| Problem 21 : |
Let p be a prime number and n is a positive integer number .We write the
following expression in the reductiable form ie..
M p1 p2 pn
--- = --- + --- + ... + ---
N 1 2 n
where gcd(M,N)=1.
We denote fp(n) := the smallest positive integer such that 2fp(n)
divides by M
n - 3
(i) Show that f2(n) >= -------
2
for all n is a positive integer number
(ii) Prove or disprove that
n-p-1
fp(n) >= -------
p
Comment : I solved for the first but the same method does not hold for the
general case !Any help are well come!
| Problem 22 : |
For any positive integer n,show that there exists a polynomial P(x)
of degree n with integer coeffecients such that P(0),P(1),...,P(n)
are all distinct powers of 2 .
| Problem 23 : |
Let p be a prime and Q(x) be a cubic polynomial with integer
coeffecients.Suppose p divides Q(xj),1<=j<=4,for 4 distinct integers
x1,...,x4 in the set {0,1,..,p-1}.Prove that p divides each of
the coeffecients of Q(x).
| Problem 24 : |
Prove that for each positive integer n there exist n consecutive
positive integers, none of which is an integral power of a prime
Comment : This is well-known that there exist a arithmetic progress
so that for any positive integer n then we can find n consecutive elements
in the sequence such that all of them are primes!.
| Problem 25 : |
For an odd positive integer n>1, let S be the set of
integers x, 1 <= x <= n such that both x and x1 are relative
primes to n. Show that:
__
|| x == 1 (mod n).
x in S
| Problem 26 : |
Given that
- a,b are odd positive integers
- 9 does not divides ab
- 7 divides ab
- a > b
Prove that there exist an odd prime number p such that p divides a2 - b2
| Problem 27 : |
Find all positive integer n>3 such that
1 2 3
1 + C + C + C divides 22000
n n n
| Problem 28 : |
Let n be a positive integer and n>=2. Let x1,...,xn be real
numbers such that
n_ __
\ xi2 + \ xixi+1 = 1
/_ /_
i=1 i=1,n-1
Let k be a fixed positive integer and k <= n. Find the maximum possible value
of |xk|.
| Problem 29 : |
Denote X={1,...,2001}. Find the smallest positive integer m
such that for every subset W of X such that |W|=m,there exist u,v in
W (u,v can be same or distinct) such that u+v is power of 2.
| Problem 30 : |
Let a,b,c,a+b-c,b+c-a,c+a-b,a+b+c be pairwise distinct prime numbers.
The sum of two numbers from a,b,c is 800. Let
d=max{a,b,c,a+b-c,b+c-a,c+a-b,a+b+c}-min{a,b,c,a+b-c,b+c-a,c+a-b,a+b+c}
Find the largest possible value of d.
Comment : I am bored with the number 800.The general problem when we repalace 800
by a given positive integer number n is great!
| Problem 31 : |
Let a be a positive integer number and A be the set of positive
integer
pairs (m,n) such that:
i) m < 2a
ii) 2n | 2am - m2 + n2
iii)n2 - m2 + 2mn <= 2a(n-m)
Consider
2am - m2 - mn
f(m,n):= --------------- where (m,n) is member of A.
n
Find min(f) and max(f).
| Problem 32 : |
The sequence x0 , x1 , x2, ..., is defined as follows:
x0 = 3 , x1 = 0 , x2 = 2
and xn = xn-2 + xn-3 for all n
>= 3
Show that for any prime p and positive integer n
xpn == xpn-1 (mod pn.So in particular, x_p == x_1 == 0 (mod p) for all p).
Comment : Some may recognize this as the Perrin Sequence.Any help to provide
the properties of this sequence are well-come!
| Problem 33 : |
1 __
---- * \ _ 2k _
25 /_ | ----- | is a natural number
k=0,2001 |_ 25 _|
| Problem 34 : |
Call a pair(x,y) of rational numbers 'good' iff
x5 + y5 = 2(xy)2.
a)Prove that there are infinitely many good pairs.
b)If (x,y) is a good pair prove that 1-xy is the square of a rational.
| Problem 35 : |
Let n be a positive integer.Let S be the set of positive integers
such that {n/m} >=1/2 where {x} denotes the fractional part of x.Prove that
sum(m belongs to S)(phi(m))= n^2 where phi is Euler's function.(i.e phi(m)
denotes the number of positive integers less than or equal to m that are
relatively prime to m)
| Problem 36 : |
i) Can we divide a square into an odd numbers of triangles having
equal areas ?
ii) Can we divide a regular hexagon into k triangles having
equal areas with gcd (k,6)=1 ?
iii) Can we divide a cube into k tetrahedons having equal
perimeters and volumes ( k odd ) ?
| Problem 37 : |
Integers from 1 to 999999 are partioned to two groups: integer for
which the nearest perfect square is odd belongs to the first group; the
integer for which the nearest perfect square is even -- to the second. In which
group the sum of its elements is greater?
| Problem 38 : |
Find all odd positive integers n>1 such that for any coprime
divisors a,b of n the natural condition holds: a+b-1 is a divisor of n, too
| Problem 39 : |
Let f(0)=f(1)=0 and f(n+2)=4n+2f(n+1) - 16n+1f(n) + n2n2 n=0,1,2,... .
Show that the numbers f(1989), f(1990), and f(1991) are divisible by 13.
| Problem 40 : |
Prove that the equation
6(6a2 + 3b2 + c2) = 5n2
has no solution in integers except a = b = c = n = 0.