For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define \[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\] where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.

Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.

Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.

Let $A$ be a non-empty subset of $\mathbb{R}$ with the property that for every real numbers $x,y$, if $x+y\in A$ then $xy\in A$. Prove that $A=\mathbb{R}$.

Find the greatest possible integer value of the side length of an equilateral triangle whose vertices belong to the interior region of a square with side length $100$.

Let $p$ be a prime number and let further $p + 1$ rational numbers $a_0,...,a_p$ with the following property given: If one removes any of the $p + 1$ numbers, then the remaining may be split in at least two groups , which all have the same mean value (for different distant numbers, however, these mean values ​​may be different). Prove that all $p + 1$ numbers are equal.

Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.

Evaluate $$\sum_{i=0}^{\infty}\frac{7^i}{(7^i+1)(7^i+7)}$$ [i]Proposed by Connor Gordon[/i]

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)

Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square. [img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img] [b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge). [b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property. (a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer. (b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer. [b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct. [img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img] [b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

Prove that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying, for all real $x$, \[ f(x)=f(2x)=f(1-x)\] are periodic.

Prove that every infinite arithmetic progression $a$, $a+d$, $a+2d$,... where $a$ and $d$ are positive integers, contains infinte geometric progression $b$, $bq$, $bq^2$,... where $b$ and $q$ are also positive integers

For all real numbers $a,b,c>0$ such that $abc=1$, prove that $\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}$.

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

Let $a,b$, be two real numbers. If for any $x>0$ holds that $|a-b|<x$, then prove that $a=b$.

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]

Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.

Write the following polynomial as a product of irreducible polynomials in $\mathbb{Z}[X]$ \[ f(X) = X^{2005} - 2005 X + 2004 . \]Justify your answer.

[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions: [b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ [b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$ [b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ [i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$

For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that $$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1). }}$$

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

Find all triplets $ (x,y,z) $ of real numbers such that \[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]