The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: $ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $

Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$. [i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.

Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

The real numbers $a_1, a_2, a_3$ are greater than $1$ and have the sum equal to $S$. If for any $i = 1, 2, 3$, holds the inequality $\frac{a_i^2}{a_i-1}>S$ , prove the inequality $$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$

Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

Prove that if $u, v$ are any nonnegative real numbers, and $a,b$ positive real numbers such that $a + b = 1$, then $$u^a v^b \le au + bv.$$

A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$ Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that $P(x,y)=G(x,y)\cdot H(x,y).$

Let $a_1,a_2,...,a_{1994}$ be real numbers in the interval $[-1,1]$, $$S=\frac{a_1+a_2+...+a_{1994}}{1994}.$$ Prove that for an arbitrary natural , $1\le n \le 1994$, holds the inequality $$| a_1+a_2+...+a_n - nS | \le 997.$$

Find all pairs of positive integers \( (a, b) \) such that \( f(x) = x \) is the only function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, \( f^n(x) \) represents the function obtained by applying \( f \) \( n \) times to \( x \). That is, \( f^1(x) = f(x) \) and \( f^{n+1}(x) = f(f^n(x))\) for all \(n \geq 1\).

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

Prove that if the sum of positive numbers $a, b, c$ is equal to $1$, then $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.$

Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression \[\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)\] does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ + & & & d & e \\ \hline & f & a & g & c \\ x & b & b & h & \\ \hline f & f & e & g & c \\ \end{tabular}$ [b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m? [b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse? [img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$ a) Prove that for all $x$ holds $f(x) \le 2x$. b) Is the inequality $f(x) \le 1.9x$ valid?