It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}$$ where $c=f(1)-1$ $a)$ Solve given functional equation for $A=\mathbb{N}$ $b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.

Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$

The triangle $ABC$ is sectioned by $AD,BE$ and $CF$ (where $D \in (BC), E \in (CA)$ and $F \in (AB)$) in seven disjoint polygons named [i]regions[/i]. In each one of the nine vertices of these regions we write a digit, such that each nonzero digit appears exactly once. We assign to each side of a region the lowest common multiple of the digits at its ends, and to each region the greatest common divisor of the numbers assigned to its sides. Find the largest possible value of the product of the numbers assigned to the regions.

What number should we put in place of the question mark such that the following statement becomes true? $$11001_? = 54001_{10}$$ A number written in the subscript means which base the number is in.

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the [i]Primle[/i]. For each guess, a digit is highlighted blue if it is in the [i]Primle[/i], but not in the correct place. A digit is highlighted orange if it is in the [i]Primle[/i] and is in the correct place. Finally, a digit is left unhighlighted if it is not in the [i]Primle[/i]. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the [i]Primle[/i]? $$\begin{array}{c} \boxed{1} \,\, \boxed{3} \\[\smallskipamount] \boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7} \end{array}$$ [i]Proposed by Andrew Wu and Jason Wang[/i]

Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$.

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers. (L Borisov)

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is $\textbf{(A) }(4+3\sqrt{2}):4\qquad\textbf{(B) }9\sqrt{2}:2\qquad\textbf{(C) }(16+12\sqrt{2}):1\qquad$ $\textbf{(D) }(2+2\sqrt{2}):1\qquad \textbf{(E) }3+2\sqrt{2}):1$

Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that $$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$ for every positive integers $m,n$. Determine the minimum possible value of $$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$ across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.

Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide. by Evangelos Psychas, Greece

Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix. Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$. [b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams. [b]p3.[/b] Find the sum of $20\%$ of $16$ and $16\%$ of $20$. [b]p4.[/b] How many times does Akmal need to roll a standard six-sided die in order to guarantee that two of the rolled values sum to an even number? [b]p5.[/b] During a period of one month, there are ten days without rain and twenty days without snow. What is the positive difference between the number of rainy days and the number of snowy days? [b]p6.[/b] Joanna has a fully charged phone. After using it for $30$ minutes, she notices that $20$ percent of the battery has been consumed. Assuming a constant battery consumption rate, for how many additional minutes can she use the phone until $20$ percent of the battery remains? [b]p7.[/b] In a square $ABCD$, points $P$, $Q$, $R$, and $S$ are chosen on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that $AP = 2PB$, $BQ = 2QC$, $CR = 2RD$, and $DS = 2SA$. What fraction of square $ABCD$ is contained within square $PQRS$? [b]p8.[/b] The sum of the reciprocals of two not necessarily distinct positive integers is $1$. Compute the sum of these two positive integers. [b]p9.[/b] In a room of government officials, two-thirds of the men are standing and $8$ women are standing. There are twice as many standing men as standing women and twice as many women in total as men in total. Find the total number of government ocials in the room. [b]p10.[/b] A string of lowercase English letters is called pseudo-Japanese if it begins with a consonant and alternates between consonants and vowels. (Here the letter "y" is considered neither a consonant nor vowel.) How many $4$-letter pseudo-Japanese strings are there? [b]p11.[/b] In a wooden box, there are $2$ identical black balls, $2$ identical grey balls, and $1$ white ball. Yuka randomly draws two balls in succession without replacement. What is the probability that the first ball is strictly darker than the second one? [b]p12.[/b] Compute the real number $x$ for which $(x + 1)^2 + (x + 2)^2 + (x + 3)^2 = (x + 4)^2 + (x + 5)^2 + (x + 6)^2$. [b]p13.[/b] Let $ABC$ be an isosceles right triangle with $\angle C = 90^o$ and $AB = 2$. Let $D$, $E$, and $F$ be points outside $ABC$ in the same plane such that the triangles $DBC$, $AEC$, and $ABF$ are isosceles right triangles with hypotenuses $BC$, $AC$, and $AB$, respectively. Find the area of triangle $DEF$. [b]p14.[/b] Salma is thinking of a six-digit positive integer $n$ divisible by $90$. If the sum of the digits of n is divisible by $5$, find $n$. [b]p15.[/b] Kiady ate a total of $100$ bananas over five days. On the ($i + 1$)-th day ($1 \le i \le 4$), he ate i more bananas than he did on the $i$-th day. How many bananas did he eat on the fifth day? [b]p16.[/b] In a unit equilateral triangle $ABC$; points $D$,$E$, and $F$ are chosen on sides $BC$, $CA$, and $AB$, respectively. If lines $DE$, $EF$, and $FD$ are perpendicular to $CA$, $AB$ and $BC$, respectively, compute the area of triangle $DEF$. [b]p17.[/b] Carlos rolls three standard six-sided dice. What is the probability that the product of the three numbers on the top faces has units digit 5? [b]p18.[/b] Find the positive integer $n$ for which $n^{n^n}= 3^{3^{82}}$. [b]p19.[/b] John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have? [b]p20.[/b] An integer $n > 1$ is conglomerate if all positive integers less than n and relatively prime to $n$ are not composite. For example, $3$ is conglomerate since $1$ and $2$ are not composite. Find the sum of all conglomerate integers less than or equal to $200$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which \[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \] for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$. Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$ [i]Proposed by Zoltán Daróczy[/i]

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$. (a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$. (b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.

Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.

Given a triangle $ABC$, in which $\angle B> 90^o$. Perpendicular bisector of the side $AB$ intersects the side $AC$ at the point $M$, and the perpendicular bisector of the side $AC$ intersects the extension of the side $AB$ beyond the vertex $B$ at point $N$. It is known that the segments $MN$ and $BC$ are equal and intersect at right angles. Find the values ​​of all angles of triangle $ABC$.

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.