For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that $$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$

[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends? [b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots? [b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$. [b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$. [b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions). [b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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 $100$.

Consider the sets $A = \{0,1,2\},$ and $B = \{1,2,3,4,5\}.$ Find the number of functions $f: A \to B$ such that $x + f(x) + xf(x)$ is odd for all $x.$ (A function $f:A \to B$ is a rule that assigns to every number in $A$ a number in $B.$) \[\mathrm a. ~15\qquad \mathrm b. ~27 \qquad \mathrm c. ~30 \qquad\mathrm d. ~42\qquad\mathrm e. ~45\]

Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and \[f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.\]

Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.

For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$

Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ [i]Mihai Opincariu[/i]

Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$.

Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}$

Determine, if they exist, the real values of $x$ and $y$ that satisfy that $$\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0$$ such that $x + y <0.$

Find all functions $f:\mathbb N\to\mathbb N$ so that $$f^{2f(b)}(2a)=f(f(a+b))+a+b$$ holds for all positive integers $a,b$.

Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that: a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$ b) $k({2^s}) = {3.2^{s - 1}}.$

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

Given a positive integer $n$, define $f_n(x)$ to be the number of square-free positive integers $k$ such that $kx \le n$. Then, define $v_(n)$ as $$v(n) =\sum^n_{i=1}\sum^n_{j=1}f_n(i^2)- 6f_n (ij) + f_n(j^2).$$ Compute the largest positive integer $2 \le n \le 100$ for which $v(n)-v(n-1)$ is negative. (Note: A square-free positive integer is a positive integer that is not divisible by the square of any prime.)

Let $f:\mathbb N\to (0,\infty)$ satisfy $\prod_{d\mid n} f(d) = 1$ for every $n$ which is not prime. Determine the maximum possible number of $n$ with $1\le n \le 100$ and $f(n)\ne 1$.

A man was born on April 1st, [b]20[/b] BCE and died on April 1st, [b]24[/b] CE. How many years did he live? Clarification: Forget about the time he's born or died, assume he is born and died at the exact precise same time on each day

Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]