Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$

Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$ $$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$

$a,b,c$ are positive real numbers which satisfy $5abc>a^3+b^3+c^3$. Prove that $a,b,c$ can form a triangle.

Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)

Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real numbers $x , y$ this equality holds : $$f(yf(x)+f(x)f(y))=xf(y)+f(xy)$$ [i]Proposed by Navid Safaei[/i]

We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number $m$ whose sum of digits is equal to $2023$. b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)

Let $a, b, c, d$ be real numbers satisfying the system of equation $\[(a+b)(c+d)=2 \\ (a+c)(b+d)=3 \\ (a+d)(b+c)=4\]$ Find the minimum value of $a^2+b^2+c^2+d^2$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$, for all real $x$.

Once upon a time, three boys visited a library for the first time. The first decided to visit the library every second day. The second decided to visit the library every third day. The third decided to visit the library every fourth day. The librarian noticed, that the library doesn't work on Wednesdays. The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases. They strictly obeyed these rules. Some Monday later I met them all in that library. What day of week was when they visited a library for the first time?

Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$. (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$. (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

[b]p1.[/b] In this problem, a half deck of cards consists of $26$ cards, each labeled with an integer from $1$ to $13$. There are two cards labeled $1$, two labeled $2$, two labeled $3$, etc. A certain math class has $13$ students. Each day, the teacher thoroughly shuffles a half deck of cards and deals out two cards to each student. Each student then adds the two numbers on the cards received, and the resulting $13$ sums are multiplied together to form a product $P$. If $P$ is an even number, the class must do math homework that evening. Show that the class always must do math homework. [b]p2.[/b] Twenty-six people attended a math party: Archimedes, Bernoulli, Cauchy, ..., Yau, and Zeno. During the party, Archimedes shook hands with one person, Bernoulli shook hands with two people, Cauchy shook hands with three people, and similarly up through Yau, who shook hands with $25$ people. How many people did Zeno shake hands with? Justify that your answer is correct and that it is the only correct answer. [b]p3.[/b] Prove that there are no integers $m, n \ge 1$ such that $$\sqrt{m+\sqrt{m+\sqrt{m+...+\sqrt{m}}}}=n$$ where there are $2006$ square root signs. [b]p4.[/b] Let $c$ be a circle inscribed in a triangle ABC. Let $\ell$ be the line tangent to $c$ and parallel to $AC$ (with $\ell \ne AC$). Let $P$ and $Q$ be the intersections of $\ell$ with $AB$ and $BC$, respectively. As $ABC$ runs through all triangles of perimeter $1$, what is the longest that the line segment $PQ$ can be? Justify your answer. [b]p5.[/b] Each positive integer is assigned one of three colors. Show that there exist distinct positive integers $x, y$ such that $x$ and $y$ have the same color and $|x -y|$ is a perfect square. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A polynomial $p(x)$ with integer coefficients satisfies the equality $$p(\sqrt{2}+\sqrt{3})=\sqrt{2}-\sqrt{3}$$ a) Find all possible values of $p(\sqrt{2}-\sqrt{3})$ b) Find an example of any polynomial $p(x)$ which satisfies the condition.

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

Let \[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3. \]Suppose that \begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*} There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.

Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?

Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$

The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.

Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$ Find all $n$ and $ a$ that satisfy the conditions.

The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 256.25\qquad\textbf{(E)}\ -16$