A rectangle is divided into several similar isosceles triangles. Determine the possible values of the angles of the triangles.

Is there a tetrahedron $ABCD$ such that $AB+BC+CD+DA=12\text{ cm}$ with volume $\mathrm V\ge2\sqrt3\text{ cm}^3?$

Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .

In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.

Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that: $\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$). [i]Edited by orl.[/i]

Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Triangle $ABC$ is given. The median $CM$ intersects the circumference of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.

Let $ABCD$ be a parallelogram ($AB < BC$). The bisector of the angle $BAD$ intersects the side $BC$ at the point K; and the bisector of the angle $ADC$ intersects the diagonal $AC$ at the point $F$. Suppose that $KD \perp BC$. Prove that $KF \perp BD$.

A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. [img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

[b]p1.[/b] The following number is the product of the divisors of $n$. $$2^63^3$$ What is $n$? [b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$. [b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards? [b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once? [b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series: $$1, 1, 2, 3, 5, 8,...$$ (Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series? [b]p6.[/b] Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$ What is $a_{15}$? [b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$? [b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$. [b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have $$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$ Compute $f(3)$. [b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? [b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$? [b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$. [b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ? [b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number $$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$ is divisible by $527$. [b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$

An acute non-isosceles $\triangle ABC$ is given. $CD, AE, BF$ are its altitudes. The points $E', F'$ are symetrical of $E, F$ with respect accordingly to $A$ and $B$. The point $C_1$ lies on $\overrightarrow{CD}$, such that $DC_1=3CD$. Prove that $\angle E'C_1F'=\angle ACB$

$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\ If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\ [i](Vlad Robu)[/i]

Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle (shaded in the diagram below) congruent to the other five is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$. [asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((2.96,2.58), 1),grey); draw(circle((-1,3), 1)); draw(circle((1,3), 1)); draw(circle((1.62,1.1), 1)); draw(circle((0,-0.08), 1)); draw(circle((-1.62,1.1), 1)); [/asy]

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $

Take any point $P_1$ on the side $BC$ of a triangle $ABC$ and draw the following chain of lines: $P_1P_2$ parallel to $AC$; $P_2P_3$ parallel to $BC$; $P_3P_4$ parallel to $AB$ ; $P_4P_5$ parallel to $CA$; and $P_5P_6$ parallel to $BC$, Here, $P_2,P_5$ lie on $AB$; $P_3,P_6$ lie on $CA$ and $P_4$ on $BC$> Show that $P_6P_1$ is parallel to $AB$.

Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.