Found problems: 10
Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.
Let $A_1A_2...A_{2010}$ be a regular $2010$-gon. Find the number of obtuse triangles whose vertices are among $A_1$, $A_2$,$ ...$, $A_{2010}$.
You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.
Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$