MathDB

1

areas of triangles are distinct primes, area of quadr. an integer, min area

A_1

and

B_1

be internal points lying on the sides

BC

and

AC

of the triangle

ABC

respectively and segments

AA_1

and

BB_1

meet at

O

. The areas of the triangles

AOB_1,AOB

and

BOA_1

are distinct prime numbers and the area of the quadrilateral

A_1OB_1C

is an integer. Find the least possible value of the area of the triangle

ABC

, and argue the existence of such a triangle.

9

1

Trapezoid and lengths

Let

ABCD

be a trapezoid with

AB\parallel CD,AB>CD

and \angle{A} \plus{} \angle{B} \equal{} 90^\circ. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.

14

1

Subsets, sums and products

Let

n\ge 5

be a positive integer. Prove that the set

\{1,2,\ldots,n\}

can be partitioned into two non-zero subsets

S_n

and

P_n

such that the sum of elements in

S_n

is equal to the product of elements in

P_n

.

13

1

Subsets and divisibility

Let

A

be a subset of the set

\{1, 2,\ldots,2006\}

, consisting of

1004

elements. Prove that there exist

3

distinct numbers

a,b,c\in A

such that

gcd(a,b)

: a) divides

c

b) doesn't divide

c

12

1

Equilateral triangle and midpoints

Let

ABC

be an equilateral triangle of center

O

, and

M\in BC

. Let

K,L

be projections of

M

onto the sides

AB

and

AC

respectively. Prove that line

OM

passes through the midpoint of the segment

KL

.

11

1

Circles and lines

Circles

\mathcal{C}_1

and

\mathcal{C}_2

intersect at

A

and

B

. Let

M\in AB

. A line through

M

(different from

AB

) cuts circles

\mathcal{C}_1

and

\mathcal{C}_2

at

Z,D,E,C

respectively such that

D,E\in ZC

. Perpendiculars at

B

to the lines

EB,ZB

and

AD

respectively cut circle

\mathcal{C}_2

in

F,K

and

N

. Prove that KF\equal{}NC.

10

1

Trapezoid and areas

Let

ABCD

be a trapezoid inscribed in a circle

\mathcal{C}

with

AB\parallel CD

, AB\equal{}2CD. Let \{Q\}\equal{}AD\cap BC and let

P

be the intersection of tangents to

\mathcal{C}

at

B

and

D

. Calculate the area of the quadrilateral

ABPQ

in terms of the area of the triangle

PDQ

.

8

1

Permutations of digits are perfect squares

Prove that there do not exist natural numbers

n\ge 10

having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

7

1

Number equation

Determine all numbers

\overline{abcd}

such that \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2.

6

1

Factorials and divisibility

Prove that for every composite number

n>4

, numbers

kn

divides (n\minus{}1)! for every integer

k

such that 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor.

5

1

Quadratic equation in naturals

Determine all pairs

(m,n)

of natural numbers for which m^2\equal{}nk\plus{}2 where k\equal{}\overline{n1}.
EDIT. [color=#FF0000]It has been discovered the correct statement is with k\equal{}\overline{1n}.

4

1

Parameter equation with unique solution

Determine the biggest possible value of

m

for which the equation 2005x \plus{} 2007y \equal{} m has unique solution in natural numbers.

3

1

Summable set

Let

n\ge 3

be a natural number. A set of real numbers

\{x_1,x_2,\ldots,x_n\}

is called summable if \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1. Prove that for every

n\ge 3

there always exists a summable set which consists of

n

elements such that the biggest element is: a) bigger than 2^{2n\minus{}2} b) smaller than

n^2

2

1

Inequality with 107/18

Let

x,y,z

be positive real numbers such that x\plus{}2y\plus{}3z\equal{}\frac{11}{12}. Prove the inequality 6(3xy\plus{}4xz\plus{}2yz)\plus{}6x\plus{}3y\plus{}4z\plus{}72xyz\le \frac{107}{18}.

1

Triangle inequality

For an acute triangle

ABC

prove the inequality: \sum_{cyclic} \frac{m_a^2}{\minus{}a^2\plus{}b^2\plus{}c^2}\ge \frac{9}{4} where

m_a,m_b,m_c

are lengths of corresponding medians.

2006 JBMO ShortLists

Subcontests

areas of triangles are distinct primes, area of quadr. an integer, min area

Trapezoid and lengths

Subsets, sums and products

Subsets and divisibility

Equilateral triangle and midpoints

Circles and lines

Trapezoid and areas

Permutations of digits are perfect squares

Number equation

Factorials and divisibility

Quadratic equation in naturals

Parameter equation with unique solution

Summable set

Inequality with 107/18

Triangle inequality