MathDB

NPC center lies on C-angle bisector - VI Soros Olympiad 1999-00 Round 1 10.3

Source:

5/21/2024

he center of the circle passing through the midpoints of all sides of triangle

ABC

lies on the bisector of its angle

C

. Find the side

AB

if

BC = a

,

AC = b

(

a

is not equal to

b

).

geometryNine Point Circle

p/q+(p+1)/(q+1) is integer (VI Soros Olympiad 1990-00 R1 10.3)

Source:

5/28/2024

Find all pairs of prime natural numbers

(p, q)

for which the value of the expression

\frac{p}{q}+\frac{p+1}{q+1}

is an integer.

number theory

f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)} (VI Soros Olympiad 1990-00 R2 10.3)

Source:

5/28/2024

Find all functions

f

that map the set of real numbers into the set of real numbers, satisfying the following conditions:
1)

|f(x)|\ge 1

,
2)

f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}

of all real values of

x

and

y

.

algebrafunctional equationfunctional

same number of airlines depart from each city of the country

Source: VI Soros Olympiad 1990-00 R3 10.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024

Some pairs of cities in the country are connected by airlines, and some are not. But every city has an airport, from which you can get to any other city, making no more than one transfer. A tourist who wants to make a round trip through several cities of the country will have to fly around at least five cities. Prove that the same number of airlines depart from each city of the country (If there is an airline from one city to another, then there is also one from the second to the first. A circular trip is a route that passes through at least three cities, starting and ending in same city, other cities are not repeated in it)

combinatorics

10.3

Problems(4)