Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that: $f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$ by Greece, Athanasios Kontogeorgis (aka socrates)

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$

Show that the equation, $2x^4+2x^2y^2+y^4=z^2$, does not have integer solution when $x \neq 0$.

Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ? Justify your answer.

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$. [b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$. [b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$? [b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive? [b]D.5[/b] Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now? [u]Set 2[/u] [b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish. [b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question? [b]D.8 / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder? [b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have? [b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make? [u]Set 3[/u] [b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs? [b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$. [b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$? [b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$. [b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the first two values of $40!(\text{mod }1763)$ [hide=Answer]1311, 3074[\hide]

For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

The following is known about the reals $ \alpha$ and $ \beta$ $ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$ Determine $ \alpha+\beta$

Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$? [b]p2.[/b] Let $A$ be the number of positive integer factors of $128$. Let $B$ be the sum of the distinct prime factors of $135$. Let $C$ be the units’ digit of $381$. Let $D$ be the number of zeroes at the end of $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$. Let $E$ be the largest prime factor of $999$. Compute $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$. [b]p3. [/b] The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of $17$ and $7$. [b]p4.[/b] A regular hexagon $ABCDEF$ has area $1$. The sides$ AB$, $CD$, and $EF$ are extended to form a larger polygon with $ABCDEF$ in the interior. Find the area of this larger polygon. [b]p5.[/b] For real numbers $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor 5.2 \rfloor = 5$. Evaluate $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$. [b]p6.[/b] The mean of five positive integers is $7$, the median is $8$, and the unique mode is $9$. How many possible sets of integers could this describe? [b]p7.[/b] How many three digit numbers x are there such that $x + 1$ is divisible by $11$? [b]p8.[/b] Rectangle $ABCD$ is such that $AD = 10$ and $AB > 10$. Semicircles are drawn with diameters $AD$ and $BC$ such that the semicircles lie completely inside rectangle $ABCD$. If the area of the region inside $ABCD$ but outside both semicircles is $100$, determine the shortest possible distance between a point $X$ on semicircle $AD$ and $Y$ on semicircle $BC$. [b]p9.[/b] $ 8$ distinct points are in the plane such that five of them lie on a line $\ell$, and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on $\ell$. How many triangles can be formed using some three of the $ 8$ points? [b]p10.[/b] Carl has $10$ Art of Problem Solving books, all exactly the same size, but only $9$ spaces in his bookshelf. At the beginning, there are $9$ books in his bookshelf, ordered in the following way. $A - B - C - D - E - F - G - H - I$ He is holding the tenth book, $J$, in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out $A$, put $J$ in its place, then take out $D$, put $A$ in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is $G$. At the end, he is holding G in his hand, and his bookshelf looks like this. $C - I - H - J - F - B - E - D - A$ Give the order (start to finish) in which Carl took out the books, expressed as a $9$-letter string (word). PS. You had better use hide for answers.

Find all real numbers $x$ such that $$x^3 = \{(x + 1)^3\}$$ where $\{y\}$ denotes the fractional part of $y$, i.e. the difference between $y$ and the largest integer less than or equal to $y$.

Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.

A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.[list][*]$a_1 = 1$ [*]$a_{3n+1} = 2a_n + 1$ [*]$a_{n+1}\ge a_n$ [*]$a_{2001} = 200$[/list]Find the value of $a_{1000}$. [i]Note[/i]. In the original statement of the problem, there was an extra condition:[list][*]every positive integer appears at least once in the sequence.[/list]However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.

Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$

Let $a,b,c$ be positive numbers such that $a+b+c \geq \dfrac 1a + \dfrac 1b + \dfrac 1c$. Prove that \[ a+b+c \geq \frac 3{abc}. \]

Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.

For $n ,p \in N^*$ , $ 1 \le p \le n$, we define $$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$ Show that: $R_n^{n-p+1} =R_n^p$ .