MathDB

1

Circumcircle is tangent to another incircle

ABC

. Its incircle is tangent to

BC, CA, AB

at

D, E, F

respectively. Let

K, L

be points on

CA, AB

respectively such that

K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF

. Prove that the circumcircle of

AKL

is tangent to the incircle of

ABC

.

7

1

Decrementing the power

Let

a,b,c \in \mathbb{R}^+

and

abc = 1

such that

a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}

. Prove that

a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}

.

6

1

2011 divides a_{k-1}a_k - k for all k = 1,2,...,2010

Let a sequence of integers

a_0, a_1, a_2, \cdots, a_{2010}

such that

a_0 = 1

and

2011

divides

a_{k-1}a_k - k

for all

k = 1, 2, \cdots, 2010

. Prove that

2011

divides

a_{2010} + 1

.

5

1

1x2xn blocks showing a total of larger than 2011

[asy] draw((0,1)--(4,1)--(4,2)--(0,2)--cycle); draw((2,0)--(3,0)--(3,3)--(2,3)--cycle); draw((1,1)--(1,2)); label("1",(0.5,1.5)); label("2",(1.5,1.5)); label("32",(2.5,1.5)); label("16",(3.5,1.5)); label("8",(2.5,0.5)); label("6",(2.5,2.5)); [/asy] The image above is a net of a unit cube. Let

n

be a positive integer, and let

2n

such cubes are placed to build a

1 \times 2 \times n

cuboid which is placed on a floor. Let

S

be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of

n

such that there exists such cuboid and its placement on the floor so

S > 2011

.

4

1

A tour route not passing the chosen city for every city

An island has

10

cities, where some of the possible pairs of cities are connected by roads. A tour route is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.

3

1

A triangle formed from unusual perpendicular lines

Given an acute triangle

ABC

, let

l_a

be the line passing

A

and perpendicular to

AB

,

l_b

be the line passing

B

and perpendicular to

BC

, and

l_c

be the line passing

C

and perpendicular to

CA

. Let

D

be the intersection of

l_b

and

l_c

,

E

be the intersection of

l_c

and

l_a

, and

F

be the intersection of

l_a

and

l_b

. Prove that the area of the triangle

DEF

is at least three times of the area of

ABC

.

2

1

Permutations of (1,...,n) such that sum of a_i/i is integer

For each positive integer

n

, let

s_n

be the number of permutations

(a_1, a_2, \cdots, a_n)

of

(1, 2, \cdots, n)

such that

\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n}

is a positive integer. Prove that

s_{2n} \ge n

for all positive integer

n

.

1

Sum of all numbers possible by removing digits of n

For a number

n

in base

10

, let

f(n)

be the sum of all numbers possible by removing some digits of

n

(including none and all). For example, if

n = 1234

,

f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979

; this is formed by taking the sums of all numbers obtained when removing no digit from

n

(1234), removing one digit from

n

(123, 124, 134, 234), removing two digits from

n

(12, 13, 14, 23, 24, 34), removing three digits from

n

(1, 2, 3, 4), and removing all digits from

n

(0). If

p

is a 2011-digit integer, prove that

f(p)-p

is divisible by

9

.
Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number

101

,

1

appears three times (by removing the first digit, giving

01

which is equal to

1

, removing the first two digits, or removing the last two digits), so it is counted three times.

2011 Indonesia MO

Subcontests

Circumcircle is tangent to another incircle

Decrementing the power

2011 divides a_{k-1}a_k - k for all k = 1,2,...,2010

1x2xn blocks showing a total of larger than 2011

A tour route not passing the chosen city for every city

A triangle formed from unusual perpendicular lines

Permutations of (1,...,n) such that sum of a_i/i is integer

Sum of all numbers possible by removing digits of n