There are 2012 lamps arranged on a table. Two persons play the following game. In each move the player flips the switch of one lamp, but he must never get back an arrangement of the lit lamps that has already been on the table. A player who cannot move loses. Which player has a winning strategy?

Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying: \[x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n \equal{} 0\] \[ny^2 \equal{} x_1^2 \plus{} x_2^2 \plus{} \ldots \plus{} x_n^2\]

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.

Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded. Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. [i]Proposed by Carl Schildkraut[/i]

Two circles of radius $ 2$ are centered at $ (2,0)$ and at $ (0,2)$. What is the area of the intersection of the interiors of the two circles? $ \textbf{(A)}\ \pi \minus{} 2\qquad \textbf{(B)}\ \frac {\pi}{2}\qquad \textbf{(C)}\ \frac {\pi\sqrt {3}}{3}\qquad \textbf{(D)}\ 2(\pi \minus{} 2)\qquad \textbf{(E)}\ \pi$

For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.

$ \lim_{n\to\infty } n\left( \sqrt[3]{n^3-6n^2+6n+1}-\sqrt{n^2-an+5} \right) $

Let $c$ be a constant. The simultaneous equations \begin{align*} x-y = &\ 2 \\ cx+y = &\ 3 \\ \end{align*} have a solution $(x, y)$ inside Quadrant I if and only if $ \textbf{(A)}\ c=-1 \qquad\textbf{(B)}\ c>-1 \qquad\textbf{(C)}\ c<\frac{3}{2} \qquad\textbf{(D)}\ 0<c<\frac{3}{2}\\ \qquad\textbf{(E)}\ -1<c<\frac{3}{2} $

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that \[\binom{m}{n-1}>\binom{m-1}{n}.\] Evaluate \[\lim_{n\to\infty}\frac{M(n)}{n}.\]

There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.

The expression $ \frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $ x=-1$ or $ x=2$, since division by zero is not allowed. For other values of $ x$: $\textbf{(A)}\ \text{The expression takes on many different values.} \\ \textbf{(B)}\ \text{The expression has only the value 2.} \\ \textbf{(C)}\ \text{The expression has only the value 1.} \\ \textbf{(D)}\ \text{The expression always has a value between } -1 \text{ and } +2. \\ \textbf{(E)}\ \text{The expression has a value greater than 2 or less than } -1.$

$\alpha,\beta$ are two different solutions to the equation $4x^2-4tx+1=0(t\in\mathbb{R})$, the domain of definition of the function $f(x)=\frac{2x-t}{x^2+1}$ is $[\alpha,\beta](\alpha<\beta)$. [b](a)[/b] Find $g(t)=\max f(x)-\min f(x)$. [b](b)[/b] Prove: for $u_i\in\left(0,\frac{\pi}{2}\right)(i=1,2,3)$, if $\sin u_1+\sin u_2+\sin u_3=1$, then $\frac{1}{g(\tan u_1)}+\frac{1}{g(\tan u_2)}+\frac{1}{g(\tan u_3)}<\frac{3}{4}\sqrt6$.

Fix positive integers $n$ and $k$, and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$, $b_1, \cdots, b_n$. An equation is written on a whiteboard: $$t=*\times*\times\cdots\times*$$ where $t$ is a fixed positive real number, with exactly $k$ asterisks. Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$, while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$, so that the equation on the whiteboard is correct. Suppose for every positive real number $t$, the number of ways for Ebi and Rubi to do so are equal. Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other. [i](Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$.)[/i] [i]Proposed by Wong Jer Ren[/i]

Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)

The side length of the largest square below is $8\sqrt{2}$, as shown. Find the area of the shaded region. [asy] size(10cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A=(0.,1.), B=(0.,0.), C=(1.,0.), D=(1.,1.), F=(1.,2.), G=(2.,3.), H=(0.,3.), I=(0.,5.), J=(-2.,3.), K=(-4.,5.), L=(-4.,1.), M=(-8.,1.), O=(-8.,-7.), P=(0.,-7.); draw(B--A--D--C--cycle); draw(A--C--(2.,1.)--F--cycle); draw(A--(2.,1.)--G--H--cycle); draw(A--G--I--J--cycle); draw(A--I--K--L--cycle); draw(A--K--M--(-4.,-3.)--cycle); draw(A--M--O--P--cycle); draw(A--O--(0.,-15.)--(8.,-7.)--cycle); filldraw(A--B--C--D--cycle,opacity(0.2)+black); filldraw(A--(2.,1.)--F--cycle,opacity(0.2)+black); filldraw(A--G--H--cycle,opacity(0.2)+black); filldraw(A--I--J--cycle,opacity(0.2)+black); filldraw(A--K--L--cycle,opacity(0.2)+black); filldraw(A--M--(-4.,-3.)--cycle, opacity(0.2)+black); filldraw(A--O--P--cycle,opacity(0.2)+black); draw(B--A); draw(A--D); draw(D--C); draw(C--B); draw(A--C); draw(C--(2.,1.)); draw((2.,1.)--F); draw(F--A); draw(A--(2.,1.)); draw((2.,1.)--G); draw(G--H); draw(H--A); draw(A--G); draw(G--I); draw(I--J); draw(J--A); draw(A--I); draw(I--K); draw(K--L); draw(L--A); draw(A--K); draw(K--M); draw(M--(-4.,-3.)); draw((-4.,-3.)--A); draw(A--M); draw(M--O); draw(O--P); draw(P--A); draw(A--O); draw(O--(0.,-15.)); draw((0.,-15.)--(8.,-7.)); draw((8.,-7.)--A); draw(A--B,black); draw(B--C,black); draw(C--D,black); draw(D--A,black); draw(A--(2.,1.),black); draw((2.,1.)--F,black); draw(F--A,black); draw(A--G,black); draw(G--H,black); draw(H--A,black); draw(A--I,black); draw(I--J,black); draw(J--A,black); draw(A--K,black); draw(K--L,black); draw(L--A,black); draw(A--M,black); draw(M--(-4.,-3.),black); draw((-4.,-3.)--A,black); draw(A--O,black); draw(O--P,black); draw(P--A,black); label("$8\sqrt{2}$",(-8,-7)--(0,-15)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] [i]Lightning 2.4[/i]

A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral. [i]Proposed byWilliam Hua[/i]

A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below. Let $r(x)$ be the ratio of the area of region I to the area of region II. (i) Find $r(1)$. Justify your answer. (ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img] [b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party. (i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$. (ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ . [b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$. (i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer. (ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer. [b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.) (a) Determine with proof the value of $f(120)$. (b) Determine with proof the smallest value $x$ for which $f(x) = 8$. (c) Show that the range of this function is the set of all positive integers. [b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.