Found problems: 970
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$
$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D)}\ 70\sqrt2 \qquad\textbf{(E)}\ 100$
Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$?
[asy]unitsize(4cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
draw(A--B--C--D--cycle);
draw(D--F);
draw(C--E);
draw(B--H);
draw(A--G);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",E,NNW);
label("$F$",F,ENE);
label("$G$",G,SSE);
label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$
Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. Let $P$ lie on $\overline{BC}$, and let $D$ and $E$ be the feet of the perpendiculars from $P$ onto $\overline{AB}$ and $\overline{AC}$ respectively. If $AD=AE$, find this common length.
[i]Proposed by Connor Gordon[/i]
A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 8\sqrt{3} \qquad\textbf{(E)}\ 6\sqrt{6} $
Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$
Let $a,b$ and $c$ be the sides of a right angled triangle. Let $\theta$ be the smallest angle of this triangle. If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are also the sides of a right angled triangle then show that $\sin\theta=\frac{\sqrt{5}-1}{2}$
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
$ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$
In triangle $ABC$, $AC=24$ inches, $BC=10$ inches, $AB=26$ inches. The radius of the inscribed circle is:
$\textbf{(A)}\ 26\text{ in} \qquad
\textbf{(B)}\ 4\text{ in} \qquad
\textbf{(C)}\ 13\text{ in} \qquad
\textbf{(D)}\ 8\text{ in} \qquad
\textbf{(E)}\ \text{None of these}$
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
Let $ABC$ be a triangle with $AB=8$, $BC=7$, and $AC=11$. Let $\Gamma_1$ and $\Gamma_2$ be the two possible circles that are tangent to $AB$, $AC$, and $BC$ when $AC$ and $BC$ are extended, with $\Gamma_1$ having the smaller radius. $\Gamma_1$ and $\Gamma_2$ are tangent to $AB$ to $D$ and $E$, respectively, and $CE$ intersects the perpendicular bisector of $AB$ at a point $F$. What is $\tfrac{CF}{FD}$?
Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.
Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled?
[i]Proposed by Evan Chen[/i]
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?
$ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $
Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that
\[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
A girls' camp is located $ 300$ rods from a straight road. On this road, a boys' camp is located $ 500$ rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is:
$ \textbf{(A)}\ 400\text{ rods} \qquad\textbf{(B)}\ 250\text{ rods} \qquad\textbf{(C)}\ 87.5\text{ rods} \qquad\textbf{(D)}\ 200\text{ rods}\\
\textbf{(E)}\ \text{none of these}$
Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.
[i]Proposed by Alexander A. Polyansky, Russia[/i]
The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.
Let $\gamma$ and $\Gamma$ be two circles such that $\gamma$ is internally tangent to $\Gamma$ at a point $X$. Let $P$ be a point on the common tangent of $\gamma$ and $\Gamma$ and $Y$ be the point on $\gamma$ other than $X$ such that $PY$ is tangent to $\gamma$ at $Y$. Let $PY$ intersect $\Gamma$ at $A$ and $B$, such that $A$ is in between $P$ and $B$ and let the tangents to $\Gamma$ at $A$ and $B$ intersect at $C$. $CX$ intersects $\Gamma$ again at $Z$ and $ZY$ intersects $\Gamma$ again at $Q$. If $AQ = 6, AB = 10$ and $\tfrac{AX}{XB} = \tfrac{1}{4}$. The length of $QZ = \tfrac{p}{q}\sqrt{r}$ where $p$ and $q$ are coprime positive integers, and $r$ is square free positive integer. Find $p + q + r$.
In triangle $ABC$, circle $\omega$ with center $O$ passes through $B$ and $C$ and it intersects segments $\overline{AB}$ and $\overline{AC}$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Suppose the circles with diameters $\overline{BB^{\prime}}$ and $\overline{CC^{\prime}}$ are externally tangent to each other at $T$ with $AB=18$, $AC=36$, and $AT=12$. Find $AO$.
[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$?
[b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other?
[b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there?
[b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks?
[b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later?
[b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression?
[b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists?
[b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute?
[b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$.
[b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle?
[b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name?
[b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland?
[b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$?
[b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$?
[b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$.
[b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ?
[b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$?
[b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds?
[b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$?
[b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The figure shows a regular heptagon with sides of length 1.
[asy]
import geometry;
unitsize(5);
real R = 1/(2 sin(pi/7));
pair A = (0, R);
pair B = rotate(360/7) * A;
pair C = rotate(360/7) * B;
pair D = rotate(360/7) * C;
pair E = rotate(360/7) * D;
pair F = rotate(360/7) * E;
pair G = rotate(360/7) * F;
pair X = B + G - A;
pair Y = (D + E) / 2;
draw(A -- B -- C -- D -- E -- F -- G -- cycle);
draw("$1$", B -- X);
draw("$1$", X -- G);
draw("$d$", X -- Y);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(X);
dot(Y);
perpendicular(Y, NW, Y - A);
[/asy]
Determine the indicated length $d$. Express your answer in simplified radical form.
[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$.
(a) Represent $29 - 12\sqrt5$ as a prefect square.
(b) Represent $10 - 6\sqrt3$ as a prefect cube.
[b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions.
$$x+cy = 1$$
$$cx+9y = 3$$
[b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$.
(a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point.
(b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation.
[b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it.
Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?”
“Well, Miranda, could you tell me something else?”
“Sure”, said Miranda, “my age is an odd number”.
[b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img]
[b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin?
(b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin?
PS. You should use hide for answers.