Let $\ast$ be the symbol denoting the binary operation on the set $S$ of all non-zero real numbers as follows: For any two numbers $a$ and $b$ of $S$, $a\ast b=2ab$. Then the one of the following statements which is not true, is $\textbf{(A) }\ast\text{ is commutative over }S \qquad\textbf{(B) }\ast\text{ is associative over }S\qquad$ $\textbf{(C) }\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad\textbf{(D) }\text{Every element of }S\text{ has an inverse for }\ast\qquad$ $\textbf{(E) }\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S$

Solve the equation $x^3 - [x] = 3$.

For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is \[ (-1)^{\alpha(i-1, j-1)}. \] Compute the remainder when $\det M$ is divided by $1000$. [i] Proposed by Evan Chen [/i]

Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that $ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

Triangle $ABC$ has incircle $(I)$ and $P,Q$ are two points in the plane of the triangle. Let $QA,QB,QC$ meet $BA,CA,AB$ respectively at $D,E,F$. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined in the same manner. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $ (I)$ at $U$. The points $V,W$ are defined in the same way. Prove that three lines $(AU,BV,CW)$ are concurrent. Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$? [i]Proposed by James Lin[/i]

Let $ p$ be the greatest prime factor of 9991. Then, the sum of the digits of $ p$ is $ \text{(A)}\ 4 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 28$

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\] where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$? [asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy] $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$

Solve in N $a^2 = 2^b3^c + 1$.

The orthocenter of triangle $ABC$ lies on its circumcircle. One of the angles of $ABC$ must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.) $$ \mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.} $$

Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$. Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying \[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \] for all $x,y \in S$. Let $N$ be the product of all possible nonzero values of $f(81)$. Find the remainder when when $N$ is divided by $p$. [i]Proposed by Yang Liu and Ryan Alweiss[/i]

Let $M$ be a set consisting of $n$ points in the plane, satisfying: i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon; ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon. Find the minimum value of $n$. [i]Leng Gangsong[/i]