For what is the smallest natural number $n$ there is a polynomial $P(x)$ with integer coefficients, having $m$ different integer roots, and at the same time the equation $P(x) = n$ has at least one integer solution if: a) $m = 5$, b) $ m = 6$?

Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$ $$f(x + y -xy) = f(x) + f(y) - f(xy).$$

Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

Find the set of all $ a \in \mathbb{R}$ for which there is no infinite sequene $ (x_n)_{n \geq 0} \subset \mathbb{R}$ satisfying $ x_0 \equal{} a,$ and for $ n \equal{} 0,1, \ldots$ we have \[ x_{n\plus{}1} \equal{} \frac{x_n \plus{} \alpha}{\beta x_n \plus{} 1}\] where $ \alpha \beta > 0.$

Let $P(x)$ be a polynomial with integer coefficients satisfying $$(x^2+1)P(x-1)=(x^2-10x+26)P(x)$$ for all real numbers $x.$ Find the sum of all possible values of $P(0)$ between $1$ and $5000,$ inclusive.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\] for all reals $x$, $y$. [i](Proposed by Lim Yun Zhe)[/i]

Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively. Let $ f: Z \rightarrow R$ be a function satisfying: (i) $ f(n) \ge 0$ for all $ n \in Z$ (ii) $ f(mn)\equal{}f(m)f(n)$ for all $ m,n \in Z$ (iii) $ f(m\plus{}n) \le max(f(m),f(n))$ for all $ m,n \in Z$ (a) Prove that $ f(n) \le 1$ for all $ n \in Z$ (b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) \equal{} 1$

Evaluate the following expression: $2013^2 + 2011^2 + … + 5^2 + 3^2 -2012^2 -2010^2-…-4^2-2^2$

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$. Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that \[ c_1 x_1 + \ldots + c_k x_k = 1. \] [i]Proposed by M. Dubashinsky[/i]

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Let $n$ be a positive integer and $P(x)$ be a polynomial with integer coefficients such that $P(1)=1,P(2)=2,\ldots,P(n)=n$. Prove that $P(0)$ is divisible by $2 \cdot 3 \cdot \ldots \cdot n$. [i]A. Voidelevich[/i]

Given is a natural number $n \geq 3$. Solve the system of equations: $\[ \begin{cases} \tan (x_1) + 3 \cot (x_1) &= 2 \tan (x_2) \\ \tan (x_2) + 3 \cot (x_2) &= 2 \tan (x_3) \\ & \dots \\ \tan (x_n) + 3 \cot (x_n) &= 2 \tan (x_1) \\ \end{cases} \]$

Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$

How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $ where $ x,y $ are non-zero integers

p1. A regular $6$-face dice is thrown three times. Calculate the probability that the number of dice points on all three throws is $ 12$? p2. Given two positive real numbers $x$ and $y$ with $xy = 1$. Determine the minimum value of $\frac{1}{x^4}+\frac{1}{4y^4}.$ p3. Known a square network which is continuous and arranged in forming corners as in the following picture. Determine the value of the angle marked with the letter $x$. [img]https://cdn.artofproblemsolving.com/attachments/6/3/aee36501b00c4aaeacd398f184574bd66ac899.png[/img] p4. Find the smallest natural number $n$ such that the sum of the measures of the angles of the $n$-gon, with $n > 6$ is less than $n^2$ degrees. p5. There are a few magic cards. By stating on which card a number is there, without looking at the card at all, someone can precisely guess the number. If the number is on Card $A$ and $B$, then the number in question is $1 + 2$ (sum of corner number top left) cards $A$ and $B$. If the numbers are in $A$, $B$, and $C$, the number what is meant is $1 + 2 + 4$ or equal to $7$ (which is obtained by adding the numbers in the upper left corner of each card $A$,$B$, and $C$). [img]https://cdn.artofproblemsolving.com/attachments/e/5/9e80d4f3bba36a999c819c28c417792fbeff18.png[/img] a. How can this be explained? b. Suppose we are going to make cards containing numbers from $1$ to with $15$ based on the rules above. Try making the cards. [hide=original wording for p5, as the wording isn't that clear]Ada suatu kartu ajaib. Dengan menyebutkan di kartu yang mana suatu bilan gan berada, tanpa melihat kartu sama sekali, seseorang dengan tepat bisa menebak bilangan yang dimaksud. Kalau bilangan tersebut ada pada Kartu A dan B, maka bilangan yang dimaksud adalah 1 + 2 (jumlah bilangan pojok kiri atas) kartu A dan B. Kalau bilangan tersebut ada di A, B, dan C, bilangan yang dimaksud adalah 1 + 2 + 4 atau sama dengan 7 (yang diperoleh dengan menambahkan bilangan-bilangan di pojok kiri atas masing-masing kartu A, B, dan C) a. Bagaimana hal ini bisa dijelaskan? b. Andai kita akan membuat kartu-kartu yang memuat bilangan dari 1 sampai dengan 15 berdasarkan aturan di atas. Coba buatkan kartu-kartunya[/hide]

Let $n$ be a given positive integer. Solve the system \[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\] \[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\] in the set of nonnegative real numbers.

Is it possible to put pairwise distinct positive integers less than $100$ in the cells of a $4 \times 4$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow))

Let $a$ be the sum and $b$ the product of the real roots of the equation $x^4-x^3-1=0$ Prove that $b < -\frac{11}{10}$ and $a > \frac{6}{11}$.

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

Let $N \ge 1$ be a positive integer and $k$ be an integer such that $1 \le k \le N$. Define the recurrence $x_n = \frac{x_{n-1} + x_{n-2} +... + x_{n-N}}{N}$ for $n > N$ and $x_k = 1$, $x_1 = x_2 = ... = x_{k-1} =x_{k+1} =.. = x_N = 0$. As $n$ approaches infinity, $x_n$ approaches some value. What is this value?

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

p1. If $x + y + z = 2$, show that $\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}$. p2. Determine the simplest form of $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}$ p3. It is known that $ABCD$ and $DEFG$ are two parallelograms. Point $E$ lies on $AB$ and point $C$ lie on $FG$. The area of $​​ABCD$ is $20$ units. $H$ is the point on $DG$ so that $EH$ is perpendicular to $DG$. If the length of $DG$ is $5$ units, determine the length of $EH$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png[/img] p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If $10$ different colors are provided and $4$ of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the $4$ rooms. [img]https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png[/img] 5. The floor of a hall is rectangular $ABCD$ with $AB = 30$ meters and $BC = 15$ meters. A cat is in position $A$. Seeing the cat, the mouse in the midpoint of $AB$ ran and tried to escape from cat. The mouse runs from its place to point $C$ at a speed of $3$ meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point $A$ the cat is chasing with a speed of $5$ meters/second. If the cat's path is also a straight line and the mouse caught before in $C$, determine an equation that can be used for determine the position and time the mouse was caught by the cat.

Define \[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \] Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals $\textbf{(A) }\frac{1}{256}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }1\qquad\textbf{(D) }-16\qquad\textbf{(E) }256$