Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$ where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers

Let $ a\in\mathbb{R}. $ Prove the following proposition: $$ \left( x,y\in\mathbb{R}\implies x^4+y^4+axy+2\ge 0 \right)\iff |a|\le 4. $$

Solve the following system of equations in positive integers \[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]

Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$. [i]Proposed by Ankan Bhattacharya[/i]

Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$. Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$.

If the equation: $f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$ holds true for all real x but $\pm 1$, find $f(x)$.

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.

Sunaina and Malay play a game on the coordinate plane. Sunaina has two pawns on $(0,0)$ and $(x,0)$, and Malay has a pawn on $(y,w)$, where $x,y,w$ are all positive integers. They take turns alternately, starting with Sunaina. In their turn they can move one of their pawns one step vertically up or down. Sunaina wins if at any point in time all the three pawns are colinear. Find all values of $x,y$ for which Sunaina has a winning strategy irrespective of the value of $w$. Proposed by NV Tejaswi

For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

We call a two-variable polynomial $P(x, y)$ [i]secretly one-variable,[/i] if there exist polynomials $Q(x)$ and $R(x, y)$ such that $\deg(Q) \ge 2$ and $P(x, y) = Q(R(x, y))$ (e.g. $x^2 + 1$ and $x^2y^2 +1$ are [i]secretly one-variable[/i], but $xy + 1$ is not). Prove or disprove the following statement: If $P(x, y)$ is a polynomial such that both $P(x, y)$ and $P(x, y) + 1$ can be written as the product of two non-constant polynomials, then $P$ is [i]secretly one-variable[/i]. [i]Note: All polynomials are assumed to have real coefficients. [/i]

[b]Problem Section #2 a) If $$ax+by=7$$ $$ax^2+by^2=49$$ $$ax^3+by^3=133$$ $$ax^4+by^4=406$$ , find the value of $2014(x+y-xy)-100(a+b).$

Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds: \[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \] [i]Proposed by Oriol Solé[/i]

Let $a$ and $b$ be two positive real numbers such that $a^{2007} = a + 1$ and $b^{4014} = b + 3a$. Determine whether $a > b$ or $b > a$.

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be positive real numbers satisfying $$\frac{a}{b} \left( \frac{a}{b}+ 2 \right) + \frac{b}{a} \left( \frac{b}{a}+ 2 \right)= 2022.$$ Find the positive integer $n$ such that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\sqrt{n}.$$

Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions: a)$f(m_{i})=-1\forall i=1,2,...,n$. b)$f(x)$ is irreducible.

The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.

Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\] [i]Proposed by Luke Robitaille[/i]

Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

Let $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$, and $(x_5, y_5)$ be the vertices of a regular pentagon centered at $(0, 0)$. Compute the product of all positive integers k such that the equality $x_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k$ must hold for all possible choices of the pentagon.