Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$. Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$, where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$.

Consider the following table where initially all squares contain zeros: $ \begin{tabular}{ | l | c | r| } \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular} $ To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations $ \begin{tabular}{ | l | c | r| } \hline 14 & & \\ \hline 19 & 36 & \\ \hline & 16 & \\ \hline \end{tabular} $

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

Let $a$ and $b$ be real numbers such that $a + b = 2$. Show that: $$\min \{|a|,|b|\} < 1 < \max \{|a|,|b|\} \Leftrightarrow a, b \in (-3,1)$$

How many finite sequances $x_1,x_2,\cdots,x_m$ are there such that $x_i=1$ or 2 and $\sum \limits_{i=1}^mx_i=10$ ? [list=1] [*] 89 [*] 73 [*] 107 [*] 119 [/list]

For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. [b](Condition)[/b] For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

For any real number $ x$ prove that: \[ x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]

Let $M$ be the set of all functions $f$ with the following properties: [b](i)[/b] $f$ is defined for all real numbers and takes only real values. [b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$ [b](iii)[/b] $f(0) \neq 0.$ Determine all functions $f \in M$ such that [b](a)[/b] $f(1)=\frac 52$, [b](b)[/b] $f(1)= \sqrt 3$.

Given positive integers $n$, $P_1$, $P_2$, …$P_n$ and two sets \[B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}\] A function $f:S \to \mathbb{Z}$ is called [b]Real[/b], if and only if for any positive integers $(y_1,y_2,…,y_n)$ and positive integer $a$ which satisfied $ 1 \leq y_i \leq P_i-a$ $\forall i \in \mathbb{N}$, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*} Find the minimum of $\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|$, where $f$ is a [b]Real[/b] function. [i]Proposed by tob8y[/i]

Let $x_0, a, b$ be reals given such that $b > 0$ and $x_0 \ne 0$. For every nonnegative integer $n$ a real value $x_{n+1}$ is chosen that satisfies $$x^2_{n+1}= ax_nx_{n+1} + bx^2_n .$$ a) Find how many different values $x_n$ can take. b) Find the sum of all possible values of $x_n$ with repetitions as a function of $n, x_0, a, b$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)+1)=y+f(f(x)f(y))$$ for all $x, y \in \mathbb{R}$. (Theresia Eisenkölbl)

Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

Find all triples of positive integers $(a,b,c)$ such that the following equations are both true: I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$

The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.

Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race. Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.

$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$

Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that: [b](i)[/b] $a_k < b_k,$ [b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$ prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit.

[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy a) $3$ footballs and $2$ tennis balls? b) $2$ footballs and $3$ tennis balls? [b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle). a) Prove that $AP \cdot PB = CP\cdot PD$. b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle. [b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$. [b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one. a) Find a formula expressing $x_n$ as a function of$ n$. b) Prove your result. [b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$. a) Prove that $AM \le \frac12 AB + \frac12 AC$. b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$