In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$, as shown. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$. [center][img]https://snag.gy/SIuOLB.jpg[/img][/center]

The absolute value of the sum of the elements of a real orthogonal matrix is at most the order of the matrix.

What is the area of the region in the complex plane consisting of all points $z$ satisfying both $|\tfrac{1}{z}-1|<1$ and $|z-1|<1$? ($|z|$ denotes the magnitude of a complex number, i.e. $|a+bi|=\sqrt{a^2+b^2}$.)

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the sides $AB$ and $CD$ of the parallelogram $ABCD$ mark points $E$ and $F$, respectively. On the diagonals $AC$ and $BD$ chose the points $M$ and $N$ so that $EM\parallel BD$ and $FN\parallel AC$. Prove that the lines $AF, DE$ and $MN$ intersect at one point. (B. Rublev)

$m, n$ are relatively prime. We have three jugs which contain $m$, $n$ and $m+n$ liters. Initially the largest jug is full of water. Show that for any $k$ in $\{1, 2, ... , m+n\}$ we can get exactly $k$ liters into one of the jugs.

Every point on a line is painted either red or blue. Prove that there always exist three points $A,B,C$ that are painted the same color and are such that the point $B$ is the midpoint of the segment $AC$.

For which commutative finite groups is the product of all elements equal to the unit element?

Let $ABCD$ be a convex quadrilateral, let $E$ and $F$ be the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ meets $DF$ in $O$. Show that if the lines $AO$ and $BO$ divide the side $CD$ in 3 equal parts, then $ABCD$ is a parallelogram.

For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove \[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]

Let $Y$ and $Z$ be the feet of the altitudes of a triangle $ABC$ drawn from angles $B$ and $C$, respectively. Let $U$ and $V$ be the feet of the perpendiculars from $Y$ and $Z$ on the straight line $BC$. The straight lines $YV$ and $ZU$ intersect at a point $L$. Prove that $AL \perp BC$.

Let $a,b,c$ be the length sides of a given triangle and $x,y,z$ be the sides length of bisectrices, respectively. Prove the following inequality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.

Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown. [img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img] On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?

The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers? [i]S. Tokarev[/i]

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \$$1 each, begonias $ \$$1.50 each, cannas $ \$$2 each, dahlias $ \$$2.50 each, and Easter lilies $ \$$3 each. What is the least possible cost, in dollars, for her garden? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((6,0)--(0,0)--(0,1)--(6,1)); draw((0,1)--(0,6)--(4,6)--(4,1)); draw((4,6)--(11,6)--(11,3)--(4,3)); draw((11,3)--(11,0)--(6,0)--(6,3)); label("1",(0,0.5),W); label("5",(0,3.5),W); label("3",(11,1.5),E); label("3",(11,4.5),E); label("4",(2,6),N); label("7",(7.5,6),N); label("6",(3,0),S); label("5",(8.5,0),S);[/asy]$ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 115 \qquad \textbf{(C)}\ 132 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 156$

$\frac{a^3}{b^3}$+$\frac{a^3+1}{b^3+1}$+...+$\frac{a^3+2015}{b^3+2015}$=2016 b - positive integer, b can't be 0 a - real Find $\frac{a^3}{b^3}$*$\frac{a^3+1}{b^3+1}$*...*$\frac{a^3+2015}{b^3+2015}$

Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon). [i]G. and L. Fejes-Toth[/i]

A rectangle of size $ m \times n$ has been filled completely by trominoes (a tromino is an L-shape consisting of 3 unit squares). There are four ways to place a tromino 1st way: let the "corner" of the L be on top left 2nd way: let the "corner" of the L be on top right 3rd way: let the "corner" of the L be on bottom left 4th way: let the "corner" of the L be on bottom right Prove that the difference between the number of trominoes placed in the 1st and the 4th way is divisible by $ 3$.

Let $n$ be a positive integer and $a_1,a_2,\dots,a_n$ be integers. Function $f: \mathbb{Z} \rightarrow \mathbb{R}$ is such that for all integers $k$ and $l$, $l \neq 0$, $$\sum_{i=1}^n f(k+a_il)=0.$$ Prove that $f \equiv 0$.

For positive $ a, b, c, d$ prove that $ (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)$

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system. Show that there is an integer $k$, with no $9$ in it's decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.