Olympiad problems

2003 All-Russian Olympiad Regional Round, 8.4

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).

1996 All-Russian Olympiad Regional Round, 8.3

Tags: pentagon , obtuse , angle , geometry

Does such a convex (all angles less than $180^o$) pentagon $ABCDE$, such that all angles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are obtuse?

1958 Kurschak Competition, 1

Tags: combinatorial geometry , obtuse , combinatorics

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 $120^o$.

Durer Math Competition CD Finals - geometry, 2015.C1

Tags: geometry , incircle , obtuse , tangent

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

1991 Tournament Of Towns, (313) 3

Tags: geometry , obtuse , equal ratio , angle , ratio

Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$ Prove that angle $C$ is obtuse. (Sergey Berlov)

2012 Tournament of Towns, 4

Tags: incenter , midpoint , geometry , cyclic , 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.

Cono Sur Shortlist - geometry, 1993.13

Tags: geometry , obtuse

Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse.

1948 Moscow Mathematical Olympiad, 155

Tags: combinatorics , geometry , angle , combinatorial geometry , obtuse , ray , maximum

What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?

2011 Saudi Arabia IMO TST, 2

Tags: geometry , angle , obtuse

Let $ABC$ be a non-isosceles triangle with circumcenter $O$, incenter $I$, and orthocenter $H$. Prove that angle $\angle OIH$ is obtuse.

2002 Estonia National Olympiad, 2

Tags: geometry , angle , acute , obtuse , orthocenter

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.

1990 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , tetrahedron , obtuse , triangle , plane

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.