Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.

The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$. Show that the marked place says $4$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/517b644caf59bbc57701662f21d57465855dc1.png[/img]

A set of $1989$ numbers is given. It is known that the sum of any $10$ of them is positive. Prove that the sum of all these numbers is positive. (Folklore)

[b]p1.[/b] Alex and Sam have a friend Pat, who is younger than they are. Alex, Sam and Pat all share a birthday. When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.) [b]p2.[/b] Let $ABCD$ be a square with side length $2$. The four sides of $ABCD$ are diameters of four semicircles, each of which lies inside the square. The set of all points which lie on or inside two of these semicircles is a four petaled flower. Find (with proof) the area of this flower. [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc724b9f74c3470434c322020997a533986d33.png[/img] [b]p3.[/b] A prime number is called [i]strongly prime[/i] if every integer obtained by permuting its digits is also prime. For example $113$ is strongly prime, since $113$, $131$, and $311$ are all prime numbers. Prove that there is no strongly prime number such that each of the digits $1, 3, 7$, and $9$ appears at least once in its decimal representation. [b]p4.[/b] Suppose $n$ is a positive integer. Let an be the number of permutations of $1, 2, . . . , n$, where $i$ is not in the $i$-th position, for all $i$ with $1 \le i \le n$. For example $a_3 = 2$, where the two permutations that are counted are $231$, and $312$. Let bn be the number of permutations of $1, 2, . . . , n$, where no $i$ is followed by $i + 1$, for all $i$ with $1 \le i \le n - 1$. For example $b_3 = 3$, where the three permutations that are counted are $132$, $213$, and $321$. For every $n \ge 1$, find (with proof) a simple formula for $\frac{a_{n+1}}{b_n}$. Your formula should not involve summations. Use your formula to evaluate $\frac{a_{2020}}{b_{2019}}$. [b]p5.[/b] Let $n \ge 2$ be an integer and $a_1, a_2, ... , a_n$ be positive real numbers such that $a_1 + a_2 +... + a_n = 1$. Prove that $$\sum^n_{k=1}\frac{a_k}{1 + a_{k+1} - a_{k-1}}\ge 1.$$ (Here $a_0 = a_n$ and $a_{n+1} = a_1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that $$ n=S(a)=S(b)=S(a+b) , $$ where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $ [i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i]

Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number \[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\] is a perfect square.

Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\). [i]Proposed by Mykhailo Shtandenko[/i]

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$ Trần Nam Dũng

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

Define the polynomials $P_0, P_1, P_2 \cdots$ by: \[ P_0(x)=x^3+213x^2-67x-2000 \] \[ P_n(x)=P_{n-1}(x-n), n \in N \] Find the coefficient of $x$ in $P_{21}(x)$.

Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$

Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that: a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$, b) $a_1$ and $n$ are relatively prime. Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.

Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials. [i]Marius Cavachi[/i]

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and $$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$ (a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$. (b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$. (c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that: - $f$ is strictly increasing, - there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$, - for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that \[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \] [i]Proposed by Pavle Martinović[/i]

Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.

The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.

Find all solutions $ (x,y)\in \mathbb{R}\times\mathbb R$ of the following system: $ \begin{cases}x^3 \plus{} 3xy^2 \equal{} 49, \\ x^2 \plus{} 8xy \plus{} y^2 \equal{} 8y \plus{} 17x.\end{cases}$

The degree-$6$ polynomial $f$ satisfies $f(7) - f(1) = 1, f(8) - f(2) = 16, f(9) - f(3) = 81, f(10) - f(4) = 256$ and $f(11) - f(5) = 625.$ Compute $f(15) - f(-3).$

Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that \[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]

Given real positive numbers $a, b, c$ and $d$, for which the equalities $a^2 + ab + b^2 = 3c^2$ and $a^3 + a^2b + ab^2 + b^3 = 4d^3$ are fulfilled. Prove that $$a + b + d \le 3c.$$

Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.

Find all functions $f\colon\mathbb R^+\to\mathbb R^+$ such that \[(f(x)+f(y)+f(z))(xf(y)+yf(z)+zf(x))>(f(x)+y)(f(y)+z)(f(z)+x)\] for all $x,y,z\in\mathbb R^+$. [i]Alexander Wang[/i] [size=59](oops)[/size]

Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once. Prove that there exist integers $\ell $ and $m,\ 1<\ell <m$, such that $a_1+a_m=2a_{\ell}$.