Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle. (I. Bogdanov, B. Frenkin)

An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$. (Grigory Filippovsky)

Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

The diagram below shows square $ABCD$ which has side length $12$ and has the same center as square $EFGH$ which has side length $6$. Find the area of quadrilateral $ABFE$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/8cfedf396cd2e86092c03fa0dcb1fb3c978965.png[/img]

Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.

In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$? $ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $

Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 \plus{} \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4$

In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.

The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?

Let $S_1$ and $S_2$ be two circles intersecting at points $A$ and $B$. Let $C$ and $D$ be points on $S_1$ and $S_2$ respectively such that line $CD$ is tangent to both circles and $A$ is closer to line $CD$ than $B$. If $\angle BCA = 52^\circ$ and $\angle BDA = 32^\circ$, determine the degree measure of $\angle CBD$. [i]Ray Li[/i]

Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon. (proposed by J. Tabov)

The sides of a triangle are $ 30$, $ 70$, and $ 80$ units. If an altitude is dropped upon the side of length $ 80$, the larger segment cut off on this side is: $ \textbf{(A)}\ 62\qquad \textbf{(B)}\ 63\qquad \textbf{(C)}\ 64\qquad \textbf{(D)}\ 65\qquad \textbf{(E)}\ 66$

Three pairwise externally tangent circles $\omega_1,\omega_2$ and $\omega_3$ are given. Let $K_{12}$ be the point of tangency between $\omega_1$ and $\omega_2$ and define $K_{23}$ and $K_{31}$ similarly. Consider the point $A_1$ on $\omega_1$. Let $A_2$ be the second intersection of the line $A_1K_{12}$ with $\omega_2$. The line $A_2K_{23}$ then intersects $\omega_3$ the second time at $A_3$, and then line $A_3K_{31}$ intersects $\omega_1$ again at $A_4$ and so on. [list=a] [*]Prove that after six steps, the process will loop; that is, $A_7=A_1$. [*]Prove that the lines $A_1A_2$ and $A_4A_5$ are perpendicular. [*]Prove that the triples of lines $A_1A_2,A_3A_4$ and $A_5A_6$ and $A_2A_3,A_4A_5$ and $A_6A_1$ intersect at two diametrically opposite points on the circle $(K_{12}K_{23}K_{31})$. [/list] [i]Proposed by E. Morozov[/i]

Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere. Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are coplanar.

Let $ABC$ be an acute scalene triangle and let $M$ be the midpoint of side $BC$. The angle bisector of the $\angle BAC$, the perpendicular bisector of the side $AB$ and the perpendicular bisector of the side $AC$ define a new triangle. Let $H$ be the point of intersection of the three altitudes of this new triangle. Prove that $H$ belongs to line segment $AM$.

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$

Let $D$ be a point inside $\triangle ABC$ such that $\angle CDA + \angle CBA = 180^{\circ}.$ The line $CD$ meets the circle $\odot ABC$ at the point $E$ for the second time. Let $G$ be the common point of the circle centered at $C$ with radius $CD$ and the arc $\overset{\LARGE \frown}{AC}$ of $\odot ABC$ which does not contain the point $B$. The circle centered at $A$ with radius $AD$ meets $\odot BCD$ for the second time at $F$. Prove that the lines $GE, FD, CB$ are concurrent or parallel.

[b]p1.[/b] Suppose $x \ne 1$ or $10$ and logarithms are computed to the base $10$. Define $y= 10^{\frac{1}{1-\log x}}$ and $z = ^{\frac{1}{1-\log y}}$ . Prove that $x= 10^{\frac{1}{1-\log z}}$ [b]p2.[/b] If $n$ is an odd number and $x_1, x_2, x_3,..., x_n$ is an arbitrary arrangement of the integers $1, 2,3,..., n$, prove that the product $$(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)$$ is an even number (possibly negative or zero). [b]p3.[/b] Prove that $\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}$ for all integers $n = 1,2,3,...$ [b]p4.[/b] Prove that if three angles of a convex polygon are each $60^o$, then the polygon must be an equilateral triangle. [b]p5.[/b] Find all solutions, real and complex, of $$4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0$$ [b]p6.[/b] A man is $\frac38$ of the way across a narrow railroad bridge when he hears a train approaching at $60$ miles per hour. No matter which way he runs he can [u]just [/u] escape being hit by the train. How fast can he run? Prove your assertion. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle and $D$ a point on its side $BC.$ Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C,$ respectively, such that $BE = BD$ and $CF = CD.$ Let $P$ be a point such that $D$ is the incenter of triangle $P EF.$ Prove that $P$ lies inside the circumcircle $\Omega$ of triangle $ABC$ or on it.

A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?

Points $A$ and $B$ are given on a circle. With the help of a compass and a ruler, construct on this circle the points $C,$ $D$, $E$ that lie on one side of the straight line $AB$ and for which the pentagon with vertices $A$, $B$, $C$, $D$, $E$ has the largest possible area