[list=a] [*] In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area. [*] In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area. [/list] [img]https://i.imgur.com/fAuxoc9.png[/img]

A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon? [asy] defaultpen(linewidth(0.7)); draw(polygon(3)); pair D=origin+1*dir(270), E=origin+1*dir(150), F=1*dir(30); draw(D--E--F--cycle);[/asy] $\textbf{(A)}\: 1:1\qquad \textbf{(B)}\: 6:5\qquad \textbf{(C)}\: 3:2\qquad \textbf{(D)}\: 2:1\qquad \textbf{(E)}\: 3:1\qquad $

In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0]; draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N); label("$A$",D2(A),plain.E); label("$B$",D2(B),NE); label("$C$",D2(C),NW); label("$D$",D2(D),W); label("$E$",D2(E),SW); label("$F$",D2(F),SE); label("$M$",D2(M),(0,-1.5)); label("$N$",D2(N),SE); [/asy]

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be: $ \textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad \\ \textbf{(C)}\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad \\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad \\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

An equilateral triangle of side length $ 10$ is completely filled in by non-overlapping equilateral triangles of side length $ 1$. How many small triangles are required? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 250 \qquad \textbf{(E)}\ 1000$

On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e $$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$ If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$. [i]K. Petrov[/i]

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? [asy]unitsize(0.8cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real r = 1 + (2/3)*(sqrt(3)); pair A = dir(47.5)*(r - 1); pair B = dir(167.5)*(r - 1); pair C = dir(-72.5)*(r - 1); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(C,1)); draw(Circle(origin,r));[/asy] $ \textbf{(A)}\ \frac{2 \plus{} \sqrt{6}}{3}\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ \frac{2 \plus{} 3\sqrt{2}}{3}\qquad \textbf{(D)}\ \frac{3 \plus{} 2\sqrt{3}}{3}\qquad \textbf{(E)}\ \frac{3 \plus{} \sqrt{3}}{2}$

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$

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.

If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0); draw(A--E--B--F--C--D--A--F^^E--D--B); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, W); label("F", F, S); label("H", (.8,0.6)); label("I", (0.4,1.4)); [/asy] $ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$. [i]Proposed by Robin Park[/i]

Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$? [asy] int i; for(i=0; i<8; i=i+1) { dot((i,0)^^(i,5)); } for(i=1; i<5; i=i+1) { dot((0,i)^^(7,i)); } draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8)); pair P=(3.5, 2.5); draw((0,4)--P--(0,3)^^(2,0)--P--(3,0)); label("$B$", (2.3,0), NE); label("$A$", (0,3.7), SE);[/asy] $\text{(A)} \ 1 \qquad \text{(B)} \ m/n \qquad \text{(C)} \ n/m \qquad \text{(D)} \ 2m/n \qquad \text{(E)} \ 2n/m$

Given a non-isoceles triangle $ABC$ inscribes a circle $(O,R)$ (center $O$, radius $R$). Consider a varying line $l$ such that $l\perp OA$ and $l$ always intersects the rays $AB,AC$ and these intersectional points are called $M,N$. Suppose that the lines $BN$ and $CM$ intersect, and if the intersectional point is called $K$ then the lines $AK$ and $BC$ intersect. $1$, Assume that $P$ is the intersectional point of $AK$ and $BC$. Show that the circumcircle of the triangle $MNP$ is always through a fixed point. $2$, Assume that $H$ is the orthocentre of the triangle $AMN$. Denote $BC=a$, and $d$ is the distance between $A$ and the line $HK$. Prove that $d\leq\sqrt{4R^2-a^2}$ and the equality occurs iff the line $l$ is through the intersectional point of two lines $AO$ and $BC$.

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

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}$

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.

Let $ABC$ be a triangle, and $M$ be the midpoint of $BC$, Let $N$ be the point on the segment $AM$ with $AN = 3NM$, and $P$ be the intersection point of the lines $BN$ and $AC$. What is the area in cm$^2$ of the triangle $ANP$ if the area of the triangle $ABC$ is $40$ cm$^2$?

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 $

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.