A square is filled with numbers .The numbers inside four squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
Let be a triangle with points lie on the perpendicular bisector of such that lie on a circle. Suppose are perpendicular to sides at points The tangent lines from points to the circumcircle of intersects at point Prove that: are parallel.
Find (AB * CD) / (AC * BD) & prove orthogonality of circles
Maverick15
Nan hour ago
by Ilikeminecraft
Source: IMO 1993, Day 1, Problem 2
Let ,,, be four points in the plane, with and on the same side of the line , such that and . Find the ratio
and prove that the circumcircles of the triangles and are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles and are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles and at the point are perpendicular.)
Let ABC be a scalene triangle. Let , and be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of intersects a second time at point different from A. Define and analogously. Prove that the circumcenter of lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
The incircle of the acute-angled triangle is tangent to its side at a point . Let be an altitude of triangle , and let be the midpoint of the segment . If is the common point of the circle and the line (distinct from ), then prove that the incircle and the circumcircle of triangle are tangent to each other at the point .
A number is called lucky if it has at least two distinct prime divisors and can be written in the form: where are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number , there exists a lucky number such that .