Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

Let \( \mathbb{R} \) denote the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \[ f(x^2) - y f(y) = f(x+y)(f(x) - y) \] for all real numbers \( x \) and \( y \).

Let $a_{1},a_{2},...,a_{n}$ be positive real numbers with $a_{1}\leq a_{2}\leq ... a_{n}$ such that the arithmetic mean of $a_{1}^{2},...,a_{n}^{2}$ is 1. If the arithmetic mean of $a_{1}, a_{2},...,a_{n}$ is $m$. Prove that if $a_{i}\leq$ m for some $1 \leq i \leq n$, then $n(m-a_{i})^2\leq n-i$

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

Let $a_1, a_2, \ldots, a_{2023}$ be positive reals such that $\sum_{i=1}^{2023}a_i^i=2023$. Show that $$\sum_{i=1}^{2023}a_i^{2024-i}>1+\frac{1}{2023}.$$

Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying: $x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$ for all $x,y \in \mathbb{R}$ [i]Proposed by Nikolay Nikolov[/i]

The sequence $x_0,x_1,x_2,.....$ of real numbers is called with period $p$,with $p$ being a natural number, when for each $p\ge2$, $x_n=x_{n+p}$. Prove that,for each $p\ge2$ there exists a sequence such that $p$ is its least period and $x_{n+1}=x_n-\frac{1}{x_n}$ $(n=0,1,....)$

Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.

Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$ (a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$ (b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$

[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$. (b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$. [b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$. (b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$. [b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$. (a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$. (b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$. [b]p4.[/b] In the Gregorian calendar: (i) years not divisible by $4$ are common years, (ii) years divisible by $4$ but not by $100$ are leap years, (iii) years divisible by $100$ but not by $400$ are common years, (iv) years divisible by $400$ are leap years, (v) a leap year contains $366$ days, a common year $365$ days. From the information above: (a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks. (b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$. [b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$ The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120.000$ Rs. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9.408$ Rs. How many centimeters long is the ruler used by the seller?

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.

It is given that $ a^2\plus{}b^2\plus{}(a\plus{}b)^2\equal{}c^2\plus{}d^2\plus{}(c\plus{}d)^2$. Prove that $ a^4\plus{}b^4\plus{}(a\plus{}b)^4\equal{}c^4\plus{}d^4\plus{}(c\plus{}d)^4$.

Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.

The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations: \[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\ xz & \equal{} & y \minus{} x \minus{} z \\ yz & \equal{} & x \minus{} y \minus{} z \end{array} \right. \]

Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$

If $a, b,c,p,q, r $are real numbers such that, for every real number $x,$ $$ax^2 - 2bx + c \ge 0 \ \ and \ \ px^2 + 2qx + r \ge 0;$$ prove that then $$apx^2 + bqx + cr \ge 0$$ for all real $x$.

Simplify $\frac{1}{\sqrt[3]{81} + \sqrt[3]{72} + \sqrt[3]{64}}$

Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ [i]Proposed by A. Golovanov[/i]

Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.

If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$