Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Let $T$ be an isosceles right triangle. Let $S$ be the circle such that the difference in the areas of $T \cup S$ and $T \cap S$ is the minimal. Prove that the centre of $S$ divides the altitude drawn on the hypotenuse of $T$ in the golden ratio (i.e., $\frac{(1 + \sqrt{5})}{2}$)

$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label("$"+lbl[i]+"'$", P, Q); label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]

Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ . .

Let $ Z_1,\,Z_2\dots,\,Z_n$ be $ d$-dimensional independent random (column) vectors with standard normal distribution, $ n \minus{} 1 > d$. Furthermore let \[ \overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i,\quad S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top\] be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $ Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z)$, $ i \equal{} 1,2,\dots,n$. Show that \[ \frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,\] and that the ratio does not depend on $ d$, only on $ n$.

Given rectangle $ R_1$ with one side $ 2$ inches and area $ 12$ square inches. Rectangle $ R_2$ with diagonal $ 15$ inches is similar to $ R_1.$ Expressed in square inches the area of $ R_2$ is: $ \textbf{(A)}\ \frac92 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ \frac{135}{2} \qquad \textbf{(D)}\ 9 \sqrt{10} \qquad \textbf{(E)}\ \frac{27 \sqrt{10}}{4}$

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

The ratio $ \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}}$ is closest to which of the following numbers? $ \text{(A)}\ 0.1\qquad \text{(B)}\ 0.2\qquad \text{(C)}\ 1\qquad \text{(D)}\ 5\qquad \text{(E)}\ 10$

The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.

Let $ ABCD$ be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a [i]tangent quadrilateral[/i], we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral $ ABCD$ touch its sides $ AB$, $ BC$, $ CD$, and $ DA$ in the points $ K$, $ L$, $ M$, and $ N$, respectively. The exterior angle bisectors of the angles $ DAB$ and $ ABC$ intersect each other at a point $ K^{\prime}$. The exterior angle bisectors of the angles $ ABC$ and $ BCD$ intersect each other at a point $ L^{\prime}$. The exterior angle bisectors of the angles $ BCD$ and $ CDA$ intersect each other at a point $ M^{\prime}$. The exterior angle bisectors of the angles $ CDA$ and $ DAB$ intersect each other at a point $ N^{\prime}$. Prove that the straight lines $ KK^{\prime}$, $ LL^{\prime}$, $ MM^{\prime}$, and $ NN^{\prime}$ are concurrent.

It is given that $\log_{6}a+\log_{6}b+\log_{6}c=6,$ where $a,$ $b,$ and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c.$

Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $ m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is $ \textbf{(A)}\ \frac {1}{2m \plus{} 1} \qquad \textbf{(B)}\ m \qquad \textbf{(C)}\ 1 \minus{} m \qquad \textbf{(D)}\ \frac {1}{4m} \qquad \textbf{(E)}\ \frac {1}{8m^2}$

In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that $P$ divides the segment $AB$ in the ratio $1 : k$, $Q$ divides the segment $BC$ in the ratio $1 : l$, and $R$ divides the segment $CA$ in the ratio $1 : m.$

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.

Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $

If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence $ \textbf{(A)}\ \text{which is a G.P} \qquad$ $ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$ $ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$ $ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$ $ \textbf{(E)}\ \text{none of these}$

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $ 1/3$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $ 1/5$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue? $ \textbf{(A)}\ \frac {1}{5}\qquad \textbf{(B)}\ \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{3}\qquad \textbf{(D)}\ \frac {2}{5}\qquad \textbf{(E)}\ \frac {1}{2}$