On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.

For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. [i](tomoyo-jung)[/i]

Let the sum of the first $ n$ primes be denoted by $ S_n$. Prove that for any positive integer $ n$, there exists a perfect square between $ S_n$ and $ S_{n\plus{}1}$.

Let $n \geq 2$ and $A, B \in \mathcal{M}_n(\mathbb{C})$ such that there exists an idempotent matrix $C \in \mathcal{M}_n(\mathbb{C})$ for which $C^*=AB-BA.$ Prove that $(AB-BA)^2=0.$ Note: $X^*$ is the [url = https://en.wikipedia.org/wiki/Adjugate_matrix]adjugate[/url] matrix of $X$ (not the conjugate transpose)

Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

Solve the equation \[ x^2 \plus{} \frac{x^2}{(x\plus{}1)^2} \equal{} 3\]

How many zeros are there at the end of $ \binom{200}{124} $?

Two sequences of positive integers, $x_1,x_2,x_3, ...$ and $y_1,y_2,y_3,..$ are given, such that $\frac{y_{n+1}}{x_{n+1}} > \frac{y_n}{x_n}$ for each $n \ge 1$. Prove that there are infinitely many values of $n$ such that $y_n > \sqrt{n}$.

An urn was filled with three balls by the following procedure: it was thrown a coin three times, inserting, each time a white ball came up heads, and every time tails came up, a black ball. We draw from this urn, four times consecutive, one ball; we return it to the urn before the next extraction. Which is the probability that in the four extractions a cue ball is obtained?

Trapezoid $ ABCD$ has $ AD\parallel{}BC$, $ BD \equal{} 1$, $ \angle DBA \equal{} 23^{\circ}$, and $ \angle BDC \equal{} 46^{\circ}$. The ratio $ BC: AD$ is $ 9: 5$. What is $ CD$? $ \textbf{(A)}\ \frac {7}{9}\qquad \textbf{(B)}\ \frac {4}{5}\qquad \textbf{(C)}\ \frac {13}{15} \qquad \textbf{(D)}\ \frac {8}{9}\qquad \textbf{(E)}\ \frac {14}{15}$

How many positive integers less than $1$ million have exactly $5$ positive divisors? $$ \mathrm a. ~ 1\qquad \mathrm b.~5\qquad \mathrm c. ~11 \qquad \mathrm d. ~23 \qquad \mathrm e. ~24 $$

Some cells of a $2007\times 2007$ table are colored. The table is [i]charrua[/i] if none of the rows and none of the columns are completely colored.[list](a) What is the maximum number $k$ of colored cells that a charrua table can have? (b) For such $k$, calculate the number of distinct charrua tables that exist.[/list]

Let $m$ be an odd positive integer greater than $1$. Let $S_m$ be the set of all non-negative integers less than $m$ which are of the form $x+y$, where $xy-1$ is divisible by $m$. Let $f(m)$ be the number of elements of $S_m$. [b](a)[/b] Prove that $f(mn)=f(m)f(n)$ if $m$, $n$ are relatively prime odd integers greater than $1$. [b](b)[/b] Find a closed form for $f(p^k)$, where $k>0$ is an integer and $p$ is an odd prime.

Find all polynomials P(x) satisfying the equation $(2x-1)P(x) = (x-1)P(2x), \forall x.$

We have $105$ coins, among which we know that there are three fake ones. Authentic coins have all the same weight, which is greater than that of the false ones, which also have the same weight. Determine from can $26$ authentic coins be selected by weighing only two in one two pan balance.

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$. Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$. Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$. Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$. Let $H_{2}$ be the orthocentre of triangle $ABD$. Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$. Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$. Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$. Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$. $\ell_{AD}$ ,$\ell_{BC}$ ,$\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$, $N=\ell_{BC} \cap \ell_{CD}$, $P=\ell_{CD} \cap \ell_{AD}$,$Q=\ell_{AD} \cap \ell_{AB}$. Prove that $MNPQ$ is cyclic.

For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$. (1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$. (2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

The function $f$ is defined by $$f(x) = \Big\lfloor \lvert x \rvert \Big\rfloor - \Big\lvert \lfloor x \rfloor \Big\rvert$$for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$? $\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\}$ $\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} $

Find all natural numbers such that each is equal to the sum of the factorials of its digits. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

Numbers $1, 2,\ldots, n$ are written on the board. By one move, we replace some two numbers $ a, b$ with the number $a^2-b{}$. Find all $n{}$ such that after $n-1$ moves it is possible to obtain $0$.