Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called [i]summable[/i] if $ \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1$. Prove that for every $ n\ge 3$ there always exists a [i]summable[/i] set which consists of $ n$ elements such that the biggest element is: a) bigger than $ 2^{2n\minus{}2}$ b) smaller than $ n^2$

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

Solve the system of equation for $(x,y) \in \mathbb{R}$ $$\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.$$ \\ Explain your answer

What is the least possible value of $$(x+1)(x+2)(x+3)(x+4)+2019$$where $x$ is a real number? $\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$: $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$) such that $a_{m(C)}>C$.

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio \[ \frac{EG}{GF} \] [b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$

Let $a, b,c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that $$a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3$$

[b]a)[/b] Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number. [b]b)[/b] Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer. [i]Paltin Ionescu[/i]

Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or $\sqrt[5]{13}$? (Use of a calculator is prohibited.)

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $

A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$. After another $15$ minutes it has temperature $77^\circ$. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)

There is a polynomial $P$ such that for every real number $x$, \[ x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x). \] When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have: $$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$

Solve the equation $|x+1|-|x-1|=2$.

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$