let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$

A complex quartic polynomial $Q$ is [i]quirky [/i] if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x) = x^4-kx^3-x^2-x-45$ is quirky. Compute the product of these four values of $k$.

Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.

Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by $$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$

Given are natural numbers $ a,b$ and $ n$ such that $ a^2\plus{}2nb^2$ is a complete square. Prove that the number $ a^2\plus{}nb^2$ can be written as a sum of squares of $ 2$ natural numbers.

For a given pair of values $x$ and $y$ satisfying $x = \sin \alpha , y = \sin \beta$ , there can be four different values of $z = \sin( \alpha +\beta )$. (a) Set up a relation between $x, y$ and $z$ not involving trigonometric functions or radicals. (b) Find those pairs of values $(x, y)$ for which $z = \sin (\alpha +\beta)$ takes on fewer than four distinct values.

In three years, Xingyou’s age in years will be twice his current height in feet. If Xingyou’s current age in years is also his current height in feet, what is Xingyou’s age in years right now?

[b]p1.[/b] The three little pigs are learning about fractions. They particularly like the number x = $1/5$, because when they add the denominator to the numerator, add the denominator to the denominator, and form a new fraction, they obtain $6/10$, which equals $3x$ (so each little pig can have his own $x$). The $101$ Dalmatians hear about this and want their own fraction. Your job is to help them. (a) Find a fraction $y$ such that when the denominator is added to the numerator and also added to the denominator, the result is $101y$. (b) Prove that the fraction $y$ (put into lowest terms) in part (a) is the only fraction in lowest terms with this property. [b]p2.[/b] A small kingdom consists of five square miles. The king, who is not very good at math, wants to divide the kingdom among his $9$ sons. He tells each son to mark out a region of $1$ square mile. Prove that there are two sons whose regions overlap by at least $1/9$ square mile. [b]p3.[/b] Let $\pi (n)$ be the number of primes less than or equal to n. Sometimes $n$ is a multiple of $\pi (n)$. It is known that $\pi (4) = 2$ (because of the two primes $2, 3$) and $\pi (64540) = 6454$. Show that there exists an integer $n$, with $4 < n < 64540$, such that $\pi (n) = n/8$. [b]p4.[/b] Two circles of radii $R$ and $r$ are externally tangent at a point $A$. Their common external tangent is tangent to the circles at $B$ and $C$. Calculate the lengths of the sides of triangle $ABC$ in terms of $R$ and $r$. [img]https://cdn.artofproblemsolving.com/attachments/e/a/e5b79cb7c41e712602ec40edc037234468b991.png[/img] [b]p5.[/b] There are $2005$ people at a meeting. At the end of the meeting, each person who has shaken hands with at most $10$ people is given a red T-shirt with the message “I am unfriendly.” Then each person who has shaken hands only with people who received red T-shirts is given a blue T-shirt with the message “All of my friends are unfriendly.” (Some lucky people might get both red and blue T-shirts, for example, those who shook no one’s hand.) Prove that the number of people who received blue T-shirts is less than or equal to the number of people who received red T-shirts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations: $$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$

Prove that for all positive real numbers $a$, $b$, and $c$, \[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \] and determine when equality occurs.

Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

Find \[ \frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771} \]

Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $f(x)$ has three real positive roots and that $f(0) < 0$, prove that $2b^3+ 9a^2 d - 7abc \le 0$.

Let’s consider the real numbers $a, b, c$ satisfying the condition \[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\] Find the minimal value of the expression \[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]

Solve the equation: $$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$

$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation \[f(x)f(y)-af(xy)=x+y\] , for every real numbers $x,y$. Find all possible real values of $a$.

[b]p1.[/b] If $$ \begin{cases} a^2 + b^2 + c^2 = 1000 \\ (a + b + c)^2 = 100 \\ ab + bc = 10 \end{cases}$$ what is $ac$? [b]p2.[/b] If a and b are real numbers such that $a \ne 0$ and the numbers $1$, $a + b$, and $a$ are, in some order, the numbers $0$, $\frac{b}{a}$ , and $b$, what is $b - a$? [b]p3.[/b] Of the first $120$ natural numbers, how many are divisible by at least one of $3$, $4$, $5$, $12$, $15$, $20$, and $60$? [b]p4.[/b] For positive real numbers $a$, let $p_a$ and $q_a$ be the maximum and minimum values, respectively, of $\log_a(x)$ for $a \le x \le 2a$. If $p_a - q_a = \frac12$ , what is $a$? [b]p5.[/b] Let $ABC$ be an acute triangle and let $a$, $b$, and $c$ be the sides opposite the vertices $A$, $B$, and $C$, respectively. If $a = 2b \sin A$, what is the measure of angle $B$? [b]p6.[/b] How many ordered triples $(x, y, z)$ of positive integers satisfy the equation $$x^3 + 2y^3 + 4z^3 = 9?$$ [b]p7.[/b] Joe has invented a robot that travels along the sides of a regular octagon. The robot starts at a vertex of the octagon and every minute chooses one of two directions (clockwise or counterclockwise) with equal probability and moves to the next vertex in that direction. What is the probability that after $8$ minutes the robot is directly opposite the vertex it started from? [b]p8.[/b] Find the nonnegative integer $n$ such that when $$\left(x^2 -\frac{1}{x}\right)^n$$ is completely expanded the constant coefficient is $15$. [b]p9.[/b] For each positive integer $k$, let $$f_k(x) = \frac{kx + 9}{x + 3}.$$ Compute $$f_1 \circ f_2\circ ... \circ f_{13}(2).$$ [b]p10.[/b] Exactly one of the following five integers cannot be written in the form $x^2 + y^2 + 5z^2$, where $x$, $y$, and $z$ are integers. Which one is it? $$2003, 2004, 2005, 2006, 2007$$ [b]p11.[/b] Suppose that two circles $C_1$ and $C_2$ intersect at two distinct points $M$ and $N$. Suppose that $P$ is a point on the line $MN$ that is outside of both $C_1$ and $C_2$. Let $A$ and $B$ be the two distinct points on $C_1$ such that AP and BP are each tangent to $C_1$ and $B$ is inside $C_2$. Similarly, let $D$ and $E$ be the two distinct points on $C_2$ such that $DP$ and $EP$ are each tangent to $C_2$ and $D$ is inside $C_1$. If $AB = \frac{5\sqrt2}{2}$ , $AD = 2$, $BD = 2$, $EB = 1$, and $ED =\sqrt2$, find $AE$. [b]p12.[/b] How many ordered pairs $(x, y)$ of positive integers satisfy the following equation? $$\sqrt{x} +\sqrt{y} =\sqrt{2007}.$$ [b]p13.[/b] The sides $BC$, $CA$, and $CB$ of triangle $ABC$ have midpoints $K$, $L$, and $M$, respectively. If $$AB^2 + BC^2 + CA^2 = 200,$$ what is $AK^2 + BL^2 + CM^2$? [b]p14.[/b] Let $x$ and $y$ be real numbers that satisfy: $$x + \frac{4}{x}= y +\frac{4}{y}=\frac{20}{xy}.$$ Compute the maximum value of $|x - y|$. [b]p15.[/b] $30$ math meet teams receive different scores which are then shuffled around to lend an aura of mystery to the grading. What is the probability that no team receives their own score? Express your answer as a decimal accurate to the nearest hundredth. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\textbf{(A) }3\qquad \textbf{(B) }4\qquad \textbf{(C) }5\qquad \textbf{(D) }6\qquad \textbf{(E) }11$

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.

Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and $$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$ [i]Gabriela Constantinescu[/i]

Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$. I. Gorodnin