Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?

Let $ABC$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.

Consider the triangle $ABC$, where $\angle B= 30^o, \angle C = 15^o$, and $M$ is the midpoint of the side $[BC]$. Let point $N \in (BC)$ be such that $[NC] = [AB]$. Show that $[AN$ is the angle bisector of $MAC$

Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.

Let $\gamma$ be a semicircle with diameter $AB$ . A creek is built with origin in $A$ , which has its vertices alternately in the diameter $AB$ and in the semicircle $\gamma$ , so that its sides make equal angles $\alpha$ with the diameter (but alternately in either direction). It is requested: a) Values of the angle $\alpha$ for the ravine to pass through the other end $B$ of the diameter. b) The total length of the ravine, in the case that it ends in $B$ , as a function of the length $d$ of the diameter and of the angle $\alpha$ . [img]https://cdn.artofproblemsolving.com/attachments/3/c/54c71cdf1bf8fbb3bdfa38f4b14a1dc961c5fe.png[/img]

A triangle with sides $a,b,c$ is called a good triangle if $a^2,b^2,c^2$ can form a triangle. How many of below triangles are good? (i) $40^{\circ}, 60^{\circ}, 80^{\circ}$ (ii) $10^{\circ}, 10^{\circ}, 160^{\circ}$ (iii) $110^{\circ}, 35^{\circ}, 35^{\circ}$ (iv) $50^{\circ}, 30^{\circ}, 100^{\circ}$ (v) $90^{\circ}, 40^{\circ}, 50^{\circ}$ (vi) $80^{\circ}, 20^{\circ}, 80^{\circ}$ $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

The volume of a certain rectangular solid is $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$?

Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear. [i]Proposed by CDMX[/i]

At the altitude $AH_1$ of an acute non-isosceles triangle $ABC$ chose a point $X$ , from which draw the perpendiculars $XN$ and $XM$ on the sides $AB$ and $AC$ respectively. It turned out that $H_1A$ is the angle bisector $MH_1N$. Prove that $X$ is the point of intersection of the altitudes of the triangle $ABC$.

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$ .

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

Let $ABC$ be an equilateral triangle with side length $2$ that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $AB$ and $AC$. Compute the length of this chord.

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)

A regular $ n\minus{}$gon $ A_1A_2A_3 \cdots A_k \cdots A_n$ inscribed in a circle of radius $ R$ is given. If $ S$ is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]

Let $ABC$ be a triangle in which $\angle B$ is obtuse.Let $\Gamma$ be its circumcircle and $O$ be the centre of $\Gamma$.Let the tangent to $\Gamma$ at $C$ intersect the line $AB$ in $B_1$.Let $O_1$ be the circumcentre of the circumcircle $\Gamma_1$ of $\triangle AB_1 C$.Take any point $B_2$ on the segment $BB_1$ different from $B,B_1$.Let $C_1$ be the point of contact of the tangent from $B_2$ to $\Gamma$ which is closer to $C$.Let $O_2$ be the circumcentre of $\triangle AB_2 C_1$.Prove that $O,O_2,O_1,C_1,C$ are concyclic if $OO_2\perp AO_1$.

The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$