MathDB

1

How many points among them should we take ?

We are given a fixed point on the circle of radius

1

, and going from this point along the circumference in the positive direction on curved distances

0, 1, 2, \ldots

from it we obtain points with abscisas

n = 0, 1, 2, .\ldots

respectively. How many points among them should we take to ensure that some two of them are less than the distance

\frac 15

apart ?

18

1

Construct triangle A'B'C'

If an acute-angled triangle

ABC

is given, construct an equilateral triangle

A'B'C'

in space such that lines

AA',BB', CC'

pass through a given point.

17

1

prove that there exists an equilateral triangle ABC iff

Given a point

O

and lengths

x, y, z

, prove that there exists an equilateral triangle

ABC

for which

OA = x, OB = y, OC = z

, if and only if

x+y \geq z, y+z \geq x, z+x \geq y

(the points

O,A,B,C

are coplanar).

20

1

Prove that the angle is not less than 120 degrees

Given

n \ (n \geq 3)

points in space such that every three of them form a triangle with one angle greater than or equal to

120^\circ

, prove that these points can be denoted by

A_1,A_2, \ldots,A_n

in such a way that for each

i, j, k, 1 \leq i < j < k \leq n

, angle

A_iA_jA_k

is greater than or equal to

120^\circ .

25

1

Find the maximum possible number of triangles

Given

k

parallel lines

l_1, \ldots, l_k

and

n_i

points on the line

l_i, i = 1, 2, \ldots, k

, find the maximum possible number of triangles with vertices at these points.

23

1

Find all complex numbers m - ISL 1968

Find all complex numbers

m

such that polynomial

x^3 + y^3 + z^3 + mxyz

can be represented as the product of three linear trinomials.

21

1

Find all positive integers k - ISL 1968

Let

a_0, a_1, \ldots , a_k \ (k \geq 1)

be positive integers. Find all positive integers

y

such that

a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).

14

1

Find the locus of the center of the circumcircle of XAY

A line in the plane of a triangle

ABC

intersects the sides

AB

and

AC

respectively at points

X

and

Y

such that

BX = CY

. Find the locus of the center of the circumcircle of triangle

XAY .

13

1

Prove that there exists a plane for congruent triangles

Given two congruent triangles

A_1A_2A_3

and

B_1B_2B_3

(

A_iA_k = B_iB_k

), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.

10

1

Find values of a/b such that conditions satisfy

Consider two segments of length

a, b \ (a > b)

and a segment of length

c = \sqrt{ab}

.
(a) For what values of

a/b

can these segments be sides of a triangle ?
(b) For what values of

a/b

is this triangle right-angled, obtuse-angled, or acute-angled ?

9

1

Geometry inequality about distances From M to AB,BC,CA

Let

ABC

be an arbitrary triangle and

M

a point inside it. Let

d_a, d_b, d_c

be the distances from

M

to sides

BC,CA,AB

;

a, b, c

the lengths of the sides respectively, and

S

the area of the triangle

ABC

. Prove the inequality

abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.

Prove that the left-hand side attains its maximum when

M

is the centroid of the triangle.

16

1

p(x) is divisible by m - ISl 1968

A polynomial

p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k

with integer coefficients is said to be divisible by an integer

m

if

p(x)

is divisible by m for all integers

x

. Prove that if

p(x)

is divisible by

m

, then

k!a_0

is also divisible by

m

. Also prove that if

a_0, k,m

are non-negative integers for which

k!a_0

is divisible by

m

, there exists a polynomial

p(x) = a_0x^k+\cdots+ a_k

divisible by

m.

12

1

Inequality holds for any integer m and positive reals a, b

If

a

and

b

are arbitrary positive real numbers and

m

an integer, prove that

\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.

11

1

Find all solutions (x1, x2, . . . , xn) of the equation

Find all solutions

(x_1, x_2, . . . , x_n)

of the equation

1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0

8

1

Find the loci of vertices B,C,D of trapezoids

Given an oriented line

\Delta

and a fixed point

A

on it, consider all trapezoids

ABCD

one of whose bases

AB

lies on

\Delta

, in the positive direction. Let

E,F

be the midpoints of

AB

and

CD

respectively. Find the loci of vertices

B,C,D

of trapezoids that satisfy the following:
(i)

|AB| \leq a

(

a

fixed);
(ii)

|EF| = l

(

l

fixed);
(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.
[hide="Remark"]
Remark. The constants are chosen so that such trapezoids exist.

7

1

Product of the radii of three circles doesn't exceed A

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed

A=\frac{3\sqrt 3}{8}

times the product of the side lengths of the triangle. When does equality hold?

6

1

The equation has at least n-1 roots

If

a_i \ (i = 1, 2, \ldots, n)

are distinct non-zero real numbers, prove that the equation

\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n

has at least

n - 1

real roots.

5

1

Prove the inequality and when it not remains - ISL 1968

Let

h_n

be the apothem (distance from the center to one of the sides) of a regular

n

-gon (

n \geq 3

) inscribed in a circle of radius

r

. Prove the inequality

(n + 1)h_n+1 - nh_n > r.

Also prove that if

r

on the right side is replaced with a greater number, the inequality will not remain true for all

n \geq 3.

1

The ships have the smallest distantance - ISL 1968

Two ships sail on the sea with constant speeds and fixed directions. It is known that at

9:00

the distance between them was

20

miles; at

9:35

,

15

miles; and at

9:55

,

13

miles. At what moment were the ships the smallest distance from each other, and what was that distance ?

1968 IMO Shortlist

Subcontests

How many points among them should we take ?

Construct triangle A'B'C'

prove that there exists an equilateral triangle ABC iff

Prove that the angle is not less than 120 degrees

Find the maximum possible number of triangles

Find all complex numbers m - ISL 1968

Find all positive integers k - ISL 1968

Find the locus of the center of the circumcircle of XAY

Prove that there exists a plane for congruent triangles

Find values of a/b such that conditions satisfy

Geometry inequality about distances From M to AB,BC,CA

p(x) is divisible by m - ISl 1968

Inequality holds for any integer m and positive reals a, b

Find all solutions (x1, x2, . . . , xn) of the equation

Find the loci of vertices B,C,D of trapezoids

Product of the radii of three circles doesn't exceed A

The equation has at least n-1 roots

Prove the inequality and when it not remains - ISL 1968

The ships have the smallest distantance - ISL 1968