Found problems: 1679
Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?
The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.
A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?
The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?
[asy]
draw((0,0)--(8,0)--(8,6)--(0,0));
draw(Circle((4.5,1),1));
draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow);
dot((0,0));
dot((8,0));
dot((8,6));
dot((4.5,1));
label("A", (0,0), SW);
label("B", (8,0), SE);
label("C", (8,6), NE);
label("8", (4,0), S);
label("6", (8,3), E);
label("10", (4,3), NW);
label("P", (4.5,1), NW);
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17 $
Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.
Find $\frac{area(CDF)}{area(CEF)}$ in the figure.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(5.75cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2, xmax = 21, ymin = -2, ymax = 16; /* image dimensions */
/* draw figures */
draw((0,0)--(20,0));
draw((13.48,14.62)--(7,0));
draw((0,0)--(15.93,9.12));
draw((13.48,14.62)--(20,0));
draw((13.48,14.62)--(0,0));
label("6",(15.16,12.72),SE*labelscalefactor);
label("10",(18.56,5.1),SE*labelscalefactor);
label("7",(3.26,-0.6),SE*labelscalefactor);
label("13",(13.18,-0.71),SE*labelscalefactor);
label("20",(5.07,8.33),SE*labelscalefactor);
/* dots and labels */
dot((0,0),dotstyle);
label("$B$", (-1.23,-1.48), NE * labelscalefactor);
dot((20,0),dotstyle);
label("$C$", (19.71,-1.59), NE * labelscalefactor);
dot((7,0),dotstyle);
label("$D$", (6.77,-1.64), NE * labelscalefactor);
dot((13.48,14.62),dotstyle);
label("$A$", (12.36,14.91), NE * labelscalefactor);
dot((15.93,9.12),dotstyle);
label("$E$", (16.42,9.21), NE * labelscalefactor);
dot((9.38,5.37),dotstyle);
label("$F$", (9.68,4.5), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
The sides of a triangle are in the ratio $ 6: 8: 9$. Then:
$ \textbf{(A)}\ \text{the triangle is obtuse} \qquad\textbf{(B)}\ \text{the angles are in the ratio } 6: 8: 9$
$ \textbf{(C)}\ \text{the triangle is acute}$
$ \textbf{(D)}\ \text{the angle opposite the largest side is double the angle opposite the smallest side}$
$ \textbf{(E)}\ \text{none of these}$
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.
Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A);
draw(B--P--D--B^^P--Q--D--C--Q);
draw(Q--A--P, linetype("4 4"));
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, E);
label("$A'$", D, S);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
$ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $
Two vertices of the rectangle are located on side $BC$ of triangle $ABC$, and the other two are on sides $AB$ and $AC$. It is known that the midpoint of the altitude of this triangle, drawn on the side $BC$, lies on one of the diagonals of the rectangle, and the side of the rectangle located on $BC$ is three times less than $BC$. In what ratio does the altitude of the triangle divide the side $BC$ ?
Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to
$ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$
(hmm,, looks familiar, isn't it? :wink: )
A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.
Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$.
Proposed by V. Proizvolov
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy]
$\textbf{(A)}\ \dfrac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \dfrac{5}{9} \qquad \textbf{(C)}\ \dfrac{2}{3} \qquad \textbf{(D)}\ \dfrac{\sqrt{5}}{3} \qquad \textbf{(E)}\ \dfrac{7}{9}$
[asy]pair A=(-2,0), O=origin, C=(2,0);
path X=Arc(O,2,0,180), Y=Arc((-1,0),1,180,0), Z=Arc((1,0),1,180,0), N=X..Y..Z..cycle;
filldraw(N, black, black);
draw(reflect(A,C)*N);
draw(A--C, dashed);
label("A",A,W);
label("C",C,E);
label("O",O,SE);
dot((-1,0));
dot(O);
dot((1,0));
label("1",(-1,0),NE);
label("1",(1,0),NW);[/asy]
The large circle has diameter $ \overline{AC}$. The two small circles have their centers on $ \overline{AC}$ and just touch at $ O$, the center of the large circle. If each small circle has radius $ 1$, what is the value of the ratio of the area of the shaded region to the area of one of the small circles?
\[ \textbf{(A)}\ \text{between }\frac{1}{2} \text{ and }1 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ \text{between 1 and }\frac{3}{2} \qquad
\textbf{(D)}\ \text{between }\frac{3}{2} \text{ and }2 \\
\textbf{(E)}\ \text{cannot be determined from the information given}
\]
A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?
[asy]
import graph;
size(6cm);
real L = 0.05;
pair A = (0,0);
pair B = (sqrt(3),0);
pair C = (sqrt(3),1);
pair D = (0,1);
pair X1 = (sqrt(3)/3,0);
pair X2= (2*sqrt(3)/3,0);
pair Y1 = (2*sqrt(3)/3,1);
pair Y2 = (sqrt(3)/3,1);
dot(X1);
dot(Y1);
draw(A--B--C--D--cycle, linewidth(2));
draw(X1--Y1,dashed);
draw(X2--(2*sqrt(3)/3,L));
draw(Y2--(sqrt(3)/3,1-L));
[/asy]
$ \textbf{(A)}\ 1:2\qquad\textbf{(B)}\ 3:5\qquad\textbf{(C)}\ 2:3\qquad\textbf{(D)}\ 3:4\qquad\textbf{(E)}\ 4:5 $
There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$.
[img]https://i.imgur.com/oxixe5q.png[/img]
[i]Proposed by Giorgi Arabidze, Georgia[/i]
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
Let $ABC$ be an acute triangle. Let $ H $ be the foot of perpendicular from $ A $ to $ BC $. $ D, E $ are the points on $ AB, AC $ and let $ F, G $ be the foot of perpendicular from $ D, E $ to $ BC $. Assume that $ DG \cap EF $ is on $ AH $. Let $ P $ be the foot of perpendicular from $ E $ to $ DH $. Prove that $ \angle APE = \angle CPE $.
Say that a (nondegenerate) triangle is [i]funny[/i] if it satisfies the following condition: the altitude, median, and angle bisector drawn from one of the vertices divide the triangle into 4 non-overlapping triangles whose areas form (in some order) a 4-term arithmetic sequence. (One of these 4 triangles is allowed to be degenerate.) Find with proof all funny triangles.
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.