Found problems: 11
Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse.
Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).
Let $ABC$ be a non-isosceles triangle with circumcenter $O$, incenter $I$, and orthocenter $H$. Prove that angle $\angle OIH$ is obtuse.
In a triangle $ABC$ two points, $C_1$ and $A_1$ are marked on the sides $AB$ and $BC$ respectively (the points do not coincide with the vertices). Let $K$ be the midpoint of $A_1C_1$ and $I$ be the incentre of the triangle $ABC$. Given that the quadrilateral $A_1BC_1I$ is cyclic, prove that the angle $AKC$ is obtuse.
Does such a convex (all angles less than $180^o$) pentagon $ABCDE$, such that all angles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are obtuse?
Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?
Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least $120^o$.
What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?
Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$
Prove that angle $C$ is obtuse.
(Sergey Berlov)
Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.
Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.