Let $B_1$ be the foot of the altitude from the vertex $B$ in the acute-angled $\triangle ABC$. Let $D$ be the midpoint of side $AB$, and $O$ be the circumcentre of $\triangle ABC$. Line $B_1D$ meets line $CO$ at $E$. Prove that the points $B, C, B_1$, and $E$ lie on a circle.

Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.) $ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $

Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.

Let $\triangle BC$ be a triangle with side lengths $AB = 9, BC = 10, CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC, E = BO \cap CA, F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

Is it possible to cut a rectangle $2001\times2003$ into pieces of the form [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS82L2RjZTZjNzc0M2YxMzM1ZDIzZTY2Zjc2NGJlMWJlMWUwMmU2ZWRlLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNS0xMyBhdCAzLjQ2LjQ2IFBNLnBuZw==[/img] each consisting of three unit squares?

The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be: $ \textbf{(A)}\ \text{equal to the second} \qquad\textbf{(B)}\ \frac {4}{3} \text{ times the second} \qquad\textbf{(C)}\ \frac {2}{\sqrt {3}} \text{ times the second} \\ \textbf{(D)}\ \frac {\sqrt {2}}{\sqrt {3}} \text{ times the second} \qquad\textbf{(E)}\ \text{indeterminately related to the second}$ [i][Note: The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.][/i]

A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents. This is represented by the figure below. [asy] import olympiad; pair A, B, C; A = dir(90); B = dir(210); C = dir(330); draw(A--B--C--cycle); draw(incircle(A,B,C)); draw(incircle(A,2/3*A+1/3*B,2/3*A+1/3*C)); draw(incircle(A,8/9*A+1/9*B,8/9*A+1/9*C)); draw(incircle(A,26/27*A+1/27*B,26/27*A+1/27*C)); draw(incircle(A,80/81*A+1/81*B,80/81*A+1/81*C)); draw(incircle(A,242/243*A+1/243*B,242/243*A+1/243*C)); draw(incircle(B,2/3*B+1/3*A,2/3*B+1/3*C)); draw(incircle(B,8/9*B+1/9*A,8/9*B+1/9*C)); draw(incircle(B,26/27*B+1/27*A,26/27*B+1/27*C)); draw(incircle(B,80/81*B+1/81*A,80/81*B+1/81*C)); draw(incircle(B,242/243*B+1/243*A,242/243*B+1/243*C)); draw(incircle(C,2/3*C+1/3*B,2/3*C+1/3*A)); draw(incircle(C,8/9*C+1/9*B,8/9*C+1/9*A)); draw(incircle(C,26/27*C+1/27*B,26/27*C+1/27*A)); draw(incircle(C,80/81*C+1/81*B,80/81*C+1/81*A)); draw(incircle(C,242/243*C+1/243*B,242/243*C+1/243*A)); [/asy] What is the ratio of the area of the largest circle to the combined area of all the other circles? $\text{(A) }\frac{8}{1}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }\frac{9}{1}\qquad\text{(D) }\frac{9}{3}\qquad\text{(E) }\frac{10}{3}$

A circle $\mathfrak{D}$ is drawn through the vertices $A$ and $B$ of $\triangle ABC$. If $\mathfrak{D}$ intersects $AC$ at a point $M$ and $BC$ at $P$ and $MP$ contains the incenter of $\triangle ABC$, then the length $MP$ is (in standard notation, where $t=\frac{1}{a+b+c}$): (A)$at(b+c)$ (B)$ct(b+a)$ (C)$bct$ (D)$abt$

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

Hassan has a piece of paper in the shape of a hexagon. The interior angles are all $120^o$, and the side lengths are $1$, $2$, $3$, $4$, $5$, $6$, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?

There are four collinear spots: $ A,B,C,D $ $ \bar{AB}=\bar{BC}=\frac{\bar{CD}}{4}=\sqrt{5} $ There are two circles; One which has $ \bar{AC} $ as a diameter, and the other having $ \bar{BD} $ as a diameter. Let's put $ \odot (AC) \cap \odot (BD) = E,F $ Let's put the area of $ EAFD $ $ S $ Find $ S^2 $.

[b]p1A.[/b] Compute $$1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ...$$ $$1 - \frac{1}{2^3} + \frac{1}{3^3} - \frac{1}{4^3} + \frac{1}{5^3} - ...$$ [b]p1B.[/b] Real values $a$ and $b$ satisfy $ab = 1$, and both numbers have decimal expansions which repeat every five digits: $$ a = 0.(a_1)(a_2)(a_3)(a_4)(a_5)(a_1)(a_2)(a_3)(a_4)(a_5)...$$ and $$ b = 1.(b_1)(b_2)(b_3)(b_4)(b_5)(b_1)(b_2)(b_3)(b_4)(b_5)...$$ If $a_5 = 1$, find $b_5$. [b]p2.[/b] $P(x) = x^4 - 3x^3 + 4x^2 - 9x + 5$. $Q(x)$ is a $3$rd-degree polynomial whose graph intersects the graph of $P(x)$ at $x = 1$, $2$, $5$, and $10$. Compute $Q(0)$. [b]p3.[/b] Distinct real values $x_1$, $x_2$, $x_3$, $x_4 $all satisfy $ ||x - 3| - 5| = 1.34953$. Find $x_1 + x_2 + x_3 + x_4$. [b]p4.[/b] Triangle $ABC$ has sides $AB = 8$, $BC = 10$, and $CA = 11$. Let $L$ be the locus of points in the interior of triangle $ABC$ which are within one unit of either $A$, $B$, or $C$. Find the area of $L$. [b]p5.[/b] Triangles $ABC$ and $ADE$ are equilateral, and $AD$ is an altitude of $ABC$. The area of the intersection of these triangles is $3$. Find the area of the larger triangle $ABC$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle inscribed in a circle ($\omega$) and $I$ is the incenter. Denote $D,E$ as the intersection of $AI,BI$ with ($\omega$). And $DE$ cuts $AC,BC$ at $F,G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of ($\omega$) at $A,B$ meet at $K$. Prove that three lines $AE,BD,KP$ are concurrent or parallel.

Consider the cyclic quadrilateral $ABCD$. Let $M$ be the point of intersection of its diagonals, and $L$ be the midpoint of the arc $AD$ (not containing other vertices of the quadrilateral). Prove that the distances from $L$ to the centers of the circles inscribed in triangles $ABM$ and $CDM$ are equal.

A cone-shaped mountain has its base on the ocean floor and has a height of $ 8000$ feet. The top $ \frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet? $ \textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4\minus{}\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000$

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

Let $ M,N,P,Q $ be the middlepoints of the segments $ AB,BC,CD,DA, $ respectively, of a convex quadrilateral $ ABCD. $ Prove that if $ ANP $ and $ CMQ $ are equilateral, then $ ABDC $ is a rhombus . Moreover, determine the angles of this rhombus.

What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) $ \textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad $

Two triangles $ABC$ and $PQR$ have the same circumcircles. Let $E_a, E_b, E_c$ be the centers of the Euler circles of triangles $PBC, QCA, RAB$. Assume that $d_a$ is a line through $Ea$ parallel to $AP$, $d_b, d_c$ are defined in the same manner. Prove that three lines $d_a, d_b, d_c$ are concurrent. Nguyễn Tiến Lâm, Trần Quang Hùng

A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle. (A. Akopyan, A. Zaslavsky)

Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.

[b]p1.[/b] An $8$-inch by $11$-inch sheet of paper is laid flat so that the top and bottom edges are $8$ inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold? [b]p2.[/b] Triangle $ABC$ is equilateral, with $AB = 6$. There are points $D$, $E$ on segment AB (in the order $A$, $D$, $E$, $B$), points $F$, $G$ on segment $BC$ (in the order $B$, $F$, $G$, $C$), and points $H$, $I$ on segment $CA$ (in the order $C$, $H$, $I$, $A$) such that $DE = F G = HI = 2$. Considering all such configurations of $D$, $E$, $F$, $G$, $H$, $I$, let $A_1$ be the maximum possible area of (possibly degenerate) hexagon $DEF GHI$ and let $A_2$ be the minimum possible area. Find $A_1 - A_2$. [b]p3.[/b] Find $$\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .