Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.

The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer. [img]6740[/img]

Find all ordered pairs $(a, b)$ of positive integers for which \[\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.\]

Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that \[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\] If \[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\] then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Tristan Shin[/i]

Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.

Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]

Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle?

At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?

Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

Let $A$, $B$, $C$, $D$ be four points in space not all lying on the same plane. The segments $AB$, $BC$, $CD$, and $DA$ are tangent to the same sphere. Prove that their four points of tangency are coplanar.

There are $n$ ($n \ge 10$) cards with numbers $1, 2, \ldots, n$ lying in a row on a table, face down, so that the numbers on any adjacent cards differ by at least $5.$ Is it always enough to turn at most $n-5$ cards to determine which of the cards has number $n$? (It is not necessary to turn the card with number $n$.)

Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$ [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(0.75,1.63),SE*lsf); dot((1,3),ds); label("$A$",(0.96,3.14),NE*lsf); dot((0,0),ds); label("$B$",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("$C$",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("$M$",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("$D$",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

There are $15$ stone placed in a line. In how many ways can you mark $5$ of these stones so that there are on odd number of stones between any two of the stones you marked?

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.

Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

Let $Q$ be a quadratic polynomial with a unique zero. Suppose $Q(12)=Q(16)$ and $Q(20)=24$. Compute $Q(28)$.

Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.

Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.

There are $100$ books in a row, numbered from $1$ to $100$ in some order. An operation is choose three books and reorder in any order between them(the others $97$ books stay at the same place). Denote that a book is in [i]correct position[/i] if the book $i$ is in the position $i$. Determine the least integer $m$ such that, for any initial configuration, we can realize $m$ operations and all the books will be in the correct position.

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.

The equation $$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that i) $K$ crosses each segment exactly once, ii) $K$ does not intersect itself iii) $K$ does not pass through $A, B , C$ or $P.$