In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P.$ Prove that $(PF_1)(PF_2)d^2$ is constant as $P$ varies on the ellipse, where $PF_1$ and $PF_2$ are distances from $P$ to the foci $F_1$ and $F_2$ of the ellipse.

Let ${ABC}$ be a triangle and ${\Gamma}$ the circle with diameter ${AB}$. The bisectors of ${\angle BAC}$ and ${\angle ABC}$ intersect ${\Gamma}$ (also) at ${D}$ and ${E}$, respectively. The incircle of ${ABC}$ meets ${BC}$ and ${AC}$ at ${F}$ and ${G}$, respectively. Prove that ${D, E, F}$ and ${G}$ are collinear.

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the child, after some operations, can get each box containing $n$ candies, no matter which the initial distribution of candies is.

Let $\ell$ be tangent to the circle $k$ at $B$. Let $A$ be a point on $k$ and $P$ the foot of perpendicular from $A$ to $\ell$. Let $M$ be symmetric to $P$ with respect to $AB$. Find the set of all such points $M.$

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number? $\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$

On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$. Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$

Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$. Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.

In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that: $ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$

Let $a$ and $b$ be positive integers such that $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$. Find $10a + b$.

The lattice points $(i, j)$ for integers $0 \le i, j \le 3$ are each being painted orange or black. Suppose a coloring is good if for every set of integers $x_1, x_2, y_1, y_2$ such that $0 \le x_1 < x_2 \le 3$ and $0 \le y_1 < y_2 \le 3$, the points $(x_1, y_1),(x_1, y_2),(x_2, y_1),(x_2, y_2)$ are not all the same color. How many good colorings are possible?

Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.

A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that \[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\] holds.

Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A [i]step[/i] is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it. [b]a)[/b] Show that, for any $ m,n\ge 5, $ there exists a succession of [i]steps[/i] that make the rectagle to be single-colored. [b]b)[/b] What about $ m=n+1=5? $

Let's construct a family $\{K_n\}$ of geometric figures following the pattern shown in pictures: [center][img]https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png[/img][/center] where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures $K_n$, for integer $n \ge 0$ . Then, we denote by $P_n$ and $A_n$ the perimeter and area of the figure $K_n$, respectively. If the side of square to build $K_0$ measures $x$: (a) Calculate $P_0$ and $A_0$ (in terms of the length $x$). (b) Find an explicit formula for $P_n$, and for $A_n$, in terms of $x$ and of $n$. Simplify your answers. (c) If $P_{2022} = A_{2022}$, find the measure of the six sides of the figure $K_0$, in its simplest form.

What is the median of the following list of $4040$ numbers$?$ $$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$ $\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.

We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

The function $f(x)$ has the property that $f(x) = -\frac{1}{f(x-1)}.$ Given that $f(0)=-\frac{1}{21},$ find the value of $f(2021).$ [i]Proposed by Ada Tsui[/i]

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

There are three different points on the line $ A $, $ B $, $ C $ and a point $ S $ outside this line; perpendicularly drawn at points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on the circle