11.4

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

circumsphere of pyramid (V Soros Olympiad 1998-99 Round 1 11.4)

Source:

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of

3

from the indicated vertex and at distances

\sqrt5, \sqrt6, \sqrt7

from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

geometry3D geometrypyramidsphere

sum of digits -= 25, 99 divides them (V Soros Olympiad 1998-99 Round 2 11.4)

Source:

Find all natural numbers less than

105

that are divisible by

1999

and whose digits sum (in decimal notation) to

25

locus of intersections of tangents

Source: V Soros Olympiad 1998-99 Round 3 11.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

AB

is drawn in a circle. On its extensions beyond points

A

B

P

Q

respectively are taken such that

AP = BQ

P

Q

two tangents to the circle are drawn, intersecting at point

M

. Find the locus of points

M

P

Q

move along a straight line and for any

P

Q

all possible pairs of tangents are taken, which determine four points from the desired locus of points) .

geometryLocusTangents