For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$. Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$. When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$

In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear. [i]Ray Li.[/i]

Let $N \ge 3$ be a positive integer. For every pair $(a,b)$ of integers with $1 \le a <b \le N$ consider the quotient $q = b/a$. Show that the pairs with $q < 2$ are equally numbered as those with $q > 2$.

The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

Evaluate \[\sum_{k=1}^{2007}(-1)^{k}k^{2}\]

There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square? [asy] size(130); defaultpen(fontsize(11)); int i, j; for(i=0; i<9; i=i+1) { for(j=0; j<9; j=j+1) if((i==4) && (j==4)) { dot((i,j),linewidth(5)); } else { dot((i,j),linewidth(3)); } } dot("$P$",(4,4),NE); draw((0,0)--(0,8)--(8,8)--(8,0)--cycle); [/asy] $\textbf{(A) } \frac{1}{5} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{2}{5} \qquad\textbf{(D) } \frac{9}{20} \qquad\textbf{(E) } \frac{1}{2}$

Prove that the inequality $$\frac{4}{a+bc+4}+\frac{4}{b+ca+4}+\frac{4}{c+ab+4}\le 1+\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}$$ holds for all positive reals $a,b,c$ such that $a^2+b^2+c^2+abc=4$.

Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers. [i]Proposed by Bogdan Rublov[/i]

Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.

Two sets $M=\{(x,y)|x^2+2y^2=3\},N=\{(x,y)|y=mx+b\}$. For all $m\in\mathbb{R}$, $M\cap N\neq\varnothing$, then the range value of $b$ is $\text{(A)}\left[-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right]\qquad\text{(B)}\left(-\frac{\sqrt6}{2},\frac{\sqrt6}{2}\right)\qquad\text{(C)}\left(-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]\qquad\text{(D)}\left[-\frac{2\sqrt3}{3},\frac{2\sqrt3}{3}\right]$

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Let $ABC$ be a triangle with area $9$, and let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let $P$ be the point in side $BC$ such that $PC = \frac{1}{3}BC$. Let $O$ be the intersection point between $PN$ and $CM$. Find the area of the quadrilateral $BPOM$.

Each one of $2001$ children chooses a positive integer and writes down his number and names of some of other $2000$ children to his notebook. Let $A_c$ be the sum of the numbers chosen by the children who appeared in the notebook of the child $c$. Let $B_c$ be the sum of the numbers chosen by the children who wrote the name of the child $c$ into their notebooks. The number $N_c = A_c - B_c$ is assigned to the child $c$. Determine whether all of the numbers assigned to the children could be positive.

The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values ​​of $S$ for which this is possible.

Let $ABC$ be a triangle such that $AB = 26, AC = 30,$ and $BC = 28$. Let $C'$ and $B'$ be the reflections of the circumcenter $O$ over $AB$ and $AC$, respectively. The length of the portion of line segment $B'C'$ inside triangle $ABC$ can be written as $\frac{p}{q}$, where $p,q$ are relatively prime positive integers. Compute $p+q$.

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[m \vert n \Longleftrightarrow f(m) \vert f(n).\]

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

An ant starts at the point $(1,1)$. It can travel along the integer lattice, only moving in the positive $x$ and $y$ directions. What is the number of ways it can reach $(5,5)$ without passing through $(3,3)$?

Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow: $x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$ $y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$ Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.

Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.

Let $F$ be a point inside a convex pentagon $ABCDE$, and let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ denote the distances from $F$ to the lines $AB$, $BC$, $CD$, $DE$, $EA$, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ are chosen on the inner bisectors of the angles $A$, $B$, $C$, $D$, $E$ of the pentagon respectively, so that $AF_{1} = AF$ , $BF_{2} = BF$ , $CF_{3} = CF$ , $DF_{4} = DF$ and $EF_{5} = EF$ . If the distances from $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ to the lines $EA$, $AB$, $BC$, $CD$, $DE$ are $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$, $b_{5}$, respectively. Prove that $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}$

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that \[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \] holds for all $ x, y \in \mathbb{R}$.

Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$. (U.S.A. - 1989 IMO Shortlist)

Let $Q_n$ be the set of all $n$-tuples $x=(x_1,\ldots,x_n)$ with $x_i \in \{0,1,2 \}$, $i=1,2,\ldots,n$. A triple $(x,y,z)$ (where $x=(x_1,x_2,\ldots,x_n)$, $y=(y_1,y_2,\ldots,y_n)$, $z=(z_1,z_2,\ldots,z_n)$) of distinct elements of $Q_n$ is called a [i]good[/i] triple, if there exists at least one $i \in \{1,2, \ldots, n \}$, for which $\{x_i,y_i,z_i \}=\{0,1,2 \}$. A subset $A$ of $Q_n$ will be called a [i]good[/i] subset, if any three elements of $A$ form a [i]good[/i] triple. Prove that every [i]good[/i] subset of $Q_n$ contains at most $2 \cdot \left(\frac{3}{2}\right)^n$ elements.