Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

The [i]minimal polynomial[/i] of a complex number $r$ is the unique polynomial with rational coefficients of minimal degree with leading coefficient $1$ that has $r$ as a root. If $f$ is the minimal polynomial of $\cos\frac\pi7$, what is $f(-1)$?

[b]p1.[/b] Given an equilateral triangle $ABC$ with area $16\sqrt3$, and an interior point $P$ with distances from vertices $|AP| = 4$ and $|BP| = 6$. (a) Find the length of each side. (b) Find the distance from point $P$ to the side $AB$. (c) Find the distance $|PC|$. [b]p2.[/b] Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them $A$ (who tosses first), $B$ (second), $C$ (third, if there is a third), etc. (a) If there are only two players, what is the probability that $A$ (the first) wins? (b) If there are exactly $3$ players, what is the probability that $A$ (the first) wins? (c) If there are exactly $3$ players, what is the probability that $B$ (the second) wins? [b]p3.[/b] A circular castle of radius $r$ is surrounded by a circular moat of width $m$ ($m$ is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat. (a) If the radius $r$ is $140$ feet and there are only $3$ life guards available, what is the minimum possible width of moat they can watch? (b) Find the minimum number of life guards needed as a function of $r$ and $m$. [img]https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png[/img] [b]p4.[/b] (a)Find all linear (first degree or less) polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all linear polynomials $g(x)$. (b) Prove your answer to part (a). (c) Find all polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all polynomials $g(x)$. (d) Prove your answer to part (c). [b]p5.[/b] A non-empty set $B$ of integers has the following two properties: i. each number $x$ in the set can be written as a sum $x = y+ z$ for some $y$ and $z$ in the set $B$. (Warning: $y$ and $z$ may or may not be distinct for a given $x$.) ii. the number $0$ can not be written as a sum $0 = y + z$ for any $y$ and $z$ in the set $B$. (a) Find such a set $B$ with exactly $6$ elements. (b) Find such a set $B$ with exactly $6$ elements, and such that the sum of all the $6$ elements is $1988$. (c) What is the smallest possible size of such a set $B$ ? (d) Prove your answer to part (c). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.

Let $Y$ be the family of all subsets of $X = \{1,2,...,n\}$ ($n > 1$) and let $f : Y \to X$ be an arbitrary mapping. Prove that there exist distinct subsets $A,B$ of $X$ such that $f(A) = f(B) = max A\triangle B$, where $A\triangle B = (A-B)\cup(B-A)$.

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$. For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds. (Walther Janous)

You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that \[(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},\] where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$, where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.

Let $\phi =\frac{1}{2019}$. Define $$g_n =\begin{cases} 0 & \text{ if} \,\,\,\, round (n\phi) = round \,\,\,\, ((n - 1)\phi) \\ 1 & \text{ otherwise} .\end{cases}.$$ where round $(x)$ denotes the round function. Compute the expected value of $g_n$ if $n$ is an integer chosen from interval $[1, 2019^2]$.

Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.

Prove that if the product of positive numbers $a,b$ and $c$ equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$

Find the sum of the elements of the set $$M = \left\{ \frac{n}{2}+\frac{m}{5} \,\, | m, n = 0, 1, 2,..., 100\right\}$$

Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$. (Walther Janous, Austria)

Two identical decks have 36 cards each. One deck is shuffled and put on top of the second. For each card of the top deck, we count the number of cards between it and the corresponding card of the bottom deck. What is the sum of these numbers? Sorry if this has been posted before but I would like to know if I solved it correctly. Thanks!

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) $\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}$

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $

Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$. [b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers. [b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$. [b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area? [img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$ for all $x,y\in\mathbb{R}$.

a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{\minus{}1,0,1\}$. b) Does there exist a multiple of $ x^2\minus{}3x\plus{}1$ such that all of its coefficient are in the set $ \{\minus{}1,0,1\}$

Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements