Found problems: 196
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$
A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$.
[list]
[b](1)[/b] Prove that the locus of the incenter of triangle $PAB$ is a segment,
[b](2)[/b] If $\angle APB=\frac{\pi}{3}$, then find the area of triangle $PAB$.[/list]
A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$ \textbf{(A)}\ 4.0\qquad
\textbf{(B)}\ 4.2\qquad
\textbf{(C)}\ 4.5\qquad
\textbf{(D)}\ 5.0\qquad
\textbf{(E)}\ 5.6$
The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is
$ \textbf{(A)}\ -\frac{3}{2}\qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]
Two points $P$ and $Q$ lying on side $BC$ of triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $.
Author:Mehdi E'tesami Fard , Iran
At most how many points with integer coordinates are there over a circle with center of $(\sqrt{20}, \sqrt{10})$ in the $xy$-plane?
$ \textbf{(A)}\ 8
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None}
$
A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.
Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is:
$\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad
\textbf{(B)}\ y = \dfrac{3}{4}x\qquad
\textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$
$
\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad
\textbf{(E)}\ y = \dfrac{3}{4}x + 2$
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.
A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$.
Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.
What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A [i]rational point[/i] is a point both of whose coordinates are rational numbers.)
Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.
[b]Q4.[/b] A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total length of each typr of road is the same, what is the average speed of his journey?
\[(A) \; 2 \text{km/h} \qquad (B) \; 2,5 \text{km/h} ; \qquad (C ) \; 3 \text{km/h} ; \qquad (D) \; 3,5 \text{km/h} ; \qquad (E) \; 4 \text{km/h}.\]
The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.
[asy]
path seg1, seg2;
seg1=(6,0)--(0,3);
seg2=(2,0)--(4,2);
dot((0,0));
dot((1,0));
fill(circle((2,0),0.1),black);
dot((3,0));
dot((4,0));
dot((5,0));
fill(circle((6,0),0.1),black);
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((5,1));
dot((6,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
fill(circle((4,2),0.1),black);
dot((5,2));
dot((6,2));
fill(circle((0,3),0.1),black);
dot((1,3));
dot((2,3));
dot((3,3));
dot((4,3));
dot((5,3));
dot((6,3));
draw(seg1);
draw(seg2);
pair [] x=intersectionpoints(seg1,seg2);
fill(circle(x[0],0.1),black);
label("$A$",(0,3),NW);
label("$B$",(6,0),SE);
label("$C$",(4,2),NE);
label("$D$",(2,0),S);
label("$E$",x[0],N);[/asy]
$\text{(A)}\ \frac{4\sqrt5}{3}\qquad\text{(B)}\ \frac{5\sqrt5}{3}\qquad\text{(C)}\ \frac{12\sqrt5}{7}\qquad\text{(D)}\ 2\sqrt5 \qquad\text{(E)}\ \frac{5\sqrt{65}}{9}$
A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$.
[i]Proposed by Lewis Chen[/i]
Let $f$ be a diff erentiable real-valued function defi ned on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?
All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?
$ \textbf{(A)}\ 14\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 18$
The solution of the equations
\begin{align*}
2x-3y&=7 \\
4x-6y &=20 \\
\end{align*} is:
$ \textbf{(A)}\ x=18, y=12 \qquad
\textbf{(B)}\ x=0, y=0 \qquad
\textbf{(C)}\ \text{There is no solution} \\
\textbf{(D)}\ \text{There are an unlimited number of solutions} \qquad
\textbf{(E)}\ x=8, y=5$
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.)
[asy]unitsize(10mm);
defaultpen(linewidth(1.2pt)+fontsize(10pt));
dotfactor=4;
pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08);
dot (A);
dot (B);
dot (C);
dot (D);
dot (E);
label("$A$",A,S);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,N);
label("$E$",E,W);
guide squiggly(path g, real stepsize, real slope=45)
{
real len = arclength(g);
real step = len / round(len / stepsize);
guide squig;
for (real u = 0; u < len; u += step){
real a = arctime(g, u);
real b = arctime(g, u + step / 2);
pair p = point(g, a);
pair q = point(g, b);
pair np = unit( rotate(slope) * dir(g,a));
pair nq = unit( rotate(0 - slope) * dir(g,b));
squig = squig .. p{np} .. q{nq};
}
squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))};
return squig;
}
pen pp = defaultpen + 2.718;
draw(squiggly(A--B, 4.04, 30), pp);
draw(squiggly(A--D, 7.777, 20), pp);
draw(squiggly(A--E, 5.050, 15), pp);
draw(squiggly(B--C, 5.050, 15), pp);
draw(squiggly(B--D, 4.04, 20), pp);
draw(squiggly(C--D, 2.718, 20), pp);
draw(squiggly(D--E, 2.718, -60), pp);
[/asy]
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.
a) Prove that $B_{n}$ does not depend on location of $P$.
b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$
$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $
Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates.
I tried to prove that any circle with center at $(0,0)$ has at most one point with coordinates $(a-\sqrt{2},b-\sqrt{3})$;$a,b \in \mathbb{Z}$.
So that when we translate the center to $(\sqrt{2},\sqrt{3})$ we have what we wanted to show.
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$