Found problems: 1679
The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.
Let $ABC$ be an acute-angled triangle having angles $\alpha,\beta,\gamma$ at vertices $A, B, C$ respectively. Let isosceles triangles $BCA_1, CAB_1, ABC_1$ be erected outwards on its sides, with apex angles $2\alpha ,2\beta ,2\gamma$ respectively. Let $A_2$ be the intersection point of lines $AA_1$ and $B_1C_1$ and let us define points $B_2$ and $C_2$ analogously. Find the exact value of the expression $$\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}$$
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors boys to the number of senior boys is equal to the ratio of the number of juniors to the number of seniors. Find the ratio of the number of junior boys to the number of junior girls.
Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment. Factory A previously filtered $80\%$ of pollutants and Factory B previously filled $72\%$ of pollutants. After installing the new scrubber system, both factories now filter $99.5\%$ of pollutants.
Jerry explains the level of pollution reduction to Michael, "Factory A is the much larger factory. It's four times as large as Factory B. Without any filters at all, it would pollute four times as much as Factory B. Even with the better pollution filtration system, Factory A was polluting nearly three times as much as Factory B."
Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find $a+b$ where $a/b$ is the ratio in lowest terms of volume of pollutants unfiltered from both factories $\textit{after}$ installation of the new scrubber system to the volume of pollutants unfiltered from both factories $\textit{before}$ installation of the new scrubber system.
In order to encourage the kids to straighten up their closets and the storage shed, Jerry offers his kids some extra spending money for their upcoming vacation. "I don't care what you do, I just want to see everything look clean and organized."
While going through his closet, Joshua finds an old bag of marbles that are either blue or red. The ratio of blue to red marbles in the bag is $17:7$. Alexis also has some marbles of the same colors, but hasn't used them for anything in years. She decides to give Joshua her marbles to put in his marble bag so that all the marbles are in one place. Alexis has twice as many red marbles as blue marbles, and when the twins get all their marbles in one bag, there are exactly as many red marbles and blue marbles, and the total number of marbles is between $200$ and $250$. How many total marbles do the twins have together?
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_{n}\le .4$ for all $n$ such that $1\le n \le 9$ is given to be $p^{a}q^{b}r/(s^{c}),$ where $p,$ $q,$ $r,$ and $s$ are primes, and $a,$ $b,$ and $c$ are positive integers. Find $(p+q+r+s)(a+b+c).$
Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.
Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?
Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $
Points $ A$ and $ C$ lie on a circle centered at $ O$, each of $ \overline{BA}$ and $ \overline{BC}$ are tangent to the circle, and $ \triangle ABC$ is equilateral. The circle intersects $ \overline{BO}$ at $ D$. What is $ \frac {BD}{BO}$?
$ \textbf{(A)}\ \frac {\sqrt2}{3} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\sqrt3}{3} \qquad \textbf{(D)}\ \frac {\sqrt2}{2} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$
In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;
pair A = (0,2);
pair B = origin;
pair C = (2,0);
pair D = (2,2);
pair E = midpoint(A--D);
pair F = foot(C,B,E);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW);
draw(A--B--C--D--cycle);
draw(B--E);
draw(C--F);
draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]
In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.
Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$.
[b]proposed by Sergej Berlov[/b]
Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.
Square $ABCD$ is shown in the diagram below. Points $E$, $F$, and $G$ are on sides $\overline{AB}$, $\overline{BC}$ and $\overline{DA}$, respectively, so that lengths $\overline{BE}$, $\overline{BF}$, and $\overline{DG}$ are equal. Points $H$ and $I$ are the midpoints of segments $\overline{EF}$ and $\overline{CG}$, respectively. Segment $\overline{GJ}$ is the perpendicular bisector of segment $\overline{HI}$. The ratio of the areas of pentagon $AEHJG$ and quadrilateral $CIHF$ can be written as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
draw((0,0)--(50,0)--(50,50)--(0,50)--cycle);
label("$A$",(0,50),NW);
label("$B$",(50,50),NE);
label("$C$",(50,0),SE);
label("$D$",(0,0),SW);
label("$E$",(0,100/3-1),W);
label("$F$",(100/3-1,0),S);
label("$G$",(20,50),N);
label("$H$",((100/3-1)/2,(100/3-1)/2),SW);
label("$I$",(35,25),NE);
label("$J$",(((100/3-1)/2+35)/2,((100/3-1)/2+25)/2),S);
draw((0,100/3-1)--(100/3-1,0));
draw((20,50)--(50,0));
draw((100/6-1/2,100/6-1/2)--(35,25));
draw((((100/3-1)/2+35)/2,((100/3-1)/2+25)/2)--(20,50));
[/asy]
A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.
Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$
[hide]If possible, don't use projective geometry[/hide]
In right triangle $ \triangle ACE$, we have $ AC \equal{} 12$, $ CE \equal{} 16$, and $ EA \equal{} 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB \equal{} 3$, $ CD \equal{} 4$, and $ EF \equal{} 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE$?
[asy]
size(200);defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
pair C = (0,0);
pair E = (16,0);
pair A = (0,12);
pair F = waypoint(E--A,0.25);
pair B = waypoint(A--C,0.25);
pair D = waypoint(C--E,0.25);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("$A$",A,NW);label("$B$",B,W);label("$C$",C,SW);label("$D$",D,S);label("$E$",E,SE);label("$F$",F,NE);
label("$3$",midpoint(A--B),W);
label("$9$",midpoint(B--C),W);
label("$4$",midpoint(C--D),S);
label("$12$",midpoint(D--E),S);
label("$5$",midpoint(E--F),NE);
label("$15$",midpoint(F--A),NE);
draw(A--C--E--cycle);
draw(B--F--D--cycle);[/asy]$ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {9}{25}\qquad \textbf{(C)}\ \frac {3}{8}\qquad \textbf{(D)}\ \frac {11}{25}\qquad \textbf{(E)}\ \frac {7}{16}$
Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.