In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$ [asy] unitsize(75); pathpen = black; pointpen=black; pair A = MP("A", D((0,0)), dir(200)); pair B = MP("B", D((2,0)), dir(-20)); pair C = MP("C", D((1/2,1)), dir(100)); pair D = MP("D", D(midpoint(B--C)), dir(30)); pair E = MP("E", D(midpoint(A--B)), dir(-90)); pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013); draw(A--B--C--cycle); draw(A--D--E--C); [/asy] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 13.5 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 14.5 \qquad \textbf{(E)}\ 15 $

Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $G$ be the centroid of $\triangle ABC$. Prove that: the distances from $G$ to the perpendicular bisectors of $TA, TB, TC$ are the same.

Prove that if for a polynomial $P(x, y)$, we have \[P(x - 1, y - 2x + 1) = P(x, y),\] then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$

An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$. Is it possible to order the real numbers so it becomes mean-free? related: [url]https://www.youtube.com/watch?v=ppaXUxsEjMQ[/url]

Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $ [b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square. [b]b)[/b] at most two of them are greater than the side of the square. [b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.

\(ABCD\) is a convex quadrilateral, \(AC\) bisects the angle \(\angle BAD\). Points \(E\) and \(F\) are on the sides \(BC\) and \(CD\) respectively such that \(EF \parallel BD\). Extend \(FA\) and \(EA\) to points \(P\) and \(Q\) respectively, such that the circle \(\omega_1\) passing through points \(A\), \(B\), \(P\) and the circle \(\omega_2\) passing through points \(A\), \(D\), \(Q\) are both tangent to line \(AC\). Prove that the points \(B\), \(P\), \(Q\), \(D\) are concyclic.

A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?

Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

Let $ABC$ be a triangle with bisectors $BE$ and $CF$ meet at $I$. Let $D$ be the projection of $I$ on the $BC$. Let M and $N$ be the orthocenters of triangles $AIF$ and $AIE$, respectively. Lines $EM$ and $FN$ meet at $P.$ Let $X$ be the midpoint of $BC$. Let $Y$ be the point lying on the line $AD$ such that $XY \perp IP$. Prove that line $AI$ bisects the segment $XY$. [i]Proposed by Tran Quang Hung - Vietnam[/i]

Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.

Which of the following statements is false? $ \textbf{(A)}\ \text{All equilateral triangles are congruent to each other.}$ $ \textbf{(B)}\ \text{All equilateral triangles are convex.}$ $ \textbf{(C)}\ \text{All equilateral triangles are equilangular.}$ $ \textbf{(D)}\ \text{All equilateral triangles are regular polygons.}$ $ \textbf{(E)}\ \text{All equilateral triangles are similar to each other.}$

Let $n>d>0$ integers. Batman, Joker, Clark play the following game in an infinite checkered board. Initially, Batman and Joker are in cells with distance $n$ and a candy is in a cell with distance $d$ to Batman. Batman is blindfold, and can only see his cell. Clark and Joker can see the whole board. The following two moves go alternately. 1 - Batman goes to an adjacent cell. If he touches Joker, Batman loses. If he touches the candy, Batman wins. If the cell is empty, Clark chooses to say loudly one of the following two words [b]hot[/b] or [b]cold[/b]. 2 - Joker goes to an adjacent cell. If he touches Batman or candy, Joker wins. Otherwise, the game continues. Determine for each $d$, the least $n$, such that Batman, and Clark can plan an strategy to ensure the Batman's win, regardless of initial positions of the Joker and of the candy. Note: Two cells are adjacent if its have a common side. The distance between two cells $X$ and $Y$ is the least $p$ such that there exist cells $X=X_0,X_1,X_2,\dots, X_p=Y$ with $X_i$ adjacent to $X_{i-1}$ for all $i=1,2,\dots,p$.

Determine all real solutions of the following system of equations: $$x+y^2=y^3$$ $$y+x^2=x^3$$

Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.

Fix $2n$ distinct reals $x_1,y_1,...,x_n,y_n$ and dene the $n\times n$ matrix where its $(i, j)$-th element is $x_i + y_j$ for all $i, j = 1,..., n$. Show that if the products of the numbers in each column is always the same, then also the products of the numbers in each row is always the same. ( Alberto Alfarano)

We will designate by $Z_{(5)}$ a certain subset of the set $Q$ of the rational numbers . A rational belongs to $Z_{(5)}$ if and only if there exist equal fraction to this rational such that $5$ is not a divisor of its denominator. (For example, the rational number $13/10$ does not belong to $Z_{(5)}$ , since the denominator of all fractions equal to $13/10$ is a multiple of $5$. On the other hand, the rational $75/10$ belongs to $Z_{(5)}$ since that $75/10 = 15/12$). Reasonably answer the following questions: a) What algebraic structure (semigroup, group, etc.) does $Z_{(5)}$ have with respect to the sum? b) And regarding the product? c) Is $Z_{(5)}$ a subring of $Q$? d) Is $Z_{(5)}$ a vector space?

Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$, different from $A$ and $B$. The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$. The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$. If the perimeter of the triangle $PEQ$ is $24$, find the length of the side $PQ$

In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent. [i]Alan Cheng[/i]

A circle with center O is inscribed in a quadrilateral ABCD and touches its non-parallel sides BC and AD at E and F respectively. The lines AO and DO meet the segment EF at K and N respectively, and the lines BK and CN meet at M. Prove that the points O,K,M and N lie on a circle.

We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?

Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$ .