[/quote]
,
(mod 
)
,
,
Point 
is chosen on the circle with radius 
is chosen on the circle with radius 
is chosen on the circle with radius 
,
in terms of 
be nonnegative real numbers such that 
.
.
and determine when the equality holds.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
of moving 
unit in the positive direction and probability 
of moving 
,
is chosen on the side 
.
and 
,
and 
are drawn, respectively. The point 
is the circumcenter of 
.
and 
lies on the line 
.
,
,
,
.
.
be a point in the plane of triangle 
,
,
are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to 
is acute.
and 
be two distinct rays not lying on the same line, and let 
be a circle with center 
that is tangent to ray 
and ray 
.
be a point on segment 
.
be the intersection of lines 
and 
be the intersection of line 
is tangent to 
lie on the plane such that 
is a parallelogram. 
is the orthocenter of 
intersects 
at 
intersects 
at 
The circle with diameter 
intersects the circumcircle of 
again at 
Prove that: The circumcircle of 
passes through the reflection point of 
,
denote the incenter and 
.
perpendicular to 
intersects the tangent from 
.
and 
.
.
such that for any positive integer number 
and any arrangement of the numbers from 1 to 
on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
Y by
p1. Within an equilateral triangle of side 
, 
points are marked. Show that there are two of these points at a distance less than or equal to 
.
p2. A certain country has
cities and 
direct flights between some city pairs. Suppose there are no two cities with more than one direct flight between them and that direct flights can be made in either direction. Show that can be reached from any city to any other in the country through some combination of flights.
p3. Consider a grid board of
cells. How many squares are crossed by a diagonal line of the board?
Clarification: We say that a line crosses a square when it passes through it's interior.
p4. The numbers are written on a blackboard
We want to place symbols 
or 
in front of each number and consider the result of the corresponding sum.
a) Is it possible to place the symbols so that the resulting sum is
?
b) Show that there is some result that can be obtained by placing the symbols or in at least 
different ways.
p5. On the squares of a board of
are written in a disorderly way the numbers from 
to 
. Show that there exists a row such that the product of its squares is divisible by 
.
p6. To promote the years of the National Mathematical Olympiad, the olympic commission made smaller posters and postcards, using exactly the same image. On Professor Cort's desk is there is a poster stretched, and on this, without sticking out, is a postal letter. Professor Cortes notices a very particular phenomenon: there is a point in both images that is located exactly in the same position on the desk. Show that such a phenomenon always exists. Determine if it is possible for there to be more than one point with such a property.
PS. Easier versions of P1, P3 were posted as Juniors P1, P3.
, 
points are marked. Show that there are two of these points at a distance less than or equal to 
.p2. A certain country has
cities and 
direct flights between some city pairs. Suppose there are no two cities with more than one direct flight between them and that direct flights can be made in either direction. Show that can be reached from any city to any other in the country through some combination of flights.p3. Consider a grid board of
cells. How many squares are crossed by a diagonal line of the board?Clarification: We say that a line crosses a square when it passes through it's interior.
p4. The numbers are written on a blackboard
We want to place symbols 
or 
in front of each number and consider the result of the corresponding sum.a) Is it possible to place the symbols so that the resulting sum is
?b) Show that there is some result that can be obtained by placing the symbols
or in at least 
different ways.p5. On the squares of a board of
are written in a disorderly way the numbers from 
to 
. Show that there exists a row such that the product of its squares is divisible by 
.p6. To promote the
years of the National Mathematical Olympiad, the olympic commission made smaller posters and postcards, using exactly the same image. On Professor Cort's desk is there is a poster stretched, and on this, without sticking out, is a postal letter. Professor Cortes notices a very particular phenomenon: there is a point in both images that is located exactly in the same position on the desk. Show that such a phenomenon always exists. Determine if it is possible for there to be more than one point with such a property.PS. Easier versions of P1, P3 were posted as Juniors P1, P3.
This post has been edited 3 times. Last edited by parmenides51, Dec 21, 2022, 11:11 PM
