MathDB

1

Geometric inequality 2

A_1,A_2,...,A_{2002}

be arbitrary points in the plane. Prove that for every circle of radius

1

and for every rectangle inscribed in this circle, there exist

3

vertices

M,N,P

of the rectangle such that

MA_1 + MA_2 + \cdots + MA_{2002} +

NA_1 + NA_2 + \cdots + NA_{2002} +

PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006

.

12

1

Areas, quadrilaterals and lots of points

Let

ABCD

be a convex quadrilateral with AB\equal{}AD and BC\equal{}CD. On the sides

AB,BC,CD,DA

we consider points

K,L,L_1,K_1

such that quadrilateral

KLL_1K_1

is rectangle. Then consider rectangles

MNPQ

inscribed in the triangle

BLK

, where

M\in KB,N\in BL,P,Q\in LK

and

M_1N_1P_1Q_1

inscribed in triangle

DK_1L_1

where

P_1

and

Q_1

are situated on the

L_1K_1

,

M

on the

DK_1

and

N_1

on the

DL_1

. Let

S,S_1,S_2,S_3

be the areas of the

ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1

respectively. Find the maximum possible value of the expression: \frac{S_1\plus{}S_2\plus{}S_3}{S}

11

1

Isosceles triangle

Let

ABC

be an isosceles triangle with AB\equal{}AC and \angle A\equal{}20^\circ. On the side

AC

consider point

D

such that AD\equal{}BC. Find

\angle BDC

.

10

1

Geometric inequality

Let

ABC

be a triangle with area

S

and points

D,E,F

on the sides

BC,CA,AB

. Perpendiculars at points

D,E,F

to the

BC,CA,AB

cut circumcircle of the triangle

ABC

at points

(D_1,D_2), (E_1,E_2), (F_1,F_2)

. Prove that: |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S

9

1

Angles in a triangle

In triangle

ABC,H,I,O

are orthocenter, incenter and circumcenter, respectively.

CI

cuts circumcircle at

L

. If AB\equal{}IL and AH\equal{}OH, find angles of triangle

ABC

.

8

1

Similar triangles and cyclic quadrilaterals

Let

ABC

be a triangle with centroid

G

and

A_1,B_1,C_1

midpoints of the sides

BC,CA,AB

. A paralel through

A_1

to

BB_1

intersects

B_1C_1

at

F

. Prove that triangles

ABC

and

FA_1A

AB_1GC_1

is cyclic.

7

1

Prove that at least 3 of the numbers are equal

Consider integers a_i,i\equal{}\overline{1,2002} such that a_1^{ \minus{} 3} \plus{} a_2^{ \minus{} 3} \plus{} \ldots \plus{} a_{2002}^{ \minus{} 3} \equal{} \frac {1}{2} Prove that at least 3 of the numbers are equal.

6

1

Easy inequality with 6 numbers

Let

a_1,a_2,...,a_6

be real numbers such that: a_1 \not \equal{} 0, a_1a_6 \plus{} a_3 \plus{} a_4 \equal{} 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2 Prove that

a_4a_6\le a_5^2

. When does equality holds?

5

1

Very easy inequality

Let

a,b,c

be positive real numbers. Prove the inequality: \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}

3

1

If $abc=9/4$ then prove...

Let

a,b,c

be positive real numbers such that abc\equal{}\frac{9}{4}. Prove the inequality: a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b} Jury's variant: Prove the same, but with abc\equal{}2

2

1

Triangular array

Positive real numbers are arranged in the form:

1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...

2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...

4 \ \ \ 8 \ \ \ 13 ...

7 \ \ \ 12 ...

11 ...

Find the number of the line and column where the number 2002 stays.

1

Product game

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations - to take 2 numbers from these, to double first one, to add the second one and to save the sum. - to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. - to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

2002 JBMO ShortLists

Subcontests

Geometric inequality 2

Areas, quadrilaterals and lots of points

Isosceles triangle

Geometric inequality

Angles in a triangle

Similar triangles and cyclic quadrilaterals

Prove that at least 3 of the numbers are equal

Easy inequality with 6 numbers

Very easy inequality

If $abc=9/4$ then prove...

Triangular array

Product game