Found problems: 196
$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?
[asy]
draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2));
label("D",(0,1),NW);label("E",(.4,1),N);label("C",(1,1),NE);
label("P",(0,.6),W);label("M",(.25,.55),E);label("Q",(1,.2),E);
label("A",(0,0),SW);label("B",(1,0),SE);
//Credit to Zimbalono for the diagram[/asy]
Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is
$\textbf{(A) }5:12\qquad\textbf{(B) }5:13\qquad\textbf{(C) }5:19\qquad\textbf{(D) }1:4\qquad \textbf{(E) }5:21$
Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$
A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer?
[asy]size(12cm);
pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75);
pair B = (-1,-1);
pair C = (-1,-7);
pair A = (-13,-7);
path inclined_plane = A--B--C--cycle;
draw(inclined_plane, p);
real r = 1; // for marking angles
draw(arc(A, r, 0, degrees(B-A))); // mark angle
label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta
draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle
real h = 1.2; // height of box
real w = 1.9; // width of box
path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box
// box on slope with label
picture box_on_slope;
filldraw(box_on_slope, box, light_grey, black);
label(box_on_slope, "$m$", (w/2,h/2));
pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB
pair T1 = dir(125); // point of tangency with pulley
pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box
draw(T1--X1); // string
add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope);
// picture for the hanging box
picture hanging_box;
filldraw(hanging_box, box, light_grey, black);
label(hanging_box, "$M$", (w/2,h/2));
pair T2 = (1,0);
pair X2 = (1,-3);
draw(T2--X2); // string
add(shift(X2-(w/2,h)) * hanging_box);
// Draws the actual pulley
filldraw(unitcircle, grey, p); // outer boundary of pulley wheel
filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel
path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley
filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm
filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]
Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.
If the normal to the curve $x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23}$ at some point makes an angle $\theta$ with the $X$-axis, show that the equation of the normal is
\[y\cos\theta-x\sin\theta=a\cos 2\theta\]
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$
Take a point $P\left(\frac 12,\ \frac 14\right)$ on the coordinate plane. Let two points $Q(\alpha ,\ \alpha ^ 2),\ R(\beta ,\ \beta ^2)$ move in such a way that 3 points $P,\ Q,\ R$ form an isosceles triangle with the base $QR$, find the locus of the barycenter $G(X,\ Y)$ of $\triangle{PQR}$.
[i]2011 Tokyo University entrance exam[/i]
A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c\plus{}s$?
[asy]unitsize(1cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);
draw((1,1)--(-1,0));
pair P=foot((1,-1),(1,1),(-1,0));
draw((1,-1)--P);
draw(rightanglemark((-1,0),P,(1,-1),4));
label("$M$",(-1,0),W);
label("$C$",(-0.1,-0.3));
label("$A$",(-0.4,0.7));
label("$B$",(0.7,0.4));[/asy]$ \textbf{(A)}\ \frac{24}{5} \qquad
\textbf{(B)}\ \frac{32}{5} \qquad
\textbf{(C)}\ 8\plus{}\sqrt5 \qquad
\textbf{(D)}\ 5\plus{}\frac{16\sqrt5}{5} \qquad
\textbf{(E)}\ 10\plus{}5\sqrt5$
In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.
Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form
\[ 12x^2 + bxy + cy^2 + d = 0. \]
Find the product $bc$.
For the simultaneous equations
\[ 2x\minus{}3y\equal{}8\]
\[ 6y\minus{}4x\equal{}9\]
$\textbf{(A)}\ x=4,y=0 \qquad
\textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad
\textbf{(C)}\ x=0,y=0 \qquad\\
\textbf{(D)}\ \text{There is no solution} \qquad
\textbf{(E)}\ \text{There are an infinite number of solutions}$
Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.
This is the one with the 8 circles?
I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...
The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is
$\text{(A)} \ \frac{\sqrt{2}}{2} \qquad \text{(B)} \ -\frac{\sqrt{2}}{2} \qquad \text{(C)} \ 2 \qquad \text{(D)} \ -2 \qquad \text{(E)} \ \text{not uniquely determined}$
In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is:
$ \textbf{(A)}\ \text{A straight line passing through the origin.} \\
\textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\
\textbf{(C)}\ \text{Two line segments forming an upright V.} \\
\textbf{(D)}\ \text{Two line segments forming an inverted V.} \\
\textbf{(E)}\ \text{None of these.}$
Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\]
$\textbf{(A) }3 + 2 \sqrt 2\qquad
\textbf{(B) } 2 + \sqrt 3\qquad
\textbf{(C ) }3 \sqrt 3\qquad
\textbf{(D) }6\qquad
\textbf{(E) }6 + 2 \sqrt 3$
Let $ S$ be the set of points $ (a,b)$ in the coordinate plane, where each of $ a$ and $ b$ may be $ \minus{} 1$, $ 0$, or $ 1$. How many distinct lines pass through at least two members of $ S$?
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$
Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.
Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.
Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.
Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$.
Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.
A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$.
Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.