In triangle $ABC$(with incenter $I$) let the line parallel to $BC$ from $A$ intersect circumcircle of $\triangle ABC$ at $A_1$ let $AI\cap BC=D$ and $E$ is tangency point of incircle with $BC$ let $ EA_1\cap \odot (\triangle ADE)=T$ prove that $AI=TI$.

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle. [b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$. [b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains? [b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other. [b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Construct triangle given all lenght of it altitudes. Please, do it elementary with Euclidian geometry (no trigonometry or coordinate geometry).

Given two points $A,B$ outside of a given plane $P,$ find the positions of points $M$ in the plane $P$ for which the ratio $\frac{MA}{MB}$ takes a minimum or maximum.

Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that: \[ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. \] Where $S_{XYZ}$ is the surface of the triangle $XYZ$.

The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.

Suppose $n$ points are uniformly chosen at random on the circumference of the unit circle and that they are then connected with line segments to form an $n$-gon. What is the probability that the origin is contained in the interior of this $n$-gon? Give your answer in terms of $n$, and consider only $n \ge 3$.

Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line $ AB$ such that $ D \minus{} A \minus{} B \minus{} E, AD \equal{} AC$ and $ BE \equal{} BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of $ ABC$ at $ M$ and $ N$. Line which connects $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.

Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$ Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide. [i]Tilek Askerbekov[/i]

Let $\Omega$ be the circumcircle of an isosceles trapezoid $ABCD$, in which $AD$ is parallel to $BC$. Let $X$ be the reflection point of $D$ with respect to $BC$. Point $Q$ is on the arc $BC$ of $\Omega$ that does not contain $A$. Let $P$ be the intersection of $DQ$ and $BC$. A point $E$ satisfies that $EQ$ is parallel to $PX$, and $EQ$ bisects $\angle BEC$. Prove that $EQ$ also bisects $\angle AEP$. [i]Proposed by Li4.[/i]

Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic. [i]Proposer[/i]: [b]Dorian Croitoru[/b]

Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon.

The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$? $\textbf{(A) } 3 : 11\qquad \textbf{(B) } 5 : 11\qquad \textbf{(C) } 1 : 2\qquad \textbf{(D) }2 : 3\qquad \textbf{(E) }25 : 33$

Given a circle $\Gamma$ with center $I$, and a triangle $\triangle ABC$ with all its sides tangent to $\Gamma$. A line $\ell$ is drawn such that it bisects both the area and the perimeter of $\triangle ABC$. Prove that line $\ell$ passes through $I$.

On $\vartriangle ABC$ let $D$ be a point on side $\overline{AB}$, $F$ be a point on side $\overline{AC}$, and $E$ be a point inside the triangle so that $\overline{DE}\parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Given that $AF = 6, AC = 33, AD = 7, AB = 26$, and the area of quadrilateral $ADEF$ is $14$, nd the area of $\vartriangle ABC$.

The triangle $ABC$ has $\angle ACB = 30^o$, $BC = 4$ cm and $AB = 3$ cm . Compute the altitudes of the triangle.

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.

For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$. Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$. Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$. [i]Proposed by Mehran Talaei[/i]

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

Prove that the points symmetric to the vertices of a triangle with respect to the opposite side are collinear if and only if the distance from the orthocenter to the circumcenter is twice the circumradius.

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.