1 A grasshopper rests on the point 
on the plane. Denote by 
the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point 
to 
then the area of 
is equal to 
Find all the positive integral poijnts 
which can be covered by the grasshopper after a finite number of steps, starting from 
If a point satisfies the above condition, then show that there exists a certain path for the grasshopper to reach from such that the number of jumps does not exceed 
2 is a point lying outside a circle 
. The tangents from drawn to meet the circle at 
Let 
be points on the rays 
respectively such that 
is tangent to 
The parallel lines drawn through parallel to 
respectively meet 
at 
respectively.
Show that the straight lines 
always pass through a fixed point 
and 
always pass through a fixed point 
Show that 
is constant.
3 Let
be a positive integer 
There are real numbers 
that satisfy:
Find the maximum and minimum value of the sum
on the plane. Denote by the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point to then the area of is equal to Find all the positive integral poijnts which can be covered by the grasshopper after a finite number of steps, starting from If a point satisfies the above condition, then show that there exists a certain path for the grasshopper to reach from such that the number of jumps does not exceed 2
is a point lying outside a circle . The tangents from drawn to meet the circle at Let be points on the rays respectively such that is tangent to The parallel lines drawn through parallel to respectively meet at respectively. Show that the straight lines always pass through a fixed point and always pass through a fixed point Show that is constant.3 Let
be a positive integer There are real numbers that satisfy:Find the maximum and minimum value of the sum
4 Let 
be a sequence of integers satisfying 
and 
Prove that
for every natural number 
5 Find all positive integers such that 
is a perfect square.
6 Let be an integer greater than 
pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.
be a sequence of integers satisfying and Prove that
for every natural number 5 Find all positive integers
such that is a perfect square.6 Let
be an integer greater than pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies. 
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου