A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible given that the line $AB$ and the lake are exterior to each other.

Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$ is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.

Let $x_1$ and $x_2$ be such that $x_1 \neq x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals $ \textbf{(A)}\ -\frac{h}{3} \qquad\textbf{(B)}\ \frac{h}{3} \qquad\textbf{(C)}\ \frac{b}{3} \qquad\textbf{(D)}\ 2b \qquad\textbf{(E)}\ -\frac{b}{3} $

Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. [i]Proposed by Kevin Sun[/i]

In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.

An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

Tony works $ 2$ hours a day and is paid $ \$0.50$ per hour for each full year of his age. During a six month period Tony worked $ 50$ days and earned $ \$630$. How old was Tony at the end of the six month period? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.

Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$, respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$, $BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to $BC$. [i]proposed by USJL[/i]

The equation $x^2+2x=i$ has two complex solutions. Determine the product of their real parts.

In a convex quadrilateral $ABCD$ it is given that $\angle{CAB} = 40^{\circ}, \angle{CAD} = 30^{\circ}, \angle{DBA} = 75^{\circ}$, and $\angle{DBC}=25^{\circ}$. Find $\angle{BDC}$.

It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?

Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?

Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.

Find the amount of different values given by the following expression: $\frac{n^2-2}{n^2-n+2}$ where $ n \in \{1,2,3,..,100\}$

Under the new AMC 10, 12 scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum? $\textbf{(A) }175\qquad\textbf{(B) }179.5\qquad\textbf{(C) }182\qquad\textbf{(D) }188.5\qquad\textbf{(E) }201$

Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Consider the grid of points $X = \{(m,n) | 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $( c = a \pm 2$ and $d = b \pm 1)$ or $( c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:

If $f(x) = x^2 + x$, prove that the equation $4f(a) = f(b)$ has no solutions in positive integers $a$ and $b$.

Assume that, for a certain school, it is true that [list]I: Some students are not honest II: All fraternity members are honest[/list] A necessary conclusion is: $\textbf{(A)}\ \text{Some students are fraternity members} \qquad\\ \textbf{(B)}\ \text{Some fraternity members are not students} \qquad\\ \textbf{(C)}\ \text{Some students are not fraternity members} \qquad\\ \textbf{(D)}\ \text{No fraternity member is a student} \qquad\\ \textbf{(E)}\ \text{No student is a fraternity member} $

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

Given a triangle $ABC$ , a circle centered at some point $O$ meets the segments $BC$ , $CA$ , $AB$ in the pairs of points $X$ and $X^{'}$ , $Y$ and $Y^{'}$ , $Z$ and $Z^{'}$ , respectively ,labelled in circular order : $X,X^{'},Y,Y^{'},Z,Z^{'}$. Let $M$ be the Miquel point of the triangle $XYZ$ and let $M^{'}$ be that of the triangle $X^{'}Y^{'}Z^{'}$ . Prove that the segments $OM$ and $OM^{'}$ have equal lehgths.

In a board with $3$ rows and $555$ columns, $3$ squares are colored red, one in each of the $3$ rows. If the numbers from $1$ to $1665$ are written in the boxes, in row order, from left to right (in the first row from $1$ to $555$, in the second from $556$ to $1110$ and in the third from $1111$ to $1665$) there are $3$ numbers that are written in red squares. If they are written in the boxes, ordered by columns, from top to bottom, the numbers from $1$ to $1665$ (in the first column from $1$ to $3$, in the second from $4$ to $6$, in the third from $7$ to $9$,... ., and in the last one from $1663$ to $1665$) there are $3$ numbers that are written in red boxes. We call [i]red[/i] numbers those that in one of the two distributions are written in red boxes. Indicate which are the $3$ squares that must be colored red so that there are only $3$ red numbers. Show all the possibilities.