Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points. Proposed by:A Kuznetsov

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?

i) Prove that $$ \lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}$$. ii) Prove that there is a positive constant $c$ such that for every $x\in [1,\infty)$ we have $$\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}$$

Seven line segments, with lengths no greater than $10$ inches, and no shorter than $1$ inch, are given. Show that one can choose three of them to represent the sides of a triangle. Give an example which shows that if only six segments are used, then such a choice may be impossible.

Let $({{x}_{n}})$ be an integer sequence such that $0\le {{x}_{0}}<{{x}_{1}}\le 100$ and $${{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{ }\forall n\ge 0.$$ a) Prove that if ${{x}_{0}}=2,{{x}_{1}}=3$ then for each positive integer $n,$ the sum of divisors of the following number is divisible by $24$ $${{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018.$$ b) Find all pairs of numbers $({{x}_{0}},{{x}_{1}})$ such that ${{x}_{n}}{{x}_{n+1}}+2019$ is a perfect square for infinitely many nonnegative integer numbers $n.$

Al wishes to label the faces of his cube with the integers $2009,2010,$ and $2011,$ with one integer per face, such that adjacent faces (faces that share an edge) have integers that differ by at most $1.$ Determine the number of distinct ways in which he can label the cube, given that two configurations that can be rotated on to each other are considered the same, and that we disregard the orientation in which each number is written on to the cube.

Calculate the definite integral $$\int_2^4 \sin ((x-3)^3) dx$$

Let $ABCD$ be a square of side 5, $E$ a point on $BC$ such that $BE=3, EC= 2$. Let $P$ be a variable point on the diagonal $BD.$ Determine the length of $PB$ if $PE+PC$ is smallest.

An open necklace can contain rubies, emeralds, and sapphires. At every step we can perform any of the following operations: [list=1] [*]We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire.[/*] [*]We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. [/*] [*]If we find two consecutive sapphires then we can remove them.[/*] [*]If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.[/*] [/list] Furthermore we can also reverse all of the above operations. For example by reversing 3. we can put two consecutive sapphires on any position we wish. Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones). [i]Remark:[/i] A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed. [i]Proposed by Demetres Christofides, Cyprus[/i]

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

Given triangle $ABC$, midpoint $M$ of the side $[BC]$, the centre $O$ of the inscribed circle. The line $(MO)$ crosses the height $AH$ in the point $E$. Prove that the distance $|AE|$ equals the inscribed circle radius.

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

Let $ a,\ b,\ c$ be positive real numbers. Prove the following inequality. \[ \int_1^e \frac {x^{a \plus{} b \plus{} c \minus{} 1}[2(a \plus{} b \plus{} c) \plus{} (c \plus{} 2a)x^{a \minus{} b} \plus{} (a \plus{} 2b)x^{b \minus{} c} \plus{} (b \plus{} 2c)x^{c \minus{} a} \plus{}(2a \plus{} b)x^{a \minus{} c} \plus{} (2b \plus{} c)x^{b \minus{} a} \plus{} (2c \plus{} a)x^{c \minus{} b}]}{(x^a \plus{} x^b)(x^b \plus{} x^c)(x^c \plus{} x^a)}\geq a \plus{} b \plus{} c.\] I have just posted 500 th post. [color=blue]Thank you for your cooperations, mathLinkers and AOPS users.[/color] I will keep posting afterwards. Japanese Communities Modeartor kunny

A player chooses one of the numbers $ 1$ through $ 4$. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered $ 1$ through $ 4$. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $ \$1$. If the number chosen appears on the bottom of both of the dice, then the player wins $ \$2$. If the number chosen does not appear on the bottom of either of the dice, the player loses $ \$1$. What is the expected return to the player, in dollars, for one roll of the dice? $ \textbf{(A)}\ \minus{}\frac{1}{8}\qquad \textbf{(B)}\ \minus{}\frac{1}{16}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac{1}{16}\qquad \textbf{(E)}\ \frac{1}{8}$

There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks? [i]Proposed by Gabriel Wu[/i]

Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.

Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most \[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]

Find the polynomials $ f(x)$ having the following properties: (i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$ (ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$

For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because \[4=3+1=2+2=2+1+1=1+1+1.\] Also define $p(0)=1$. Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1. [i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]

If $x_1$ and $x_2$ are the solutions of the equation $x^2-(m+3)x+m+2=0$ Find all real values of $m$ such that the following inequations are valid $\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$ and $x_1^2+x_2^2<5$

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

Let $a, b, c, d, e$ be positive and [b]different [/b] divisors of $n$ where $n \in Z^{+}$. If $n=a^4+b^4+c^4+d^4+e^4$ let's call $n$ "marvelous" number. $a)$ Prove that all "marvelous" numbers are divisible by $5$. $b)$ Can count of "marvelous" numbers be infinity?