Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let $AC$ be a diameter of a circle $ \omega $ of radius $1$, and let $D$ be a point on $AC$ such that $CD=\frac{1}{5}$. Let $B$ be the point on $\omega$ such that $DB$ is perpendicular to $AC$, and $E$ is the midpoint of $DB$. The line tangent to $\omega$ at $B$ intersects line $CE$ at the point $X$. Compute $AX$.

Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home? [asy]pair J=(0,6), K=origin, L=(10,0), M=(10,6); draw(J--K--L--M--cycle); label("$J$", J, dir((5,3)--J)); label("$K$", K, dir((5,3)--K)); label("$L$", L, dir((5,3)--L)); label("$M$", M, dir((5,3)--M));[/asy] $\textbf{(A)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(15,15)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(B)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw((0,6)--(1,6)--(1,12)--(2,12)--(2,11)--(3,11)--(3,1)--(12,1)--(12,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(C)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2.7,8)--(3,9)^^(11,9)--(14,0)); draw(Arc((4,9), 1, 0, 180)); draw(Arc((10,9), 1, 0, 180)); draw(Arc((7,9), 2, 180,360)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(D)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(2,6)--(7,14)--(10,12)--(14,0)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy] $\textbf{(E)}$ [asy]size(80);defaultpen(linewidth(0.8)); draw((16,0)--origin--(0,16)); draw(origin--(3,6)--(7,6)--(10,12)--(14,12)); label("time", (8,0), S); label(rotate(90)*"distance", (0,8), W); [/asy]

In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.

A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: $\textbf{(A)}\ y+3x-4=0 \qquad \textbf{(B)}\ y+3x+4=0 \qquad \textbf{(C)}\ y-3x-4=0 \qquad\\ \textbf{(D)}\ 3y+x-12=0 \qquad \textbf{(E)}\ 3y-x-12=0 $

Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

Where is AMC 10a No.19? Thanks

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$, [b](a)[/b] $|f(x)| \leq 5/4$, [b](b)[/b] $|g(x)| \leq 2$.

Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]

Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x)=f(2001)$.