Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.) [i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions: [list] [*] $f$ maps the zero polynomial to itself, [*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and [*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots. [/list] [i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]

Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that $$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$

Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that \[ f'\left(\frac ax\right)=\frac x{f(x)} \] for all $x>0.$

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

[b]p1.[/b] Compute the value of $N$, where $$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$ [b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there? [b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$ [b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$. [b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img] [b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$ Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of $m + n$. [b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$ [b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$. [b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers? [b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that $$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$ Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Consider a cube with side length $2$. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints? [b]p2.[/b] Digits $H$, $M$, and $C$ satisfy the following relations where $\overline{ABC}$ denotes the number whose digits in base $10$ are $A$, $B$, and $C$. $$\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1$$ $$\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1$$ $$\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1$$ Find $\overline{HMC}$. [b]p3.[/b] Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins? [b]p4.[/b] Cyclic quadrilateral $[BLUE]$ has right $\angle E$. Let $R$ be a point not in $[BLUE]$. If $[BLUR] =[BLUE]$, $\angle ELB = 45^o$, and $\overline{EU} = \overline{UR}$, find $\angle RUE$. [b]p5.[/b] There are two tracks in the $x, y$ plane, defined by the equations $$y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}$$ A baton of length $1$ has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out? [b]p6.[/b] For integers $1 \le a \le 2$, $1 \le b \le 10$,$ 1 \le c \le 12$, $1 \le d \le 18$, let $f(a, b, c, d)$ be the unique integer between $0$ and $8150$ inclusive that leaves a remainder of a when divided by $3$, a remainder of $b$ when divided by $11$, a remainder of $c$ when divided by $13$, and a remainder of $d$ when divided by $19$. Compute $$\sum_{a+b+c+d=23}f(a, b, c, d).$$ [b]p7.[/b] Compute $\cos ( \theta)$ if $$\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.$$ [b]p8.[/b] How many solutions does this equation $$\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2$$ have in positive integers $a, b, c$ that are all less than $2019^2$? [b]p9.[/b] Consider a square grid with vertices labeled $1, 2, 3, 4$ clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label $1$, and at any given vertex he jumps to the vertex diagonally across from him with probability $\frac12$ and the vertices adjacent to him each with probability $\frac14$ . After $2019$ jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is $3$ can be written as $2^{-m} -2^{-n}$ for positive integers $m,n$. Find $m + n$. [b]p10.[/b] The base ten numeral system uses digits $0-9$ and each place value corresponds to a power of $10$. For example, $$2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.$$ Let $\phi =\frac{1 +\sqrt5}{2}$. We can define a similar numeral system, base , where we only use digits $0$ and $1$, and each place value corresponds to a power of . For example, $$11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2}$$ Note that base  representations are not unique, because, for example, $100_{\phi} = 11_{\phi}$. Compute the base $\phi$ representation of $7$ with the fewest number of $1$s. [b]p11.[/b] Let $ABC$ be a triangle with $\angle BAC = 60^o$ and with circumradius $1$. Let $G$ be its centroid and $D$ be the foot of the perpendicular from $A$ to $BC$. Suppose $AG =\frac{\sqrt6}{3}$ . Find $AD$. [b]p12.[/b] Let $f(a, b)$ be a function with the following properties for all positive integers $a \ne b$: $$f(1, 2) = f(2, 1)$$ $$f(a, b) + f(b, a) = 0$$ $$f(a + b, b) = f(b, a) + b$$ Compute: $$\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)$$ [b]p13.[/b] You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules: 1. There is a tower of height $2^n$ at the origin. 2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tower and a new tower of height $2^{i-1}$. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall. If one unit of tower height costs $\$9$ and one unit of wall length costs $\$3$ and $n = 1000$, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other. [b]p14.[/b] For $n$ digits, $(a_1, a_2, ..., a_n)$ with $0 \le a_i < n$ for $i = 1, 2,..., n$ and $a_1 \ne 0$ define $(\overline{a_1a_2 ... a_n})_n$ to be the number with digits $a_1$, $a_2$, $...$, $a_n$ written in base $n$. Let $S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}$ be the set of $n$-tuples such that $(\overline{a_1a_2 ... a_n})_n$ is divisible by $n + 1$. Find all $n > 1$ such that $n$ divides $|S_n| + 2019$. [b]p15.[/b] Let $P$ be the set of polynomials with degree $2019$ with leading coefficient $1$ and non-leading coefficients from the set $C = \{-1, 0, 1\}$. For example, the function $f = x^{2019} - x^{42} + 1$ is in $P$, but the functions $f = x^{2020}$, $f = -x^{2019}$, and $f = x^{2019} + 2x^{21}$ are not in $P$. Define a [i]swap [/i]on a polynomial $f$ to be changing a term $ax^n$ to $bx^n$ where $b \in C$ and there are no terms with degree smaller than $n$ with coefficients equal to $a$ or $b$. For example, a swap from $x^{2019} + x^{17} - x^{15} + x^{10}$ to $x^{2019} + x^{17} - x^{15} - x^{10}$ would be valid, but the following swaps would not be valid: $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}$$ $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2$$ $$x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1$$ Let $B$ be the set of polynomials in $P$ where all non-leading terms have the same coefficient. There are $p$ polynomials that can be reached from each element of $B$ in exactly $s$ swaps, and there exist $0$ polynomials that can be reached from each element of $B$ in less than $s$ swaps. Compute $p \cdot s$, expressing your answer as a prime factorization. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a real number $ k\ge 2, $ solve the equation $ \frac{\{x\}[x]}{x} =k. $

Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.

Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.

Let be given a positive integer $n > 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that $$P(x)P(x^2) ... P(x^n) = P\left( x^{\frac{n(n+1)}{2}}\right)$$ for all $x \in R$

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Prove that there is no function $f: \mathbb{Z}\to\mathbb{Z}$ such that $f(f(x))=x+1$, for all $x\in\mathbb{Z}$.

Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.

Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that $$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.

Find all values of $a$ such that absolute value of one of the roots of the equation $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$ is twice of absolute value of the other root.

The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.

Prove that the equation $ z^3\plus{}z^2\plus{}z\minus{}6\equal{}0$ doesn't have roots $ x$ with $ |x|\equal{}2$.

$f:R->R$ such that : $f(1)=1$ and for any $x\in R$ i) $f(x+5)\geq f(x)+5$ ii)$f(x+1)\leq f(x)+1$ If $g(x)=f(x)+1-x$ find g(2016)

Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)\] for all $x\in\mathbb{R}$.

The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.