MathDB

1

About the quadrilateral MNPQ

ABCD

be a convex quadrilateral and draw regular triangles

ABM, CDP, BCN, ADQ

, the first two outward and the other two inward. Prove that

MN = AC

. What can be said about the quadrilateral

MNPQ

?

19

1

Prove that z exists

Let

M

be the set of real numbers of the form

\frac{m+n}{\sqrt{m^2+n^2}}

, where

m

and

n

are positive integers. Prove that for every pair

x \in M, y \in M

with

x < y

, there exists an element

z \in M

such that

x < z < y.

18

1

Lines in 3D space

Let

O

be a point of three-dimensional space and let

l_1, l_2, l_3

be mutually perpendicular straight lines passing through

O

. Let

S

denote the sphere with center

O

and radius

R

, and for every point

M

of

S

, let

S_M

denote the sphere with center

M

and radius

R

. We denote by

P_1, P_2, P_3

the intersection of

S_M

with the straight lines

l_1, l_2, l_3

, respectively, where we put

P_i \neq O

if

l_i

meets

S_M

at two distinct points and

P_i = O

otherwise (

i = 1, 2, 3

). What is the set of centers of gravity of the (possibly degenerate) triangles

P_1P_2P_3

as

M

runs through the points of

S

?

17

1

Right triangles

The right triangles

ABC

and

AB_1C_1

C

and

C_1

, and we also have

\angle CAB = \angle C_1AB_1

. Let

M

be the point of intersection of the lines

BC_1

and

B_1C

. Prove that if the lines

AM

and

CC_1

exist, they are perpendicular.

15

1

Prove the inequality for all s

Show that

\frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}

holds for every

1 \neq s > 0

real and

0 < a \leq 1

rational.

14

1

IMO ShortList 1982 C2

Let

ABCD

be a convex plane quadrilateral and let

A_1

denote the circumcenter of

\triangle BCD

. Define

B_1, C_1,D_1

in a corresponding way.
(a) Prove that either all of

A_1,B_1, C_1,D_1

coincide in one point, or they are all distinct. Assuming the latter case, show that

A_1

, C1 are on opposite sides of the line

B_1D_1

, and similarly,

B_1,D_1

are on opposite sides of the line

A_1C_1

. (This establishes the convexity of the quadrilateral

A_1B_1C_1D_1

.)
(b) Denote by

A_2

the circumcenter of

B_1C_1D_1

, and define

B_2, C_2,D_2

in an analogous way. Show that the quadrilateral

A_2B_2C_2D_2

is similar to the quadrilateral

ABCD.

12

1

Distance between circle's center and L

Four distinct circles

C,C_1, C_2

, C3 and a line L are given in the plane such that

C

and

L

are disjoint and each of the circles

C_1, C_2, C_3

touches the other two, as well as

C

and

L

. Assuming the radius of

C

to be

1

, determine the distance between its center and

L.

11

1

Find the rearrangement

(a) Find the rearrangement

\{a_1, \dots , a_n\}

of

\{1, 2, \dots, n\}

that maximizes

a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.

(b) Find the rearrangement that minimizes

Q.

10

1

White balls and Black balls

A box contains

p

white balls and

q

black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

9

1

Prove that DL=DM

Let

ABC

be a triangle, and let

P

be a point inside it such that

\angle PAC = \angle PBC

. The perpendiculars from

P

to

BC

and

CA

meet these lines at

L

and

M

, respectively, and

D

is the midpoint of

AB

. Prove that

DL = DM.

8

1

A figure

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

7

1

Prove that 2p(-1) is a multiple of p(1)+p(-1)-2(1+p(0))

Let

p(x)

be a cubic polynomial with integer coefficients with leading coefficient

1

and with one of its roots equal to the product of the other two. Show that

2p(-1)

is a multiple of

p(1)+p(-1)-2(1+p(0)).

4

1

Determine a such that the equationhas four real roots in G.P

Determine all real values of the parameter

a

for which the equation

16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0

has exactly four distinct real roots that form a geometric progression.

2

1

Area and convex polygon

Let

K

be a convex polygon in the plane and suppose that

K

is positioned in the coordinate system in such a way that

\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),

where the

Q_i

denote the quadrants of the plane. Prove that if

K

contains no nonzero lattice point, then the area of

K

is less than

4.

1982 IMO Shortlist

Subcontests

About the quadrilateral MNPQ

Prove that z exists

Lines in 3D space

Right triangles

Prove the inequality for all s

IMO ShortList 1982 C2

Distance between circle's center and L

Find the rearrangement

White balls and Black balls

Prove that DL=DM

A figure

Prove that 2p(-1) is a multiple of p(1)+p(-1)-2(1+p(0))

Determine a such that the equationhas four real roots in G.P

Area and convex polygon