$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$

Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$. Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that $$ p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1. $$ [i](I. Gorodnin)[/i]

Prove that: [b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and [b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.

Prove that there exist integers $a, b, c$ all greater than $2011$ such that \[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\] [Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]

Let $x$, $y$, $z$ be real numbers such that $x+y+z \ne 0$. Find the minimum value of $\frac{|x|+|x+4y|+|y+7z|+2|z|}{|x+y+z|}$

Suppose you are given a fair coin and a sheet of paper with the polynomial $x^m$ written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial $x^r$ (where $r$ is going to change with time - initially it is $m$) written on the paper and replace it with $x^{r-1}$. If tails show up, replace it with $x^{r+1}$. What is the expected value of the polynomial I get after $m$ such tosses? (Note: this is a different concept from the most probable value)

Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$, let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that (i) $g_n(x) > g_n(y)$ if $x > y > 0$. (ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.

Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.

Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$.

[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].

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Let $x \geq y \geq z$ be positive real numbers such that \begin{align*}x^2+y^2+z^2 \geq 2xy+2yz+2zx.\end{align*} What is the minimum value of $\frac{x}{z}$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt 2\qquad\textbf{(C)}\ \sqrt 3\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is $ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $

Let $p$ be an odd prime and $S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}$, where $q = \frac{3p-5}{2}$. We write $\frac{1}{2}-2S_{q}$ in the form $\frac{m}{n}$, where $m$ and $n$ are integers. Prove that $m \equiv n (mod p)$

[u]Round 5[/u] [b]p13.[/b] The expression $\sqrt2 \times \sqrt[3]{3} \times \sqrt[6]{6}$ can be expressed as a single radical in the form $\sqrt[n]{m}$, where $m$ and $n$ are integers, and $n$ is as small as possible. What is the value of $m + n$? [b]p14.[/b] Bertie Bott also produces Bertie Bott’s Every Flavor Pez. Each package contains $6$ peppermint-, $2$ kumquat-, $3$ chili pepper-, and $5$ garlic-flavored candies in a random order. Harold opens a package and slips it into his Dumbledore-shaped Pez dispenser. What is the probability that of the first four candies, at least three are garlic-flavored? [b]p15.[/b] Quadrilateral $ABCD$ with $AB = BC = 1$ and $CD = DA = 2$ is circumscribed around and inscribed in two different circles. What is the area of the region between these circles? [u] Round 6[/u] [b]p16.[/b] Find all values of x that satisfy $\sqrt[3]{x^7} + \sqrt[3]{x^4} = \sqrt[3]{x}$. [b]p17.[/b] An octagon has vertices at $(2, 1)$, $(1, 2)$, $(-1, 2)$, $(-2, 1)$, $(-2, -1)$, $(-1, -2)$, $(1, -2)$, and $(2, -1)$. What is the minimum total area that must be cut off of the octagon so that the remaining figure is a regular octagon? [b]p18.[/b] Ron writes a $4$ digit number with no zeros. He tells Ronny that when he sums up all the two-digit numbers that are made by taking 2 consecutive digits of the number, he gets 99. He also reveals that his number is divisible by 8. What is the smallest possible number Ron could have written? [u]Round 7[/u] [b]p19.[/b] In a certain summer school, 30 kids enjoy geometry, 40 kids enjoy number theory, and 50 kids enjoy algebra. Also, the number of kids who only enjoy geometry is equal to the number of kids who only enjoy number theory and also equal to the number of kids who only enjoy algebra. What is the difference between the maximum and minimum possible numbers of kids who only enjoy geometry and algebra? [b]p20.[/b] A mouse is trying to run from the origin to a piece of cheese, located at $(4, 6)$, by traveling the shortest path possible along the lattice grid. However, on a lattice point within the region $\{0 \le x \le 4, 0 \le y \le 6$, $(x, y) \ne (0, 0),(4, 6)\}$ lies a rock through which the mouse cannot travel. The number of paths from which the mouse can choose depends on where the rock is placed. What is the difference between the maximum possible number of paths and the minimum possible number of paths available to the mouse? [b]p21.[/b] The nine points $(x, y)$ with $x, y \in \{-1, 0, 1\}$ are connected with horizontal and vertical segments to their nearest neighbors. Vikas starts at $(0, 1)$ and must travel to $(1, 0)$, $(0, -1)$, and $(-1, 0)$ in any order before returning to $(0, 1)$. However, he cannot travel to the origin $4$ times. If he wishes to travel the least distance possible throughout his journey, then how many possible paths can he take? [u]Round 8[/u] [b]p22.[/b] Let $g(x) = x^3 - x^2- 5x + 2$. If a, b, and c are the roots of g(x), then find the value of $((a + b)(b + c)(c + a))^3$. [b]p23.[/b] A regular octahedron composed of equilateral triangles of side length $1$ is contained within a larger tetrahedron such that the four faces of the tetrahedron coincide with four of the octahedron’s faces, none of which share an edge. What is the ratio of the volume of the octahedron to the volume of the tetrahedron? [b]p24.[/b] You are the lone soul at the south-west corner of a square within Elysium. Every turn, you have a $\frac13$ chance of remaining at your corner and a $\frac13$ chance of moving to each of the two closest corners. What is the probability that after four turns, you will have visited every corner at least once? PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Will the number $1/1996$ decrease or increase and by how many times if in the decimal notation of this number the first non-zero digit after the decimal point is crossed out?

Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.

Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$. When does the equality occur?

Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set \[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \] has exactly $n+1$ elements.

Prove that if $ a_1 < a_2 < \ldots < a_n $ and $ b_1 < b_2 < \ldots < b_n $, where $ n \geq 2 $, then $$\qquad (a_1 + a_2 + \ldots + a_n)(b_1 + b_2 + \ldots + b_n) < n(a_1b_1 + a_2b_2 + \ldots + a_nb_n).$$

Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$ Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$ [i]D. Zmiaikou[/i]