$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of parabola $y^2=x$. Compute the sum of all possible areas of $ABCD$.

Let $T$ be a triangle with side lengths $26$, $51$, and $73$. Let $S$ be the set of points inside $T$ which do not lie within a distance of $5$ of any side of $T$. Find the area of $S$.

In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.

In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

The equation $x^3y+xy^3+xy=0$ represents $\textbf{(A)}~\text{a circle}$ $\textbf{(B)}~\text{a circle and a pair of straight lines}$ $\textbf{(C)}~\text{a rectangular hyperbola}$ $\textbf{(D)}~\text{a pair of straight lines}$

For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal [i]bisector[/i] if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon? [i]Proposed by Morteza Saghafian[/i]

A point inside a triangle is called "[i]good[/i]" if three cevians passing through it are equal. Assume for an isosceles triangle $ABC \ (AB=BC)$ the total number of "[i]good[/i]" points is odd. Find all possible values of this number.

Let $ABCD$ a convex quadrilateral where: $|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$ What form does the quadrilateral have?

Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel. Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

On a right triangle of paper, two points $A$ and $B$ have been painted. You have scissors and you have the right to make cuts (on paper) as follows: cut through a height of the given triangle. In doing so, remove, without the respective altitude, one of the two triangles and continue the process. Prove that after a finite number of cuts you can separate points $A$ and $B$ leaving one of them outside the remaining triangles.

The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

Let $BC$ be a fixed segment in the plane, and let $A$ be a variable point in the plane not on the line $BC$. Distinct points $X$ and $Y$ are chosen on the rays $CA^\to$ and $BA^\to$, respectively, such that $\angle CBX = \angle YCB = \angle BAC$. Assume that the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet line $XY$ at $P$ and $Q$, respectively, such that the points $X$, $P$, $Y$ and $Q$ are pairwise distinct and lie on the same side of $BC$. Let $\Omega_1$ be the circle through $X$ and $P$ centred on $BC$. Similarly, let $\Omega_2$ be the circle through $Y$ and $Q$ centred on $BC$. Prove that $\Omega_1$ and $\Omega_2$ intersect at two fixed points as $A$ varies. [i]Daniel Pham Nguyen, Denmark[/i]

[u]Round 5[/u] [b]p13.[/b] Five different schools are competing in a tournament where each pair of teams plays at most once. Four pairs of teams are randomly selected and play against each other. After these four matches, what is the probability that Chad's and Jordan's respective schools have played against each other, assuming that Chad and Jordan come from different schools? [b]p14.[/b] A square of side length $1$ and a regular hexagon are both circumscribed by the same circle. What is the side length of the hexagon? [b]p15.[/b] From the list of integers $1,2, 3,...,30$ Jordan can pick at least one pair of distinct numbers such that none of the $28$ other numbers are equal to the sum or the difference of this pair. Of all possible such pairs, Jordan chooses the pair with the least sum. Which two numbers does Jordan pick? [u]Round 6[/u] [b]p16.[/b] What is the sum of all two-digit integers with no digit greater than four whose squares also have no digit greater than four? [b]p17.[/b] Chad marks off ten points on a circle. Then, Jordan draws five chords under the following constraints: $\bullet$ Each of the ten points is on exactly one chord. $\bullet$ No two chords intersect. $\bullet$ There do not exist (potentially non-consecutive) points $A, B,C,D,E$, and $F$, in that order around the circle, for which $AB$, $CD$, and $EF$ are all drawn chords. In how many ways can Jordan draw these chords? [b]p18.[/b] Chad is thirsty. He has $109$ cubic centimeters of silicon and a 3D printer with which he can print a cup to drink water in. He wants a silicon cup whose exterior is cubical, with five square faces and an open top, that can hold exactly $234$ cubic centimeters of water when filled to the rim in a rectangular-box-shaped cavity. Using all of his silicon, he prints a such cup whose thickness is the same on the five faces. What is this thickness, in centimeters? [u]Round 7[/u] [b]p19.[/b] Jordan wants to create an equiangular octagon whose side lengths are exactly the first $8$ positive integers, so that each side has a different length. How many such octagons can Jordan create? [b]p20.[/b] There are two positive integers on the blackboard. Chad computes the sum of these two numbers and tells it to Jordan. Jordan then calculates the sum of the greatest common divisor and the least common multiple of the two numbers, and discovers that her result is exactly $3$ times as large as the number Chad told her. What is the smallest possible sum that Chad could have said? [b]p21.[/b] Chad uses yater to measure distances, and knows the conversion factor from yaters to meters precisely. When Jordan asks Chad to convert yaters into meters, Chad only gives Jordan the result rounded to the nearest integer meters. At Jordan's request, Chad converts $5$ yaters into $8$ meters and $7$ yaters into $12$ meters. Given this information, how many possible numbers of meters could Jordan receive from Chad when requesting to convert $2014$ yaters into meters? [u]Round 8[/u] [b]p22.[/b] Jordan places a rectangle inside a triangle with side lengths $13$, $14$, and $15$ so that the vertices of the rectangle all lie on sides of the triangle. What is the maximum possible area of Jordan's rectangle? [b]p23.[/b] Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$. Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. [b]p24.[/b] We say a polynomial $P$ in $x$ and $y$ is $n$-[i]good [/i] if $P(x, y) = 0$ for all integers $x$ and $y$, with $x \ne y$, between $1$ and $n$, inclusive. We also define the complexity of a polynomial to be the maximum sum of exponents of $x$ and $y$ across its terms with nonzero coeffcients. What is the minimal complexity of a nonzero $4$-good polynomial? In addition, give an example of a $4$-good polynomial attaining this minimal complexity. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2915803p26040550]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$. by E.Bakaev

Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.

Let $\Omega$ be the circumcircle of triangle $ABC$ such that the tangents to $\Omega$ in points $B$ and $C$ intersect in $P$. The squares $ABB_1B_2$ and $ACC_1C_2$ are constructed on the sides $AB$ and $AC$ in the exterior of triangle $ABC$, such that the lines $B_1B_2$ and $C_1C_2$ intersect in point $Q$. Prove that $P$, $A$, and $Q$ are collinear.

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space? [i]Alexey Tolpygo[/i]

The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.

Given a cuboid $Q$ with dimensions $a, b, c$, $a < b < c$. Find the length of the edge of a cube $K$ , which has parallel faces and a common center with the given cuboid so that the volume of the difference of the sets $Q \cup K$ and $Q \cap K$ is minimal.