Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? $ \textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$. A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$. Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$, respectively. Let $P$ be the intersection of the lines $AM$ and $CD$. The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$. The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$. Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$. Prove that $MNPQ$ is a parallelogram.

Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? [asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy] $\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that: $a)$ Points $M,N,A$ and $B$ lie on a circle $b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.

[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle. [b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$. [b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains? [b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other. [b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that a) The triangle $ABC$ is equilateral if and only if $A' =B'.$ b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$

Let $ABC$ be a triangle. The tangent in $A$ of the circumcircle of $ABC$ cuts the line $(BC)$ in $X$. Let $A'$ be the symetric of $A$ by $X$ and $C'$ the symetric of $C$ by the line $(AX)$ Prove that the points $A, C', A'$ and $B$ are concyclic.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$. [i]James Tao[/i]

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute. b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Omega$ such that $AB<CD$. Suppose that $AC$ meets $BD$ at $E$, $AD$ meets $BC$ at $F$, and $\Omega$ meets $(FAE)$, $(FBE)$ at $X$, $Y$, respectively. Prove that if $XY$ is diameter of $\Omega$, then $XY$ is perpendicular to $EF$.

Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet? [asy]defaultpen(linewidth(0.7)); size(7cm); D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle, black); D((16,-8)--(24,-8), black); label('Dog', (24, -8), SE); label('I', (8,-8), (0,0)); MP('8', (16,-4), W); MP('8', (16,-12),W); MP('8', (20,-8), N); label('Rope', (20,-8),S); D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle, black); D((16,-24)--(24,-24), black); label("II", (8,-28), (0,0)); MP('4', (16,-22), W); MP('8', (20,-24), N); label("Dog",(24,-24),SE); label("Rope", (20,-24), S); dot((24,-24)^^(24,-8));[/asy] $ \textbf{(A)}\text{ I, by }8\pi\qquad\textbf{(B)}\text{ I, by }6\pi\qquad\textbf{(C)}\text{ II, by }4\pi\qquad\textbf{(D) }\text{II, by }8\pi\qquad\textbf{(E)}\text{ II, by }10\pi $