The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

Let $ 2n$ distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these $ n$ chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)

Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.

Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.

In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$.

Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? $\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$

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}$

In the adjoining figure triangle $ ABC$ is inscribed in a circle. Point $ D$ lies on $ \stackrel{\frown}{AC}$ with $ \stackrel{\frown}{DC} \equal{} 30^\circ$, and point $ G$ lies on $ \stackrel{\frown}{BA}$ with $ \stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}$. Side $ AB$ and side $ AC$ each have length equal to the length of chord $ DG$, and $ \angle CAB \equal{} 30^\circ$. Chord $ DG$ intersects sides $ AC$ and $ AB$ at $ E$ and $ F$, respectively. The ratio of the area of $ \triangle AFE$ to the area of $ \triangle ABC$ is [asy] size(200); defaultpen(linewidth(.8pt)); pair C = origin; pair A = 2.5*dir(75); pair B = A + 2.5*dir(-75); path circ =circumcircle(A,B,C); pair D = waypoint(circ,(7/12)); pair G = waypoint(circ,(1/6)); pair E = intersectionpoint(D--G,A--C); pair F = intersectionpoint(A--B,D--G); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,SW); label("$G$",G,NE); label("$E$",E,NW); label("$F$",F,W); label("$30^\circ$",A,12S+E,fontsize(6pt)); draw(A--B--C--cycle); draw(circ); draw(Arc(A,0.25,-75,-105)); draw(D--G);[/asy]$ \textbf{(A)}\ \frac {2 \minus{} \sqrt {3}}{3}\qquad \textbf{(B)}\ \frac {2\sqrt {3} \minus{} 3}{3}\qquad \textbf{(C)}\ 7\sqrt {3} \minus{} 12\qquad \textbf{(D)}\ 3\sqrt {3} \minus{} 5\qquad$ $ \textbf{(E)}\ \frac {9 \minus{} 5\sqrt {3}}{3}$

Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains $N$ astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in a gravitational field of strength $g$. Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational field of strength $g'$ . What is the ratio $g'/g$ in terms of $N$?[asy] import roundedpath; size(300); path a = roundedpath((0,-0.3)--(4,-0.3)--(4,-1)--(5,-1)--(5,0),0.1); draw(scale(+1,-1)*a); draw(scale(+1,+1)*a); draw(scale(-1,-1)*a); draw(scale(-1,+1)*a); filldraw(circle((0,0),1),white,black); filldraw(box((-2,-0.27),(2,0.27)),white,white); draw(arc((0,0),1.5,+35,+150),dashed,Arrow); draw(arc((0,0),1.5,-150,-35),dashed,Arrow);[/asy] $ \textbf{(A)}\ 2N/(N-1) $ $ \textbf{(B)}\ N/(N-1) $ $ \textbf{(C)}\ \sqrt{(N-1)/N} $ $ \textbf{(D)}\ \sqrt{N/(N-1)} $ $ \textbf{(E)}\ \text{none of the above} $

Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$ \[\angle QXA=\angle QKP\]

The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? $\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: $ \textbf{(A)}\ \frac{1}{1} \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{2}{1} \qquad \textbf{(E)}\ \sqrt{\frac{5}{4}}$

Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.

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$.

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 $ ABCDEF$ be a regular hexagon. Let $ G$, $ H$, $ I$, $ J$, $ K$, and $ L$ be the midpoints of sides $ AB$, $ BC$, $ CD$, $ DE$, $ EF$, and $ AF$, respectively. The segments $ AH$, $ BI$, $ CJ$, $ DK$, $ EL$, and $ FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ ABCDEF$ be expressed as a fraction $ \frac {m}{n}$ where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

In triangle $ABC$, point $I$ is the center, point $I_a$ is the center of the excircle tangent to the side $BC$. From the vertex $A$ inside the angle $BAC$ draw rays $AX$ and $AY$. The ray $AX$ intersects the lines $BI$, $CI$, $BI_a$, $CI_a$ at points $X_1$, $...$, $X_4$, respectively, and the ray $AY$ intersects the same lines at points $Y_1$, $...$, $Y_4$ respectively. It turned out that the points $X_1,X_2,Y_1,Y_2$ lie on the same circle. Prove the equality $$\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.$$