Found problems: 1679
Let $ABCDEF$ be a convex hexagon, $P, Q, R$ be the intersection points of $AB$ and $EF$, $EF$ and $CD$, $CD$ and $AB$. $S, T,UV$ are the intersection points of $BC$ and $DE$, $DE$ and $FA$, $FA$ and $BC$, respectively. Prove that if $$\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},$$ then $$\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.$$
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.
Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.
Let $ABCD$ be a parallelogram and let $K$ and $L$ be points on the segments $BC$ and $CD$, respectively, such that $BK\cdot AD=DL\cdot AB$. Let the lines $DK$ and $BL$ intersect at $P$. Show that $\measuredangle DAP=\measuredangle BAC$.
Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.
A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]
Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively.
Find all possible values of the ratio $d_1:d_2$.
[i](I. Voronovich)[/i]
Let $ABC$ be a triangle and $D$ be a point on the segment $BC$ such that $DC = 2BD$. Let $E$ be the mid-point of $AC$. Let $AD$ and $BE$ intersect in $P$. Determine the ratios $BP:PE$ and $AP:PD$.
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
$ \textbf{(A)}\ \frac{\pi}{16}\qquad
\textbf{(B)}\ \frac{\pi}{8}\qquad
\textbf{(C)}\ \frac{3\pi}{16}\qquad
\textbf{(D)}\ \frac{\pi}{4}\qquad
\textbf{(E)}\ \frac{\pi}{2}$
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.
Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{33}{7} \qquad
\textbf{(C)}\ \frac{39}{7} \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ \frac{99}{10} $
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle $I$ to that of circle $II$, then $r$ equals:
$\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3$
The triangle $ABC$ is $| BC | = a$ and $| AC | = b$. On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$. Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$, respectively, $K$ and $L$. Find the ratio $| DK | : | DL |$.
Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.
A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following?
[asy]import math;
unitsize(7mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3));
fill((0,0)--(4,0)--(4,3)--cycle,mediumgray);
label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW);
label(rotate(90)*"Height",(4,1.5),E);
label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 20.5 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 21.5 \qquad
\textbf{(E)}\ 22$
In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that
$$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$
Let $ABC$ be a right triangle at $B$. Let $D$ be a point such that $BD\perp AC$ and $DC = AC$. Find the ratio $\frac{AD}{AB}$.
Suppose that a triangle $ABC$ with incenter $I$ satisfies $CA+AI=BC$. Find the ratio between the measures of the angles $\angle BAC$ and $\angle CBA$.
Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that
\[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]
A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the line $BC$ at $A_1$. $C_2$ is the reflection of $C_1$ in the midpoint of $AB$, and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$. The lines $B_2C_2$ and $BC$ intersect at $A_2$. Prove that $$\frac{sen \, \, B_1A_1C}{sen\, \, C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$
[img]https://cdn.artofproblemsolving.com/attachments/3/8/774da81495df0a0f7f2f660ae9f516cf70df06.png[/img]
Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set
$$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$
Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.