It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?

Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.

If $a=b-c, b=c-d, c=d-a$ and $abcd\ne 0$, then what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$?

Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$

[b]p1.[/b] Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls $1$, $2$, $3$, $4$, $5$, and then $6$ in that order is $p$. The probability that he rolls $2$, $2$, $4$, $4$, $6$, and then $6$ in that order is $q$. What is $p - q$? [b]p2.[/b] What is the smallest positive integer $x$ such that $x \equiv 2017$ (mod $2016$) and $x \equiv 2016$ (mod $2017$) ? [b]p3.[/b] The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^o$. If $|AB| = 10$, then what is the distance between $O$ and the line $AB$? [b]p4.[/b] A “word“ is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$? [b]p5.[/b] How many distinct complex roots does the polynomial $p(x) = x^{12} - x^8 - x^4 + 1$ have? [b]p6.[/b] Notice that $1 = \frac12 + \frac13 + \frac16$ , that is, $1$ can be expressed as the sum of the three fractions $\frac12 $, $\frac13$ , and $\frac16$ , where each fraction is in the form $\frac{1}{n}$, with each $n$ different. Give a $6$-tuple of distinct positive integers $(a, b, c, d, e, f)$ where $a < b < c < d < e < f$ such that $\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1$ and explain how you arrived at your $6$-tuple. Multiple answers will be accepted. [b]p7.[/b] You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard $6$-sided dice. Let $S$ be the set of squares on the board such that if you are initially on a square in $S$, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in $S$ as your starting position. What is the probability that you land on Go? [b]p8.[/b] Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps? [b]p9.[/b] Does there exist a pair of positive integers $(x, y)$, where $x < y$, such that $x^2 + y^2 = 1009^3$? If so, give a pair $(x, y)$ and explain how you found that pair. If not, explain why. [b]p10.[/b] Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_As_Bs_C$. [b]p11.[/b] Julia and Dan want to divide up $256$ dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until $4$ rejections have been made; once $4$ rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round? [b]p12.[/b] A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of $3$ are $\{1, 1, 1\}$ and $\{1, 2\}$. $\{1, 1, 3, 4\}$ is NOT a perfect partition of $9$ because the sum $4$ can be achieved in two different ways: $4$ and $1 + 3$. How many integers $1 \le N \le 40$ each have exactly one perfect partition? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation \[ a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2 \, ? \]

Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy \[ f(m + f(n)) = f(m) + n \] for all $m,n \in \mathbb{Z}$.

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic. Proposed by Navid Safaei

Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$: $$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$ [i]Proposed by Sina Saleh[/i]

Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number). (a) What is the smallest and biggest representable number? (b) Construct 45 representable numbers.

Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$

Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$. Determine $f(2014)$.

Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.

A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. [img]http://i14.tinypic.com/4tk6mnr.png[/img] a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.