MathDB

1

Part of 1999 Tournament Of Towns

Problems(8)

a father and his son are skating around a circular skating rink

Source: Tournament Of Towns Spring 1999 Junior 0 Level p1

5/7/2020
A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?
(Tairova)
algebrageometryspeed
TOT 1999 Spring AJ1 500 dollars in bank, withdraw 300, deposit 198

Source:

5/11/2020
There is 500500 dollars in a bank. Two bank operations are allowed: to withdraw 300300 dollars from the bank or to deposit 198198 dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done?
(AK Tolpygo)
combinatoricsnumber theory
TOT 1999 Spring OS1 1999 numbers in a row, each equals sum of neighbours

Source:

5/11/2020
In a row are written 19991999 numbers such that except the first and the last , each is equal to the sum of its neighbours. If the first number is 11, find the last number.
(V Senderov)
Sumalgebra
TOT 1999 Spring AS1 convex polyhedron is floating in a sea

Source:

5/11/2020
A convex polyhedron is floating in a sea. Can it happen that 90%90\% of its volume is below the water level, while more than half of its surface area is above the water level?
(A Shapovalov)
convex polyhedrongeometry3D geometry
TOT 1999 Autumn OJ1 folding a right triangle

Source:

5/11/2020
A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is 11 .
(A Shapovalov)
geometryright trianglefoldingratioarea
n consecutives integers in a row , sum of any 3 successive is divided by ..

Source: Tournament Of Towns Spring 1999 Junior A Level p1

7/19/2024
nn consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of nn if the last number in the row is odd?
(A Shapovalov)
combinatoricsnumber theory
TOT 1999 Autumn OS1 triangle by incenter similar to original triangle

Source:

5/11/2020
The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles.
(A Shapovalov)
geometryincentersimilar trianglessimilarangles
TOT 1999 Autumn AS1 numbers 1 to n inclusive on a circle

Source:

5/11/2020
For what values o f nn is it possible to place the integers from 11 to nn inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order?
(A Shapovalov)
combinatoricsdivisibledividesSum