Let $a,b,c>0$ and $a+b+c+abc=4$. Prove that $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).$$ [i](Zhuge Liang)[/i]

Find all functions $f : Z \to Z$ satisfying the following two conditions: (i) for all integers $x$ we have $f(f(x)) = x$, (ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.

Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$ [i]Proposed by Mihai Baluna, Romania[/i]

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test? [b]p2.[/b] You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either $20$ heads, or $16$ tails. What is the maximum number of times that you could have flipped the coin? [b]p3.[/b] The width of a rectangle is half of its length. Its area is $98$ square meters. What is the length of the rectangle, in meters? [b]p4.[/b] Carol is twice as old as her younger brother, and Carol's mother is $4$ times as old as Carol is. The total age of all three of them is $55$. How old is Carol's mother? [b]p5.[/b] What is the sum of all two-digit multiples of $9$? [b]p6.[/b] The number $2016$ is divisible by its last two digits, meaning that $2016$ is divisible by $16$. What is the smallest integer larger than $2016$ that is also divisible by its last two digits? [b]p7.[/b] Let $Q$ and $R$ both be squares whose perimeters add to $80$. The area of $Q$ to the area of $R$ is in a ratio of $16 : 1$. Find the side length of $Q$. [b]p8.[/b] How many $8$-digit positive integers have the property that the digits are strictly increasing from left to right? For instance, $12356789$ is an example of such a number, while $12337889$ is not. [b]p9.[/b] During a game, Steve Korry attempts $20$ free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with $84\%$ accuracy, assuming he makes them all? [b]p10.[/b] How many dierent ways are there to arrange the letters $MILKTEA$ such that $TEA$ is a contiguous substring? For reference, the term "contiguous substring" means that the letters $TEA$ appear in that order, all next to one another. For example, $MITEALK$ would be such a string, while $TMIELKA$ would not be. [b]p11.[/b] Suppose you roll two fair $20$-sided dice. What is the probability that their sum is divisible by $10$? [b]p12.[/b] Suppose that two of the three sides of an acute triangle have lengths $20$ and $16$, respectively. How many possible integer values are there for the length of the third side? [b]p13.[/b] Suppose that between Beijing and Shanghai, an airplane travels $500$ miles per hour, while a train travels at $300$ miles per hour. You must leave for the airport $2$ hours before your flight, and must leave for the train station $30$ minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities? [b]p14.[/b] How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of $16$? Two triangles are considered distinct if they are not congruent. [b]p15.[/b] John can drive $100$ miles per hour on a paved road and $30$ miles per hour on a gravel road. If it takes John $100$ minutes to drive a road that is $100$ miles long, what fraction of the time does John spend on the paved road? [b]p16.[/b] Alice rolls one pair of $6$-sided dice, and Bob rolls another pair of $6$-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice? [b]p17.[/b] When $20^{16}$ is divided by $16^{20}$ and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included. [b]p18.[/b] Suppose you have a $20 \times 16$ bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than $5$. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a $2\times 2$ bar and wish to break it entirely into $1\times 1$ bars. We can break it once to get two $2\times 1$ bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size $1\times 1$, and we end up using $3$ breaks in total. [b]p19.[/b] A class of $10$ students decides to form two distinguishable committees, each with $3$ students. In how many ways can they do this, if the two committees can have no more than one student in common? [b]p20.[/b] You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from $0$ to $10$ inclusive, and that no pair of vertices can share the same $x$ or $y$ coordinate value (so for example, you could not use both $(1, 2)$ and $(1, 4)$ in your polygon, but $(1, 2)$ and $(2, 1)$ is fine). What is the largest possible area that your polygon can have? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $ t \ne 0$. Compute $|f(0)|$.

Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$ for any complex number $x.$

Given is a sequence $a_1, a_2, \ldots$, such that $a_1=1$ and $a_{n+1}=\frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?

Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations: $$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition $$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$ Prove that $$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$ [i] (Nora Gavrea)[/i]

[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.

Prove that for all $a,b,c \in \mathbb{R}^+_0$ we have \[\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab} \ge \frac2a+\frac2b-\frac2c\] and determine when equality occurs.

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?

Let $n, k, \ell$ be positive integers and $\sigma : \lbrace 1, 2, \ldots, n \rbrace \rightarrow \lbrace 1, 2, \ldots, n \rbrace$ an injection such that $\sigma(x)-x\in \lbrace k, -\ell \rbrace$ for all $x\in \lbrace 1, 2, \ldots, n \rbrace$. Prove that $k+\ell|n$.

Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions: $f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$ Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$

Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^2-2ax+b=0$ has real roots, given that $P(0)\cdot P(1)\cdot P(2)\neq 0$ and $P(0),P(1),P(2)$ form a geometric progression.

$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$, $az+cx=b$, $ay+bx=c$. Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$

If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? $ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$

Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference. [i]Proposed by Giorgi Arabidze, Georgia [/i]

$100$ numbers were written around a circle. The sum of the $100$ numbers is equal to $100$ and the sum of six consecutive numbers is always less than or equal to $6$. The first number is $6$. Find all the numbers.