Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$ $$ \mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3} $$

Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that [b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$ [b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that $$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$

Prove that \[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \] for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.

Let $a$ and $b$ be two natural numbers whose greatest common divisor is $d$. Prove that exactly $d$ of the numbers $$a, 2a, 3a, ..., (b-1)a, ba$$ is divisible by $b$.

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

A sequence of integers $a_{1}, a_{2}, \cdots$ is called $good$ if: • $a_{1}=1$, and; • $a_{i+1}-a_{i}$ is either $1$ or $2$ for all $i \geq 1$. Find all positive integers n that cannot be written as a sum $n = a_{1} + a_{2} + \cdots + a_{k}$, such that the integers $a_{1} , a_{2} , \cdots , a_{k}$ forms a good sequence.

Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$? $ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad \textbf{(B)}\ x \plus{} 80 \qquad \textbf{(C)}\ \minus{}x \plus{} 118 \qquad \textbf{(D)}\ x \plus{} 118 \qquad \textbf{(E)}\ 0$

[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions: $$ax + 2y = 1$$ $$2x + ay = 1$$ [b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$. [b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$. [b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer? [b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces. PS. You should use hide for answers.

This morning, Emi dropped the watch and from there it started to move more slowly. When, according to the clock, $2$ minutes have passed, in reality it has already been $3$. Now it is $6:25$ pm and the clock says it is $3:30$ pm. What time did Emi drop the watch?

Let $p(x)$ be a polynomial of degree $2022$ such that: $$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$ Find $p(2023)$.

An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference $\frac56$: $$\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92.$$ The sequence of seven natural numbers $60$, $70$, $84$, $105$, $140$, $210$, $420$ has the property that the sequence inverted numbers (i.e. the row $\frac{1}{60}$, $\frac{1}{70}$, $\frac{1}{84}$, $\frac{1}{105}$, $\frac{1}{140}$, $\frac{1}{210}$,$\frac{1}{420}$ ) is an arithmetic sequence. (a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence? (b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?

Evaluate the sum $$\sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$$

Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid $$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$ and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$? (Serbia)

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have \[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]

Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a > 2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$. Find the value of $w =\frac{a}{b}+\frac{c}{d}$ .

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?

Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.

Let $ p(x) \equal{} x^6 \plus{} ax^5 \plus{} bx^4 \plus{} cx^3 \plus{} dx^2 \plus{} ex \plus{} f$ be a polynomial such that $ p(1) \equal{} 1, p(2) \equal{} 2, p(3) \equal{} 3, p(4) \equal{} 4, p(5) \equal{} 5,$ and $ p(6) \equal{} 6.$ What is $ p(7)$? A. 0 B. 7 C. 14 D. 49 E. 727

Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition: $$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$ for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$

There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?