MathDB

1

Prove that 45< x_1000 <45.1

x_0 = 5

and

x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )

. Prove that

45 < x_{1000} < 45. 1.

13

1

Prove that the segments do not intersect for each k, n

Let

A_0,A_1, \ldots , A_n

be points in a plane such that (i)

A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n

and (ii)

0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,

where all these angles have the same orientation. Prove that the segments

A_kA_{k+1},A_mA_{m+1}

do not intersect for each

k

and

n

such that

0 \leq k \leq m - 2 < n- 2.

12

1

Inequality on trigonometric circle the arcs - ISL 1975

Consider on the first quadrant of the trigonometric circle the arcs

AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v

, such that

x_1 < x_2 < x_3 < \cdots < x_v

. Prove that

\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})

9

1

Is it possible to find a function f such that G(f)=G_a (f) ?

Let

f(x)

be a continuous function defined on the closed interval

0 \leq x \leq 1

. Let

G(f)

denote the graph of

f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq

x \leq 1, y = f(x) \}

. Let

G_a(f)

denote the graph of the translated function

f(x - a)

(translated over a distance

a

), defined by

G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}

. Is it possible to find for every

a, \ 0 < a < 1

, a continuous function

f(x)

, defined on

0 \leq x \leq 1

, such that

f(0) = f(1) = 0

and

G(f)

and

G_a(f)

are disjoint point sets ?

7

1

For every x,y such that x+y=1 prove the sum is equal to 1

Prove that from

x + y = 1 \ (x, y \in \mathbb R)

it follows that

x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).

5

1

Prove that the sum is less than 80 - n variables inequality

Let

M

be the set of all positive integers that do not contain the digit

9

(base

10

). If

x_1, \ldots , x_n

are arbitrary but distinct elements in

M

, prove that

\sum_{j=1}^n \frac{1}{x_j} < 80 .

4

1

Sequence inequality - a_n - 2a_{n+1} + a_{n+2} ≤ 0

Let

a_1, a_2, \ldots , a_n, \ldots

be a sequence of real numbers such that

0 \leq a_n \leq 1

and

a_n - 2a_{n+1} + a_{n+2} \geq 0

for

n = 1, 2, 3, \ldots

. Prove that

0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots

3

1

Find the floor

Find the integer represented by

\left[ \sum_{n=1}^{10^9} n^{-2/3} \right]

. Here

[x]

denotes the greatest integer less than or equal to

x.

1

Six ports on a lake

There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? (i) Every route includes exactly three ports; (ii) No two routes contain the same three ports; (iii) The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

1975 IMO Shortlist

Subcontests

Prove that 45< x_1000 <45.1

Prove that the segments do not intersect for each k, n

Inequality on trigonometric circle the arcs - ISL 1975

Is it possible to find a function f such that G(f)=G_a (f) ?

For every x,y such that x+y=1 prove the sum is equal to 1

Prove that the sum is less than 80 - n variables inequality

Sequence inequality - a_n - 2a_{n+1} + a_{n+2} ≤ 0

Find the floor

Six ports on a lake

1975 IMO Shortlist

Subcontests

Prove that 45&lt; x_1000 &lt;45.1

Prove that the segments do not intersect for each k, n

Inequality on trigonometric circle the arcs - ISL 1975

Is it possible to find a function f such that G(f)=G_a (f) ?

For every x,y such that x+y=1 prove the sum is equal to 1

Prove that the sum is less than 80 - n variables inequality

Sequence inequality - a_n - 2a_{n+1} + a_{n+2} ≤ 0

Find the floor

Six ports on a lake

Prove that 45< x_1000 <45.1