For what real values of $k>0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but noncongruent, polygons?

A convex quadrilateral $ABCD$ is circumscribed about a circle $\omega$. Let $PQ$ be the diameter of $\omega$ perpendicular to $AC$. Suppose lines $BP$ and $DQ$ intersect at point $X$, and lines $BQ$ and $DP$ intersect at point $Y$. Show that the points $X$ and $Y$ lie on the line $AC$. [i]Géza Kós[/i]

[b]p1.[/b] Suppose $A$, $B$ and $C$ are the angles of a triangle. Prove that $$1 - 8 \cos A\cos B \cos C = sin^2(B - C) + (cos(B - C) - 2 cosA)^2.$$ [b]p2.[/b] Let $x_1, x_2,..., x_{100}$ be integers whose values are either $0$ or $1$. (a) Show that $$x_1 + x_2 + ... + x_{100} - (x_1x_2 + x_2x_3 + ... + x_{99}x_{100} + x_{100}x_1)\le 50.$$ (b) Give specific values for $x_1, x_2,..., x_{100}$ that give equality. [b]p3.[/b] Let $ABCD$ be a trapezoid whose area is $32$ square meters. Suppose the lengths of the parallel segments $AB$ and $DC$ are $2$ meters and $6$ meters, respectively, and $P$ is the intersection of the diagonals $AC$ and $BD$. If a line through $P$ intersects $AD$ and $BC$ at $E$ and $F$, respectively, determine, with a proof, the minimum possible area for quadrilateral $ABFE$. [b]p4.[/b] Let $n$ be a positive integer and $x$ be a real number. Show that $$\lfloor nx \rfloor = \lfloor x \rfloor +\left\lfloor x + \frac{1}{n} \right\rfloor + \left\lfloor x + \frac{2}{n} \right\rfloor + ... + \left\lfloor x + \frac{n - 1}{n} \right\rfloor$$ where $\lfloor a \rfloor$ is the greatest integer less than or equal to $a$. (For example, $\lfloor 4.5\rfloor = 4$ and $\lfloor - 4.5 \rfloor = -5$.) [b]p5.[/b] A $3n$-digit positive integer (in base $10$) containing no zero is said to be [i]quad-perfect[/i] if the number is a perfect square and each of the three numbers obtained by viewing the first $n$ digits, the middle $n$ digits and the last $n$ digits as three $n$-digit numbers is in itself a perfect square. (For example, when $n = 1$, the only quad-perfect numbers are $144$ and $441$.) Find all $9$-digit quad-perfect numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a regular polygon with 2007 vertices. The natural numbers $ 1,2, \ldots, 4014$ shall be distributed across the vertices and edge midpoints such that for each side the sum of its three numbers (two end points, one side center) has the same value. Show that such an assignment is possible.

Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$. (E Turkevich)

[b]a)[/b] Show that , if $ a,b>1 $ are two distinct real numbers, then $ \log_a\log_a b >\log_b\log_a b. $ [b]b)[/b] Show that if $ a_1>a_2>\cdots >a_n>1 $ are $ n\ge 2 $ real numbers, then $$ \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0. $$

Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$, respectively, of an acute triangle $ABC$, $AB \neq AC$. Let $\omega_B$ be the circle centered at $M$ passing through $B$, and let $\omega_C$ be the circle centered at $N$ passing through $C$. Let the point $D$ be such that $ABCD$ is an isosceles trapezoid with $AD$ parallel to $BC$. Assume that $\omega_B$ and $\omega_C$ intersect in two distinct points $P$ and $Q$. Show that $D$ lies on the line $PQ$.

The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

Find all pairs $(a, b)$ of real numbers with the following property: [list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]

Sidelines of an acute-angled triangle $T$ are colored in red, green, and blue. These lines were rotated about the circumcenter of $T$ clockwise by $120^\circ$ (we assume that the line has the same color after rotation). Prove that three points of pairs of lines of the same color are the vertices of a triangle which is congruent to $T$.

There are $2n$ different numbers in a row. By one move we can interchange any two numbers or interchange any $3$ numbers cyclically (choose $a,b,c$ and place $a$ instead of $b$, $b$ instead of $c$, $c$ instead of $a$). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

We have a stick of length $2n{}$ and a machine which cuts sticks of length $k\in\mathbb{N}$ with $k>1$ into two sticks with arbitrary positive integer lengths. What is the smallest number of cuts after which we can always find some sticks whose lengths sum up to $n{}$?

We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$. Show that the line $DE$ bisects the angle $\angle BFC$.

If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.

From an arbitrary point $O$ inside a convex $n$-gon, perpendiculars are drawn on (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is drawn, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.

$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take? A. Gribalko

Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$

Solve equation \[2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|\]

Morteza marks six points in the plane. He then calculates and writes down the area of every triangle with vertices in these points ($20$ numbers). Is it possible that all of these numbers are integers, and that they add up to $2019$?

Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$? $ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad \textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad \textbf{(C)}\ \frac{\sqrt3}{18} \qquad \textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad \textbf{(E)}\ \frac{\sqrt3}{12}$

How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct