Investigate the properties of the tetrahedron $ABCD$ for which there is equality $$\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}$$ where $\alpha, \beta, \gamma$ are the values ​​of the dihedral angles at the edges $AD, BD$ and $CD$, respectively.

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

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? [hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]

Let $ABC$ be an acute-angled triangle in which $\angle BAC = 60^o$ and $AB> AC$. Let $H$ and $I$ denote the points of intersection of the altitudes and angle bisectors of this triangle, respectively. Find the ratio $\angle ABC: \angle AHI$.

Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.

In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$. (A Shapovalov)

Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.

a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different? b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.

In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values ​​of all angles of triangle $ABC$.

Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$

Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$. [asy] unitsize(1.5 cm); pair A, B, C, D, E, S; A = (0,0); B = (1,0); C = (2.5,0); D = dir(60); E = B + 1.5*dir(60); S = extension(C,D,A,E); fill(A--B--D--cycle, gray(0.8)); fill(B--C--E--cycle, gray(0.8)); draw(interp(A,C,-0.1)--interp(A,C,1.1)); draw(A--D--B--E--C); draw(A--E); draw(C--D); draw(anglemark(D,S,A,5)); dot("$A$", A, dir(270)); dot("$B$", B, dir(270)); dot("$C$", C, dir(270)); dot("$D$", D, N); dot("$E$", E, N); dot("$S$", S, N); [/asy]

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle