A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.

Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule: (a) $g$ is nondecrasing (b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$, Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$

[b]p1.[/b] Find all positive integers $n$ such that $n^3 - 14n^2 + 64n - 93$ is prime. [b]p2.[/b] Let $a, b, c$ be real numbers such that $0 \le a, b, c \le 1$. Find the maximum value of $$\frac{a}{1 + bc}+\frac{b}{1 + ac}+\frac{c}{1 + ab}$$ [b]p3.[/b] Find the maximum value of the function $f(x, y, z) = 4x + 3y + 2z$ on the ellipsoid $16x^2 + 9y^2 + 4z^2 = 1$ [b]p4.[/b] Let $x_1,..., x_n$ be numbers such that $x_1+...+x_n = 2009$. Find the minimum value of $x^2_1+...+x^2_n$ (in term of $n$). [b]p5.[/b] Find the number of odd integers between $1000$ and $9999$ that have at least 3 distinct digits. [b]p6.[/b] Let $A_1,A_2,...,A_{2^n-1}$ be all the possible nonempty subsets of $\{1, 2, 3,..., n\}$. Find the maximum value of $a_1 + a_2 + ... + a_{2^n-1}$ where $a_i \in A_i$ for each $i = 1, 2,..., 2^n - 1$. [b]p7.[/b] Find the rightmost digit when $41^{2009}$ is written in base $7$. [b]p8.[/b] How many integral ordered triples $(x, y, z)$ satisfy the equation $x+y+z = 2009$, where $10 \le x < 31$, $100 < z < 310$ and $y \ge 0$. [b]p9.[/b] Scooby has a fair six-sided die, labeled $1$ to $6$, and Shaggy has a fair twenty-sided die, labeled $1$ to $20$. During each turn, they both roll their own dice at the same time. They keep rolling the die until one of them rolls a 5. Find the probability that Scooby rolls a $5$ before Shaggy does. [b]p10.[/b] Let $N = 1A323492110877$ where $A$ is a digit in the decimal expansion of $N$. Suppose $N$ is divisible by $7$. Find $A$. [b]p11.[/b] Find all solutions $(x, y)$ of the equation $\tan^4(x+y)+\cot^4(x+y) = 1-2x-x^2$, where $-\frac{\pi}{2} \le x; y \le \frac{\pi}{2}$ [b]p12.[/b] Find the remainder when $\sum^{50}_{k=1}k!(k^2 + k - 1)$ is divided by $1008$. [b]p13.[/b] The devil set of a positive integer $n$, denoted $D(n)$, is defined as follows: (1) For every positive integer $n$, $n \in D(n)$. (2) If $n$ is divisible by $m$ and $m < n$, then for every element $a \in D(m)$, $a^3$ must be in $D(n)$. Furthermore, call a set $S$ scary if for any $a, b \in S$, $a < b$ implies that $b$ is divisible by $a$. What is the least positive integer $n$ such that $D(n)$ is scary and has at least $2009$ elements? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length?

Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$

If $ \sqrt{x^2+2y+1} +\sqrt[3]{y^3+3x^2+3x+1} $ is rational, then $ x=y. $

Find the coefficient of $x^2$ in the following polynomial $$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and: $i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer $ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer $iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer $b)$ Does there exist infinite set $M$ of positive integers such that arithmetic mean of arbitrary non-empty subset of $M$ is integer

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

Non-zero numbers $a$ and $b$ satisfy the equality $$a^2b^2(a^2b^2 + 4) = 2(a^6 + b^6).$$ Prove that at least one of them is irrational.

For positive real $x$, let $$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of $$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$

Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that: \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

Find all functions $f : R_{>0} \to R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers. Nguyễn Duy Thái Sơn

Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality $$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$

A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].