Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent. [i]L. Lempert[/i]

Define $ P(\tau ) = (\tau + 1)^3$ . If $x + y = 0$, what is the minimum possible value of $P(x) + P(y)$?

Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds. $$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$ [i]Cosmin Nițu[/i]

Let $S$ be the set of [b]lattice[/b] points whose both coordinates are positive integers no larger than $2022$. i.e., $S=\{(x, y) \mid x, y\in \mathbb{N}, \, 1\leq x, y\leq 2022\}$. We put a card with one gold side and one black side on each point in $S$. We call a rectangle [i]"good"[/i] if: (i) All of its sides are parallel to the axes and have positive integer coordinates no larger than $2022$. (ii) The cards on its top-left and bottom-right corners are showing gold, and the cards on its top-right and bottom-left corners are showing black. Each [i]"move"[/i] consists of choosing a good rectangle and flipping all cards simultaneously on its four corners. Find the maximum possible number of moves one can perform, or show that one can perform infinitely many moves. [i]Proposed by CSJL[/i]

Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero. Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem. [i]Sutanay Bhattacharya[/i]

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

There are $8$ cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than $k$ roads go out of any city. For what values of $k$ is this possible? (D. Fomin, Leningrad)

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

Given a sequence $a_0, a_1, a_2, ... , a_n$, let its [i]arithmetic approximant[/i] be the arithmetic sequence $b_0, b_1, ... , b_n$ that minimizes the quantity $\sum_{i=0}^{n}(b_i -a_i)^2$, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence $4$, $8$, $12$, $16$ and whose anti-arithmeticity is at most $20$.

Let $\omega$ be a circle. Points $A,B,C,X,D,Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB,BC$, respectively. Prove that points $B,E,F,Y$ lie on a single circle.

We say that $ n$ squares in a $ n\times n$ board are scattered if no two of them are in the same row or column.In every square of this board is witten a natural number so that the sum of numbrs in $ n$ scattered squares is always the same and no row or no column contains two equal numbers .It turned out that the numbers on the main diagonal are arranged in the increasing order ,and that their product is the smallest among all products of $ n$ scattered numbers .Prove that scattered numbers with the greatest product are exactly those on the other diagonal.

The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart. Prove that $4 \leq \chi \leq 7$.

Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$ \[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]

Let $a_n$ and $b_n$ to be two sequences defined as below: $i)$ $a_1 = 1$ $ii)$ $a_n + b_n = 6n - 1$ $iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Determine $a_{2009}$.

Let $R(x)$ be a function that takes a natural number as input and returns a rectangle. $R(1)$ is known to have integer side lengths. Let $p(x)$ be the perimeter of $R(x)$ and let $a(x)$ be the area of $R(x)$. Suppose that $p(x+5)=6 p(x)$ for all $x$ in the domain of $R$ and that $a(x+2)=12a(x)$ for all $x> 6$ in the domain of $R$. For $x \leq 6$, $a(x+1)=a(x)+2$. Suppose $p(16)=1296$, and let the side lengths of $R(11)$ be $a$ and $b$ with $a\leq b$. Find the ordered pair $(a,b)$. [i]Proposed by Matthew Weiss

On hyperbola $y=\frac{1}{x}$ points $A_1,\ldots,A_{10}$ are chosen such that $(A_i)_x=2^{i-1}a$, where $a$ is some positive constant. Find the area of $A_1A_2 \ldots A_{10}$

If \[ \sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n} \] for relatively prime positive integers $m,n$, compute $m+n$. [i]2020 CCA Math Bonanza Lightning Round #2.4[/i]

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all infinite sequences of real numbers $ \left( a_n \right)_{n\ge 1} $ that verify, for any natural number $ n, $ the inequalities $$ \frac{1}{2\sqrt{a_{n+1}}} <\sqrt{n+1} -\sqrt{n} <\frac{1}{ 2\sqrt{a_n}} . $$

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.