In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Prove that there exists a series of concentric circles, satisfying: (1)Exery itengral point is on the concentric circles. (2)On each circle, there is exactly one itengral point.

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?

Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$ [i]Proposed by Ma Zhao Yu

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$. a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$. b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$. b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular. Valentin Vornicu

Let $ABC$ be a triangle and $\ell$ be a line parallel to $BC$ that passes through vertex $A$. Draw two circles congruent to the circle inscribed in triangle $ABC$ and tangent to line $\ell$, $AB$ and $BC$ (see picture). Lines $DE$ and $FG$ intersect at point $P$. Prove that $P$ lies on $BC$ if and only if $P$ is the midpoint of $BC$. (Mykhailo Plotnikov) [img]https://cdn.artofproblemsolving.com/attachments/8/b/2dacf9a6d94a490511a2dc06fbd36f79f25eec.png[/img]

The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$. $(a)$ Prove that the quadrilateral $AICG$ is cyclic. $(b)$ Prove that the points $B,I,G$ are collinear.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let x; y; z be real numbers, satisfying the relations $x \ge 20$ $y \ge 40$ $z \ge 1675$ x + y + z = 2015 Find the greatest value of the product P = $x
y z$

$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$ $(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$ $(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

How many three-digit numbers leave remainder $2$ when divided by $5$ and leave remainder $7$ when divided by $9$? $\mathrm{(A) \ } 20 \qquad \mathrm{(B) \ } 21 \qquad \mathrm {(C) \ } 22 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ } 24$

In a certain population the ratio of the number of women to the number of men is 11 to 10. If the average (arithmetic mean) age of the women is 34 and the average age of the men is 32, then the average age of the population is $ \textbf{(A)}\ 32\frac{9}{10}\qquad\textbf{(B)}\ 32\frac{20}{21}\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 33\frac{1}{21}\qquad\textbf{(E)}\ 33\frac{1}{10} $

Let $r$ be the inradius of triangle $ABC$. Take a point $D$ on side $BC$, and let $r_1$ and $r_2$ be the inradii of triangles $ABD$ and $ACD$. Prove that $r$, $r_1$, and $r_2$ can always be the side lengths of a triangle.

Let $a, b, c, d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c)$. Prove that $ab+cd$ is not prime.

There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.

Point $M$ is the midpoint of the base $AD$ of trapezoid $ABCD$ inscribed in circle $S$. Rays $AB$ and $DC$ intersect at point $P$, and ray $BM$ intersects $S$ at point $K$. The circumscribed circle of triangle $PBK$ intersects line $BC$ at point $L$. Prove that $\angle{LDP} = 90^{\circ}$.