Let $ABC$ be a triangle. Point $E,F$ moves on the opposite ray of $BA,CA$ such that $BF=CE$. Let $M,N$ be the midpoint of $BE,CF$. $BF$ cuts $CE$ at $D$ a) Suppost that $I$ is the circumcenter of $(DBE)$ and $J$ is the circumcenter of $(DCF)$, Prove that $MN \parallel IJ$ b) Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of triangle $AEF$. Prove that when $E$ varies on the opposite ray of $BA$, $HK$ go through a fixed point

Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

Martin has two boxes $A$ and $B$. In the box $A$ there are $100$ red balls numbered from $1$ to $100$, each one with one of these numbers. In the box $B$ there are $100$ blue balls numbered from $101$ to $200$, each one with one of these numbers. Martin chooses two positive integers $a$ and $b$, both less than or equal to $100$, and then he takes out $a$ balls from box $A$ and $b$ balls from box $B$, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.\\ What is the least possible value of $a+b$ so that Martin achieves his goal for sure? For such a minimum value of $a+b$, give an example of $a$ and $b$ satisfying the goal and explain why every $a$ and $b$ with smaller sum cannot accomplish the aim.

Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If \[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\] then \[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

In an acute angled triangle $ABC$, let $H$ be the orthocenter, and let $D,E,F$ be the feet of altitudes from $A,B,C$ to the opposite sides, respectively. Let $L,M,N$ be the midpoints of the segments $AH, EF, BC$ respectively. Let $X,Y$ be the feet of altitudes from $L,N$ on to the line $DF$ respectively. Prove that $XM$ is perpendicular to $MY$.

In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which: [b](i)[/b] $a_1 < a_2 < \cdots < a_n,$ [b](ii) [/b] $b_1 < b_2 <\cdots < b_n,$ [b](iii) [/b]$a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$

Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter. $i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation. $ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.

Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

In convex quadrilateral $KALE$, angles $\angle KAL$, $\angle AKL$, and $\angle ELK$ measure $110^\circ$, $50^\circ$, and $10^\circ$, respectively. Given that $KA = LE$ and that $\overline{KL}$ and $\overline{AE}$ intersect at point $X$, compute the value of $\tfrac{KX^2}{AL\cdot EX}$.

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

$ABC$ is a right triangle with $\angle C = 90^o$. Let $N$ be the middle of arc $BAC$ of the circumcircle and $K$ be the intersection point of $CN$ and $AB$. Assume $T$ is a point on a line $AK$ such that $TK=KA$. Prove that the circle with center $T$ and radius $TK$ is tangent to $BC$. (Mykhailo Sydorenko)

Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

$\omega_A, \omega_B, \omega_C$ are three concentric circles with radii $2,3,$ and $7,$ respectively. We say that a point $P$ in the plane is [i]nice[/i] if there are points $A, B,$ and $C$ on $\omega_A, \omega_B,$ and $\omega_C,$ respectively, such that $P$ is the centroid of $\triangle{ABC}.$ If the area of the smallest region of the plane containing all nice points can be expressed as $\tfrac{a\pi}{b},$ where $a$ and $b$ are relatively prime positive integers , what is $a+b$?

Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$ for all real $x, y$.

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

Determine all triples $(m , n , p)$ satisfying : \[n^{2p}=m^2+n^2+p+1\] where $m$ and $n$ are integers and $p$ is a prime number.

[i]Mediovagio[/i] is a computer game that consists in a $3 \times 3$ table in which each of the nine cells has a integer number from $1$ to $n$. When one clicks a cell, the numbers in the clicked cell and in the cells that share an edge with it are increased by $1$ and the sum is evaluated${}\bmod n$. Determine the values of $n$ for which it's possible, with a finite number of clicks, obtain any combination of numbers from an given initial combination. EDIT: I corrected the statement.

A train car held $6000$ pounds of mud which was $88$ percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only $82$ percent water. How many pounds does the mud weigh now?

Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that $a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$

As shown in the figure, it is known that the quadrilateral $ABCD$ satisfies $\angle ADB = \angle ACB = 90^o$. Suppose $AC$ and $BD$ intersect at point $P$, point $R$ lies on $CD$ and $RP \perp AB$. $M$ and $N$ are the midpoints of $AB$ and $CD$ respectively. Point $K$ is a point on the extension line of $NM$, the circumscribed circles of $\vartriangle DKC$ and $\vartriangle AKB$ intersect at point $S$. Prove that $KS \perp SR$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/fc0a391f8ebcdee792e9b226cbf55a058251a1.png[/img]

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$. The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$. The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$. The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$. Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$. Prove that $r_A + r_B + r_C = r$.

How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.

Let $G$ be a finite simple graph and let $k$ be the largest number of vertices of any clique in $G$. Suppose that we label each vertex of $G$ with a non-negative real number, so that the sum of all such labels is $1$. Define the [i]value of an edge[/i] to be the product of the labels of the two vertices at its ends. Define the [i]value of a labelling[/i] to be the sum of values of the edges. Prove that the maximum possible value of a labelling of $G$ is $\frac{k-1}{2k}$. (A [i]finite simple graph[/i] is a graph with finitely many vertices, in which each edge connects two distinct vertices and no two edges connect the same two vertices. A [i]clique[/i] in a graph is a set of vertices in which any two are connected by an edge.)

Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$