Found problems: 1679
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
$ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$
Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle
Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.
$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$)
Prove that $P$ and $I$ and $D$ are on a line.
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$
Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear.
[i]Valentin Vornicu[/i]
In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is
$\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$
Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?
$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$
A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given triangle. Prove that $f\ge\frac{1}{3}$ and determine those points $P$ for which $f =\frac{1}{3}$ .
A square and a circle have equal perimeters. The ratio of the area of the circle to the area of the square is:
$\textbf{(A) }\frac{4}{\pi}\qquad\textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad\textbf{(C) }\frac{4}{1}\qquad\textbf{(D) }\frac{\sqrt{2}}{\pi}\qquad \textbf{(E) }\frac{\pi}{4}$
For $\pi\leq\theta<2\pi$, let
\[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots
\] and
\[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta
+\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
The top of one tree is $16$ feet higher than the top of another tree. The height of the $2$ trees are at a ratio of $3:4$. In feet, how tall is the taller tree?
$ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112 $
$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$.
(1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$.
(2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.
Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
$ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$
Rectangle $ ABCD$ has $ AB \equal{} 4$ and $ BC \equal{} 3$. Segment $ EF$ is constructed through $ B$ so that $ EF$ is perpendicular to $ DB$, and $ A$ and $ C$ lie on $ DE$ and $ DF$, respectively. What is $ EF$?
$ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ \frac {125}{12}\qquad \textbf{(D)}\ \frac {103}{9}\qquad \textbf{(E)}\ 12$
Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?
A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.
Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$
Given that in the expansion of $(2+3x)^n$, the coefficients of $x^3$ and $x^4$ are in the ratio $8:15$. Find the value of $n$.