Call a function $\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}$ $\emph{M-rugged}$ if it is unbounded and satisfies the following two conditions: $(1)$ If $f(n)|f(m)$ and $f(n)<f(m)$ then $n|m$. $(2)$ $|f(n+1)-f(n)|\leq M$. a. Find all $1-rugged$ functions. b. Determine if the number of $2-rugged$ functions is smaller than $2019$.

Let $a, b$ and $c$ be the sides of a triangle, that is: $a + b > c$, $b + c > a$ and $c + a > b$. Show that: $$\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32$$

Let $x_1, x_2, ... , x_n$ and $y_1, y_2, ..., y_n$ be positive real numbers so that $$x_1 + x_2 + ...+ x_n \ge x_1y_1 + x_2y_2 + ... + x_ny_n.$$ Show that for any non-negative integer $p$ the following inequality holds $$\frac{x_1}{y_1^p} +\frac{ x_2}{y_2^p} + ...+ \frac{x_n}{y_n^p} \ge x_1 + x_2 + ...+ x_n.$$

There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.

Find all real $a$ such that the equation $3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7$ has a real root. [hide=Remark] This was the statement given at the contest, but there was actually a typo and the intended equation was $3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7$, which is much easier.

Find all functions ${f : (0, +\infty) \rightarrow R}$ satisfying $f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y)$ for all $x, y > 0$. (Austria)

(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold? (b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold?

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 1 Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.

A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

Let $ a_1,a_2,...,a_n$ with arithmetic mean equals zero; what is the value of: $ \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}$ , where $ a_{n+k}=a_k$ ?

Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) [hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]'' I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url] [/hide]

Let $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$ be points on the plane such that $P_0P_1P_2\Delta$ contains the origin $O$. Show that the areas of triangles $P_0OP_1,P_0OP_2,P_1OP_2$ form a geometric sequence in that order if and only if there exists a real number $x$, such that $$ a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0 $$

You start with a number. Every second, you can add or subtract any number of the form $n!$ to your current number to get a new number. In how many ways can you get from $0$ to $100$ in $4$ seconds? ($n!$ is dened as $n\times (n -1)\times(n - 2) ... 2\times1$, so $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, etc.)

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time. b) If nobody had the immunity on the first day, the epidemic influenza will stop some day.

Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$

Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.

Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$

Do there exist polynomials $p(x)$ and $q(x)$ with real coefficients such that $p^3(x)-q^2(x)$ is linear but not constant?

Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$

Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing.

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.