Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

Given a triangle $ABC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.

Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.

Find all natural numbers $ k\ge 1 $ and $ n\ge 2, $ which have the property that there exist two matrices $ A,B\in M_n\left(\mathbb{Z}\right) $ such that $ A^3=O_n $ and $ A^kB +BA=I_n. $

An acute-angled triangle $ABC$ is given, through the vertices $B$ and $C$ of which a circle $\Omega$, $A \notin \Omega$, is drawn. We consider all points $P \in \Omega$, that do not lie on none of the lines $AB$ and $AC$ and for which the common tangents of the circumscribed circles of triangles $APB$ and $APC$ are not parallel. Let $X_P$ be the point of intersection of such two common tangents. a) Prove that the locus of points $X_P$ lies to some two lines. b) Prove that if the circle $\Omega$ passes through the orthocenter of the triangle $ABC$, then one of these lines is the line $BC$.

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/b2f58b6d59cb26e2fe29d0df59c1a42639a496.png[/img]

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

Let $x, y, z$ be nonnegative real numbers such that $x + y + z = 3$. Prove that $$\frac{x}{4-y}+\frac{y}{4-z}+\frac{z}{4-x}+\frac{1}{16}(1-x)^2(1-y)^2(1-z)^2\leq 1,$$ and determine all such triples $(x, y, z)$ where the equality holds.

Which of these four numbers $ \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{\minus{}1}\cdot \sqrt{(.09)^{\minus{}1}}$, is (are) rational: $ \textbf{(A)}\ \text{none}\qquad \textbf{(B)}\ \text{all}\qquad \textbf{(C)}\ \text{the first and fourth}\qquad \textbf{(D)}\ \text{only the fourth}\qquad \textbf{(E)}\ \text{only the first}$

Let $n>1$ be an integer. Find the number of permutations $(a_1,a_2,\ldots,a_n)$ of the numbers $1,2,\ldots,n$ such that $a_i>a_{i+1}$ holds for exactly one $i\in\{1,2,\ldots,n-1\}$.

There are 5 points on plane. Prove that you can chose some of them and shift them such that distances between shifted points won't change and as a result there will be symetric by some line set of 5 points.

Five points $A$, $B$, $C$, $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$. Let $k$ be a circle tangent to $AD$, $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$, $B$ and $C$. Let $F \neq A$ be the intersection of $\tau$ and the tangent line to $k$ passing through $A$ different from $AD$. Prove that there exists a circle tangent to $BD$, $BF$, $CE$ and $\tau$.

Hossna is playing with a $m*n$ grid of points.In each turn she draws segments between points with the following conditions. **1.** No two segments intersect. **2.** Each segment is drawn between two consecutive rows. **3.** There is at most one segment between any two points. Find the maximum number of regions Hossna can create.

$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $ [i]Taclit Daniela Nadia[/i]

Convex quadrilateral $ ABCD$ has $ AB\equal{}9$ and $ CD\equal{}12$. Diagonals $ AC$ and $ BD$ intersect at $ E$, $ AC\equal{}14$, and $ \triangle AED$ and $ \triangle BEC$ have equal areas. What is $ AE$? $ \textbf{(A)}\ \frac{9}{2}\qquad \textbf{(B)}\ \frac{50}{11}\qquad \textbf{(C)}\ \frac{21}{4}\qquad \textbf{(D)}\ \frac{17}{3}\qquad \textbf{(E)}\ 6$

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

Let $ABC$ be a triangle inscribed in circle $(O)$ with diamter $KL$ passes through the midpoint $M$ of $AB$ such that $L, C$ lie on the different sides respect to $AB$. A circle passes through $M, K $cuts $LC$ at$ P, Q $(point $P$ lies between$ Q, C$). The line $KQ $cuts $(LMQ)$ at $R$. Prove that $ARBP$ is cyclic and$ AB$ is the symmedian of triangle $APR$. Please help :)

Seven students count from $1$ to $1000$ as follows: [list] [*]Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$ [*]Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. [*]Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. [*]Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. [*]Finally, George says the only number that no one else says. [/list] What number does George say? $ \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 $

Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?

Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.

We call a natural number 3-partite if the set of its divisors can be partitioned into 3 subsets each with the same sum. Show that there exist infinitely many 3-partite numbers.

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.