Found problems: 970
Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$
[i]Proposed by Andria Gvaramia, Georgia [/i]
A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
Let $\Omega$ be a sphere of radius $4$ and $\Gamma$ be a sphere of radius $2.$ Suppose that the center of $\Gamma$ lies on the surface of $\Omega.$ The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumfrence.
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.
A rectangular tea bag $PART$ has a logo in its interior at the point $Y$ . The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution[/i]. $\boxed{78}$
Using the area and the height in $\triangle PYT$, we see that $PT = 14$, and thus $AR = 14$, meaning the height from $Y$ to $AR$ is $6$. This means $PA = TR = 18$. By the Pythagorean Theorem $PY=\sqrt{12^2+9^2} = 15$ and $YT =\sqrt{12^2 +5^2} = 13$. Combining all of these gives us an answer of $18+14+18+13+15 = \boxed{78}$.
[/hide]
Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
[img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]
In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
[b]p1. [/b] If $f(x) = 3x - 1$, what is $f^6(2) = (f \circ f \circ f \circ f \circ f \circ f)(2)$?
[b]p2.[/b] A frog starts at the origin of the $(x, y)$ plane and wants to go to $(6, 6)$. It can either jump to the right one unit or jump up one unit. How many ways are there for the frog to jump from the origin to $(6, 6)$ without passing through point $(2, 3)$?
[b]p3.[/b] Alfred, Bob, and Carl plan to meet at a café between noon and $2$ pm. Alfred and Bob will arrive at a random time between noon and $2$ pm. They will wait for $20$ minutes or until $2$ pm for all $3$ people to show up after which they will leave. Carl will arrive at the café at noon and leave at $1:30$ pm. What is the probability that all three will meet together?
[b]p4.[/b] Let triangle $ABC$ be isosceles with $AB = AC$. Let $BD$ be the altitude from $ B$ to $AC$, $E$ be the midpoint of $AB$, and $AF$ be the altitude from $ A$ to $BC$. If $AF = 8$ and the area of triangle $ACE$ is $ 8$, find the length of $CD$.
[b]p5.[/b] Find the sum of the unique prime factors of $(2018^2 - 121) \cdot (2018^2 - 9)$.
[b]p6.[/b] Compute the remainder when $3^{102} + 3^{101} + ... + 3^0$ is divided by $101$.
[b]p7.[/b] Take regular heptagon $DUKMATH$ with side length $ 3$. Find the value of $$\frac{1}{DK}+\frac{1}{DM}.$$
[b]p8.[/b] RJ’s favorite number is a positive integer less than $1000$. It has final digit of $3$ when written in base $5$ and final digit $4$ when written in base $6$. How many guesses do you need to be certain that you can guess RJ’s favorite number?
[b]p9.[/b] Let $f(a, b) = \frac{a^2+b^2}{ab-1}$ , where $a$ and $b$ are positive integers, $ab \ne 1$. Let $x$ be the maximum positive integer value of $f$, and let $y$ be the minimum positive integer value of f. What is $x - y$ ?
[b]p10.[/b] Haoyang has a circular cylinder container with height $50$ and radius $5$ that contains $5$ tennis balls, each with outer-radius $5$ and thickness $1$. Since Haoyang is very smart, he figures out that he can fit in more balls if he cuts each of the balls in half, then puts them in the container, so he is ”stacking” the halves. How many balls would he have to cut up to fill up the container?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.
[i]Proposed by David Altizio[/i]
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, as shown, is $\tfrac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
[asy]
size(5cm);
pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);
real t = .385, s = 3.5*t-1;
pair R = A*t+B*(1-t), P=B*s;
pair Q = dir(-60) * (R-P) + P;
fill(P--Q--R--cycle,gray);
draw(A--B--C--A^^P--Q--R--P);
dot(A--B--C--P--Q--R);
[/asy]
Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.
On a blank piece of paper, two points with distance $1$ is given. Prove that one can use (only) straightedge and compass to construct on this paper a straight line, and two points on it whose distance is $\sqrt{2021}$ such that, in the process of constructing it, the total number of circles or straight lines drawn is at most $10.$
Remark: Explicit steps of the construction should be given. Label the circles and straight lines in the order that they appear. Partial credit may be awarded depending on the total number of circles/lines.
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?
[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 3.5;
pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0),
R = intersectionpoint(B--P,C--Q);
draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7));
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
dot("$Q$",Q,S);
dot("$P$",P,W);
dot("$R$",R,1.3*S);
label("$7$",(P+R)/2,NE);
label("$6$",(R+B)/2,NE);
[/asy]
$\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125$
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.
[b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b]
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color.
([i]Edited by Orlando Döhring[/i])
[i]Author: Radu Gologan and Dan Schwarz, Romania[/i]
A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$
[asy]
size(8cm);
draw((0,0)--(10,0));
draw((0,0)--(0,10));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((1,6)--(0,9));
draw((0,9)--(3,10));
draw((3,10)--(4,7));
draw((4,7)--(1,6));
draw((0,3)--(1,6));
draw((1,6)--(10,3));
draw((10,3)--(9,0));
draw((9,0)--(0,3));
draw((6,13/3)--(10,22/3));
draw((10,22/3)--(8,10));
draw((8,10)--(4,7));
draw((4,7)--(6,13/3));
label("$3$",(9/2,3/2),N);
label("$3$",(11/2,9/2),S);
label("$1$",(1/2,9/2),E);
label("$1$",(19/2,3/2),W);
label("$1$",(1/2,15/2),E);
label("$1$",(3/2,19/2),S);
label("$1$",(5/2,13/2),N);
label("$1$",(7/2,17/2),W);
label("$R$",(7,43/6),W);
[/asy]
$(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67$
The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters?
[asy]/* AMC8 2003 #21 Problem */
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$