The function $f$ sends sequences to sequences in the following way: given a sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers, $f$ sends $\{a_n\}_{n=0}^{\infty}$ to the sequence $\{b_n\}_{n=0}^{\infty},$ where $b_n=\sum_{k=0}^n a_k \tbinom{n}{k}$ for all $n \ge 0.$ Let $\{F_n\}_{n=0}^{\infty}$ be the Fibonacci sequence, defined by $F_0=0, F_1=1,$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 0.$ Let $\{c_n\}_{n=0}^{\infty}$ denote the sequence obtained by applying the function $f$ to the sequence $\{F_n\}_{n=0}^{\infty}$ $2022$ times. Find $c_5 \pmod{1000}.$

Consider the sequence of integer such that: $ a_1 = 2$ $ a_2 = 5$ $ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$ Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.

For any function $f(x)$, in the same rectangular coordinates, figures of function $y=f(x-1)$ and $y=f(-x+1)$ $\text{(A)}$ are symmetrical about $x$-axis $\text{(B)}$ are symmetrical about line $x=1$ $\text{(C)}$ are symmetrical about line $x=-1$ $\text{(D)}$ are symmetrical about $y$-axis

If $\angle A = 60^\circ $, $\angle E = 40^\circ $ and $\angle C = 30^\circ $, then $\angle BDC = $ [asy] pair A,B,C,D,EE; A = origin; B = (2,0); C = (5,0); EE = (1.5,3); D = (1.75,1.5); draw(A--C--D); draw(B--EE--A); dot(A); dot(B); dot(C); dot(D); dot(EE); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NE); label("$E$",EE,N); [/asy] $\text{(A)}\ 40^\circ \qquad \text{(B)}\ 50^\circ \qquad \text{(C)}\ 60^\circ \qquad \text{(D)}\ 70^\circ \qquad \text{(E)}\ 80^\circ$

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

Let $k$ be a positive integer and $u, v$ be real numbers. Consider $P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c$. (a) For $k = 2$ prove that if $a, b, c$ are rational then so is $uv$. (b) Is that also true for $k = 3$?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$?

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic. Proposed by [i]Theoklitos Parayiou, Cyprus[/i]

Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$.

Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is 10. The area of $A 'B 'C'D'$ is $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }45\qquad\textbf{(D) }50\qquad \textbf{(E) }60$

Is it possible to place a triangle with area $1999$ and perimeter $19992$ in the interior of a triangle with area $2000$ and perimeter $20002$?

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

What is the radius of a circle inscribed in a rhombus with diagonals of length $ 10$ and $ 24$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 58/13 \qquad \textbf{(C)}\ 60/13 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

$ABC$ is an equilateral triangle with side length $ 1$. $BCDE$ is a square. Some point $F$ is equidistant from $A, D$, and $E$. Find the length of $AF$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/194318955f7ed5fed1c58633cb29c33011371a.jpg[/img]

In some company, consisting of $n$ people, any two have at most $k \geq 2$ common friends. Lets call group of people working in the company unsocial if everyone in the group has at most one friend from the group. Prove that there exists an unsocial group consisting of at least $\sqrt{\frac{2n}{k}}$ people [i]M. Zorka[/i]

The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? [asy] size(6cm); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,mediumgray); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,mediumgray); fill((1,3)--(1,4)--(2,4)--(2,3)--cycle,mediumgray); fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,mediumgray); label(scale(.9)*"$1$", (3.5,3.5)); label(scale(.9)*"$2$", (4.5,3.5)); label(scale(.9)*"$3$", (4.5,4.5)); label(scale(.9)*"$4$", (3.5,4.5)); label(scale(.9)*"$5$", (2.5,4.5)); label(scale(.9)*"$6$", (2.5,3.5)); label(scale(.9)*"$7$", (2.5,2.5)); draw((1,0)--(1,7)--(2,7)--(2,0)--(3,0)--(3,7)--(4,7)--(4,0)--(5,0)--(5,7)--(6,7)--(6,0)--(7,0)--(7,7),gray); draw((0,1)--(7,1)--(7,2)--(0,2)--(0,3)--(7,3)--(7,4)--(0,4)--(0,5)--(7,5)--(7,6)--(0,6)--(0,7)--(7,7),gray); draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(1.25)); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Let $ n$ be a $ 5$-digit number, and let $ q$ and $ r$ be the quotient and remainder, respectively, when $ n$ is divided by $ 100$. For how many values of $ n$ is $ q \plus{} r$ divisible by $ 11$? $ \textbf{(A)}\ 8180 \qquad \textbf{(B)}\ 8181 \qquad \textbf{(C)}\ 8182 \qquad \textbf{(D)}\ 9000 \qquad \textbf{(E)}\ 9090$

Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$. [i]Proposed by Andrew Wu[/i]

Let $ABC$ be a given triangle with circumcenter $O$ and orthocenter $H$. Let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to the opposite sides, respectively. Let $A'$ be the reflection of $A$ with respect to $EF$. Prove that $HOA'D$ is a cyclic quadrilateral. [i]Proposed by Imad Uddin Ahmad Hasin[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $n$ be a positive integer. A German set in an $n \times n$ square grid is a set of $n$ cells which contains exactly one cell in each row and column. Given a labelling of thecells with the integers from $1$ to $n^2$ using each integer exactly once, we say that an integer is a German product if it is the product of the labels of the cells in a German set. (a) Let $n=8$. Determine whether there exists a labelling of an $8 \times 8$ grid such that the following condition is fulfilled: The difference of any two German products is alwaysdivisible by $65$. (b) Let $n=10$. Determine whether there exists a labelling of a $10 \times 10$ grid such that the following condition is fulfilled: The difference of any two German products is always divisible by $101$.