MathDB

1

Numbers of Problem and Students

x

students took an exam with

y

problems. Each student solved a half of the problem and the number of person those who solved the problem is equal for each problem.For any two students exactly three problems could be solved by the two persons. Find all pairs of

(x,\ y)

satisfying this condition, then show an example to prove that it is possiible the case in terms of

a,\ b

as follows: Note that every student solve

(a)

or don't solve

(b)

.

4

1

The number of choices such that 5 points are in a hemisphere

Given five radii

l_1,\ \cdots l_5

of a sphere

S,

no three of these radii are on a same plane. Choose a pair to endpoint from each radius

l_1,\ \cdots l_5.

Find the number of choices such that five points are in a hemisphere among 32 choices of an endpoint.

2

1

Proof of existence of infinite n such that 3d(n)=2t(n)

Denote by

d(n)

the largest odd divisor of positive integers

n

and define

D(n),\ T(n)

as follows.

D(n)=d(1)+d(2)+\cdots +d(n),\ \ T(n)=1+2+\cdots +n.

Prove that there existed infinitely positive integers

n

such that

3D(n)=2T(n).

1

Prove that w_1 or w_2 is not periodic.

Call a word forming by alphabetical small letters

a,b,c,\ \cdots x,y,z

and a periodic word arranging by a certain word more than two times repeatedly.For example kyonkyon is eight-letter periodic word. Given two words

W_1,\ W_2

which have the same number of letters and have a different first letter, if you will remove the letter,

W_1

and

W_2

will be same word.Prove that either

W_1

or

W_2

is not periodic.

5

1

Japan mathematical olympiad finals 1993, problem 5

Prove that there existed a positive number

C,

irrelevant to

n

and

a_1,\ a_2,\ \cdots a_n,

satisfying the following condition. Condition: For arbiterary positive numbers

n

and arbiterary real numbers

a_1,\ \cdots a_n,

the following inequality holds.

\max_{0\leq x\leq 2}\prod_{j=1}^n |x-a_j|\leq C^n \max_{0\leq x\leq 1}\prod_{j=1}^n |x-a_j|.

1993 Japan MO Finals

Subcontests

Numbers of Problem and Students

The number of choices such that 5 points are in a hemisphere

Proof of existence of infinite n such that 3d(n)=2t(n)

Prove that w_1 or w_2 is not periodic.

Japan mathematical olympiad finals 1993, problem 5