Prove that for positive real numbers $x$ and $y$ smaller than or equal to $1/2$, \[\frac{(x+y)^2}{xy} \geq \frac{(2-xy)^2}{(1-x)(1-y)}.\]

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?

It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$ $$ 2<a^2<3$$ Find the value of $$a^{12}-\frac{144}{a}$$

Let $\dbinom{n}{k}$ denote the binomial coefficient $\frac{n!}{k!(n-k)!}$ , and $F_m$ be the $m^{th}$ Fibonacci number given by $F_1=F_2=1$ and $F_{m+2}=F_m+F_{m+1}$ for all $m\geq 1$. Show that $\sum \dbinom{n}{k}=F_{m+1}$ for all $m\geq 1$ . Here the above sum is over all pairs of integers $n\geq k\geq 0$ with $n+k=m$ .

$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\] is a rational number. Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]

Let $a,b,c,d$ be the area of four faces of a tetrahedron, satisfying $a+b+c+d=1$. Show that $$\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2}$$ holds for all positive integers $n$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $

There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.

Solve the equation: \[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\] Note: $\lg x=\log_{10}x$. [i]Daniel Jinga [/i]

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$? $ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$ $\textbf{(B)}\ \text{The discriminant is 9}\qquad$ $\textbf{(C)}\ \text{The roots are imaginary}\qquad$ $\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$ $\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $

Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

(a) Show that the equation \[\lfloor x \rfloor (x^2 + 1) = x^3,\] where $\lfloor x \rfloor$ denotes the largest integer not larger than $x$, has exactly one real solution in each interval between consecutive positive integers. (b) Show that none of the positive real solutions of this equation is rational.

Solve the system of equation $$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.