For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.

Consider all line segments of length $4$ with one end-point on the line $y = x$ and the other end-point on the line $y = 2x$. Find the equation of the locus of the midpoints of these line segments.

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$. [i]Proposed by Nikola Velov[/i]

The ``Manhattan distance" between two cells is the shortest distance between those cells when traveling up, down, left, or right, as if one were traveling along city blocks rather than as the crow flies. Place numbers from $1$-$6$ in some cells so the following criteria are satisfied: $1.$ A cell contains at most one number. Cells can be left empty. $2.$ For each cell containing a number $N$ in the grid, exactly two other cells containing $N$ are at a Manhattan distance of $N.$ $3.$ For each cell containing a number $N$ in the grid, no other cells containing $N$ are at a Manhattan distance less than $N.$ [asy] //credits to fruitmonster97 for the diagram unitsize(1.25cm); //The gridlines for(int i=-3;i<4;++i){ draw((i,3)--(i,-3),lightgray+linewidth(1)); } for(int j=-3;j<4;++j){ if (j==0){ draw((4,j)--(-4,j),lightgray+linewidth(1)); }else{ draw((3,j)--(-3,j),lightgray+linewidth(1)); } } //The outline draw((-4,-1)--(-4,1)--(-3,1)--(-3,3)--(3,3)--(3,1)--(4,1)--(4,-1)--(3,-1)--(3,-3)--(-3,-3)--(-3,-1)--cycle); //The numbers label("$1$",(-0.5,0.5)); label("$1$",(0.5,-0.5)); label("$2$",(-1.5,1.5)); label("$2$",(-2.5,-1.5)); label("$3$",(2.5,1.5)); label("$3$",(-3.5,0.5)); label("$4$",(3.5,-0.5)); label("$4$",(1.5,-2.5)); label("$4$",(-1.5,2.5)); label("$5$",(-2.5,1.5)); label("$5$",(2.5,-1.5)); label("$6$",(1.5,-1.5)); [/asy]

A square $2n\times 2n$ grid is given. Let us consider all possible paths along grid lines, going from the centre of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its centre at the grid centre is passed through only once. (a) Prove that the number of all such paths is equal to $4\prod_{i=2}^n(16i-9)$. (b) Find the number of pairs of such paths that divide the grid into two congruent figures. (c) How many quadruples of such paths are there that divide the grid into four congruent parts?

Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ? [i](4 points)[/i]

The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$? $\textbf{(A) } -2 \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{24}{5} \qquad \textbf{(E) } 8$

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

Let $f : R \to R$, and let $P$ be a nonzero polynomial with degree no more than $2015$. For any nonnegative integer $n$, $f^{(n)}(x)$ denotes the function defined as $f$ composed with itself $n$ times. For example, $f^{(0)}(x) = x$, $f^{(1)}(x) = f(x)$, $f^{(2)}(x) = f(f(x))$, etc. Show that there always exists a real number $q$ such that $$f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).$$

Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps. Find those pair of integers $(k,n)$ for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

If $xy:yz:zx=6:8:12,$ and $x^3+y^3+z^3:xyz$ is $m:n$ where $m$ and $n$ are relatively prime positive integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

In the acute-angled triangle $ABC$, on the lines $BC$, $AC$, $AB$ we consider the points $D$, $E$ and, respectively, $F$, such that $AD\perp AC, BE\perp AB, CF\perp AC$. Let the point $A', B', C'$ be such that $\{A'\}=BC\cap EF, \{B'\}=AC\cap DF, \{C'\}=AB\cap DE$. Prove that the following inequality is true $$\frac{A'F}{A'E} \cdot \frac{B'D}{B'F} \cdot \frac{C'E}{C'D}\geq8$$

A semicircle is inscribed in another semicircle if the smaller semicircle’s diameter is a chord of the larger semicircle, and the smaller semicircle’s arc is tangent to the diameter of the larger semicircle. Semicircle $S_1$ is inscribed in a semicircle $S_2,$ which is inscribed in another semicircle $S_3.$ The radii of $S_1$ and $S_3$ are $1$ and $10,$ respectively, and the diameters of $S_1$ and $S_3$ are parallel. The endpoints of the diameter of of $S_3$ are $A$ and $B,$ and $S_2$'s arc is tangent to $AB$ at $C.$ Compute $AC \cdot CB.$ [center] [img] https://cdn.artofproblemsolving.com/attachments/9/6/ad8c82afe131103793cb2684b45c6d20b00ef0.png [/img] [/center]

If $A, B, C$, and $D$ are four distinct points in space, prove that there is a plane $P$ on which the orthogonal projections of $A, B, C$, and $D$ form a parallelogram (possibly degenerate).

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise? $ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$

Let $n > 1$ be a natural number. There are $n$ bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of $n$ there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.

Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a\plus{}b\minus{}c$, where $ a\equal{}BC$, $ b\equal{}CA$, and $ c\equal{}AB$.

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\] Show that $a+b=c=0$.

In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.

Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

Prove that for coprime each positive integers $a, c$ there is a positive integer $b$ such that $c$ divides $\underbrace{b^{b^{b^{\ldots^b}}}}_\text{b times}-a$

The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$? $\textbf{(A) } 3 : 11\qquad \textbf{(B) } 5 : 11\qquad \textbf{(C) } 1 : 2\qquad \textbf{(D) }2 : 3\qquad \textbf{(E) }25 : 33$

Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation: $|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.

The perpendicular line issued from the center of the circumcircle to the bisector of angle $C$ in a triangle $ABC$ divides the segment of the bisector inside $ABC$ into two segments with ratio of lengths $\lambda$. Given $b = AC$ and $a = BC$, find the length of side $c.$

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.