equal{} 6$ inches, $ \overline{AB} \equal{} 12$ inches, and $ \overline{BC} \equal{} 8$ inches, the side of the rhombus, in inches, is: $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 3 \frac {1}{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.

Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that: (a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$ (b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$ ($P \circ Q$ indicates $P(Q(x))$.)

Suppose a table with one row and infinite columns. We call each $1\times1$ square a [i]room[/i]. Let the table be finite from left. We number the rooms from left to $\infty$. We have put in some rooms some coins (A room can have more than one coin.). We can do $2$ below operations: $a)$ If in $2$ adjacent rooms, there are some coins, we can move one coin from the left room $2$ rooms to right and delete one room from the right room. $b)$ If a room whose number is $3$ or more has more than $1$ coin, we can move one of its coins $1$ room to right and move one other coin $2$ rooms to left. $i)$ Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more. $ii)$ Suppose that there is exactly one coin in each room from $1$ to $n$. Prove that by doing the allowed operations, we cannot put any coins in the room $n+2$ or the righter rooms.

Does there exist a real number $r$ such that the equation $$x^3-2023x^2-2023x+r=0$$ has three distinct rational roots?

The area of the quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and $(-4,2,5)$ is the number $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?

Ten distinct numbers are chosen at random from the set $\{1, 2, 3, ... , 37\}$. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.

Let $a,b,c $ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$ and $S=2x^3-9xy+9z .$ $(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any real number $t$. $(b)$ Prove that $ (3y-x^2)^3\ge S^2 .$

Consider a $1 \times 1$ grid of squares. Let $A,B,C,D$ be the vertices of this square, and let $E$ be the midpoint of segment $CD$. Furthermore, let $F$ be the point on segment $BC$ satisfying $BF = 2CF$, and let $P$ be the intersection of lines $AF$ and $BE$. Find $\frac{AP}{PF}$.

For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.