Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

Suppose that a professor has $n \ge 4$ students. Let $P$ denote the set of all ordered pairs $(n, k)$ such that the number of ways for the professor to choose one pair of students equals the number of ways for the professor to choose $k > 1$ pairs of students. For each such ordered pair $(n, k) \in P$, consider the sum $n+k=s$. Find the sum of all $s$ over all ordered pairs $(n, k)$ in $P$. [i]If the same value of $s$ appears in multiple distinct elements $(n, k)$ in $P$, count this value multiple times.[/i]

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$. [b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that \[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\] [b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.

Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$ Find the sum of all possible values of $k$

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.

Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.

[u]Round 5[/u] [b]p13.[/b] Two closed disks of radius $\sqrt2$ are drawn centered at the points $(1,0)$ and $(-1, 0)$. Let P be the region belonging to both disks. Two congruent non-intersecting open disks of radius $r$ have all of their points in $P$ . Find the maximum possible value of $r$ . [b]p14.[/b] A rectangle has positive integer side lengths. The sum of the numerical values of its perimeter and area is $2017$. Find the perimeter of the rectangle. [b]p15.[/b] Find all ordered triples of real numbers $(a,b,c)$ which satisfy $$a +b +c = 6$$ $$a \cdot (b +c) = 6$$ $$(a +b) \cdot c = 6$$ [u]Round 6[/u] [b]p16.[/b] A four digit positive integer is called confused if it is written using the digits $2$, $0$, $1$, and $7$ in some order, each exactly one. For example, the numbers $7210$ and $2017$ are confused. Find the sum of all confused numbers. [b]p17.[/b] Suppose $\vartriangle ABC$ is a right triangle with a right angle at $A$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle CAD$. Suppose that $AB = 20$ and $AC = 17$. Compute $AD$. [b]p18.[/b] Let $x$ be a real number. Find the minimum possible positive value of $\frac{|x -20|+|x -17|}{x}$. [u]Round 7[/u] [b]p19.[/b] Find the sum of all real numbers $0 < x < 1$ that satisfy $\{2017x\} = \{x\}$. [b]p20.[/b] Let $a_1,a_2, ,,, ,a_{10}$ be real numbers which sum to $20$ and satisfy $\{a_i\} <0.5$ for $1 \le i\le 10$. Find the sum of all possible values of $\sum_{ 1 \le i <j\le 10} \lfloor a_i +a_j \rfloor .$ Here, $\lfloor x \rfloor$ denotes the greatest integer $x_0$ such that $x_0 \le x$ and $\{x\} =x -\lfloor x \rfloor$. [b]p21.[/b] Compute the remainder when $20^{2017}$ is divided by $17$. [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with a right angle at $B$. Additionally, letM be the midpoint of $AC$. Suppose the circumcircle of $\vartriangle BCM$ intersects segment $AB$ at a point $P \ne B$. If $CP = 20$ and $BP = 17$, compute $AC$. [b]p23.[/b] Two vertices on a cube are called neighbors if they are distinct endpoints of the same edge. On a cube, how many ways can a nonempty subset $S$ of the vertices be chosen such that for any vertex $v \in S$, at least two of the three neighbors of $v$ are also in $S$? Reflections and rotations are considered distinct. [b]p24.[/b] Let $x$ be a real number such that $x +\sqrt[4]{5-x^4}=2$. Find all possible values of $x\sqrt[4]{5-x^4}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satises the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

Determine whether $ \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.

Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$ Do they all have to be equal? [i](Proposed by Oleksii Masalitin)[/i]

Let $P(x)$ be a non-constant polynomial with integer coefficients and let $n$ be a positive integer. The sequence $a_0,a_1,\ldots$ is defined as follows: $a_0=n$ and $a_k=P(a_{k-1})$ for all positive integers $k.$ Assume that for every positive integer $b$ the sequence contains a $b$th power of an integer greater than $1.$ Show that $P(x)$ is linear.

Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \] holds for all $x,y \in \mathbb{R}$.

The sequence $ \{x_n\}$ satisfies $ x_1 \equal{} \frac {1}{2}, x_{n \plus{} 1} \equal{} x_n \plus{} \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.

Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ?

The value of $$\left\lfloor\frac1{3!}+\frac4{4!}+\frac9{5!}+\ldots\right\rfloor$$

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.

An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers. [i]Proposed by Kyle Lee[/i]

Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$