Let $a, b, c$ be positive numbers such that $abc \le 8$. Prove that $$\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1$$

If the value of the infinite sum $$\frac{1}{2^2-1^2}+\frac{1}{4^2-2^2}+\frac{1}{8^2-4^2}+\frac{1}{16^2-8^2}+\dots.$$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\] holds for any real number $ x$.

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$

Calculate \[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]

$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.

Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

p1. It is known that $f$ is a function such that $f(x)+2f\left(\frac{1}{x}\right)=3x$ for every $x\ne 0$. Find the value of $x$ that satisfies $f(x) = f(-x)$. p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point $O$. Point $P$ lies on side $BC$ so that $AP$ is the altitude of triangle ABC. If $\angle ABC + 30^o \le \angle ACB$, prove that $\angle COP + \angle CAB < 90^o$. p3. Find all natural numbers $a, b$, and $c$ that are greater than $1$ and different, and fulfills the property that $abc$ divides evenly $bc + ac + ab + 2$. p4. Let $A, B$, and $ P$ be the nails planted on the board $ABP$ . The length of $AP = a$ units and $BP = b$ units. The board $ABP$ is placed on the paths $x_1x_2$ and $y_1y_2$ so that $A$ only moves freely along path $x_1x_2$ and only moves freely along the path $y_1y_2$ as in following image. Let $x$ be the distance from point $P$ to the path $y_1y_2$ and y is with respect to the path $x_1x_2$ . Show that the equation for the path of the point $P$ is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. [img]https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png[/img] p5. There are three boxes $A, B$, and $C$ each containing $3$ colored white balls and $2$ red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box $A$. 2. Step 2 $\bullet$ If the ball drawn from box $A$ in step 1 is white, then the ball is put into box $B$. Next from box $B$ one ball is drawn, if it is a white ball, then the ball is put into box $C$, whereas if the one drawn is red ball, then the ball is put in box $A$. $\bullet$ If the ball drawn from box $A$ in step 1 is red, then the ball is put into box $C$. Next from box $C$ one ball is taken. If what is drawn is a white ball then the ball is put into box $A$, whereas if the ball drawn is red, the ball is placed in box $B$. 3. Step 3 Take one ball each from squares $A, B$, and $C$. What is the probability that all the balls drawn in step 3 are colored red?

Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.

How many ten-digit whole numbers satisfy the following property: they have only $2$ and $5$ as digits, and there are no consecutive $2$'s in the number (i.e. any two $2$'s are separated by at least one $5$)?

Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$

A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

Set $\mathbb Q_1=\{x\in\mathbb Q\mid x\ge1\}$. Suppose that a function $f:\mathbb Q_1\to\mathbb R$ satisfies the inequality $\left|f(x+y)-f(x)-f(y)\right|<\epsilon$ for all $x,y\in\mathbb Q_1$, where $\epsilon>0$ is given. Prove that there exists a real number $q$ such that $$\left|\frac{f(x)}x-q\right|<2\epsilon\qquad\text{for all }x\in\mathbb Q_1.$$

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

For every positive integer $ n$ determine the highest value $ C(n)$, such that for every $ n$-tuple $ (a_1,a_2,\ldots,a_n)$ of pairwise distinct integers $ (n \plus{} 1)\sum_{j \equal{} 1}^n a_j^2 \minus{} \left(\sum_{j \equal{} 1}^n a_j\right)^2\geq C(n)$

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Author: Alex Zhu[/i]

In the beginning, the number 1 is written on the board 9999 times. We are allowed to perform the following actions: [list] [*] Erase four numbers of the form $x,x,y,y$, and instead write the two numbers $x+y,x-y$. (The order or location of the erased numbers does not matter) [*] Erase the number 0 from the board, if it's there. [/list] Is it possible to reach a state where: [list=a] [*] Only one number remains on the board? [*] At most three numbers remain on the board? [/list]

Rachel the unicorn lives on the numberline at the number $0$. One day, Rachel decides she’d like to travel the world and visit the numbers $1, 2, 3, \ldots, 31$. She starts off at the number $0$, with a list of the numbers she wants to visit: $1, 2, 3, \ldots , 31$. Rachel then picks one of the numbers on her list uniformly at random, crosses it off the list, and travels to that number in a straight line path. She repeats this process until she has crossed off and visited all thirty-one of the numbers from her original list. At the end of her trip, she returns to her home at $0$. What is the expected length of Rachel’s round trip?

How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$? $\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{None}$

Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.

Let $G=(V,E)$ be a tree graph with $n$ vertices, and let $P$ be a set of $n$ points in the plane with no three points collinear. Is it true that for any choice of graph $G$ and set $P$, we can embed $G$ in $P$, i.e., we can find a bijection $f:V\to P$ such that when we draw line segment $[f(x),f(y)]$ for all $(x,y)\in E$, no two such segments intersect each other?