The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression? $$\square \times \square +\square \times \square -\square \times \square$$

Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.

The following problem is open in the sense that the answer to part (b) is not currently known. Let $n$ be an odd positive integer and let \[ f_n(x,y,z) = x^n + y^n + z^n + (x+y+z)^n. \] $(a)$ Prove that there exist infinitely many values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \] for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. $(b)$ Determine all values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \] for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. (Two integer polynomials are $\emph{congruent modulo 2}$ if every coefficient of their difference is even. A polynomial is $\emph{constant modulo 2}$ if it is congruent to a constant polynomial modulo 2.)

The function $f(x, y)$ is defined for all real numbers $x, y$. It satisfies $f(x,0) = ax$ (where $a$ is a non-zero constant) and if $(c, d)$ and $(h, k)$ are distinct points such that $f(c, d) = f(h, k)$, then $f(x, y)$ is constant on the line through $(c, d)$ and $(h, k)$. Show that for any real $b$, the set of points such that $f(x, y) = b$ is a straight line and that all such lines are parallel. Show that $f(x, y) = ax + by$, for some constant $b$.

$P(x)$ is a fifth degree polynomial. $P(2018)=1$, $P(2019)=2$ $P(2020)=3$, $P(2021)=4$, $P(2022)=5$. $P(2017)=?$

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

Each of the thirty squares in the diagram below contains a number $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to $S$. [img]https://cdn.artofproblemsolving.com/attachments/8/0/3e056ebc252aee9ade1f45fd337cc6a2f84302.png[/img]

Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(0)=0,f(1)=2013$ and \[(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))\] Note: $f^2(x)=(f(x))^2$

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds $$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$ Under what conditions does equality occur?

Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$ is an integer and find its value

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

The real numbers $x_1,\ldots ,x_{2011}$ satisfy \[x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'\] where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$. Prove that $x_1=x_2=\ldots =x_{2011}$ .

Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$

If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.

In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions $ \{(0,0),(0,1),(1,0),(1,1)\}$. Given a finite number of moves can you yield the configuration $ \{(0,0), (1,1), (3,0), (2, \minus{} 1)\}$ ?

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}$

What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation $$ x^3 + ax^2 + bx + c = 0$$ has three distinct real roots forming a geometric progression?

Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.

From a fixed set formed by the first consecutive natural numbers, find the number of subsets having exactly three elements, and these in arithmetic progression.

Let \(f:\mathbb R\to\mathbb R\) be a function such that for all real numbers \(x\neq1\), \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\] Find all possible values of \(f(2023)\). [i]Proposed by Linus Tang[/i]

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that $$f(x^2 + y^2) = g(xy)$$ holds for all $x, y \in \mathbb{R}^{+}$.

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.