MathDB

1

\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j, sequences

m, n

be positive integers. Consider a sequence of positive integers

a_1, a_2, ... , a_n

that satisfies m = a_1 \ge a_2\ge ... \ge a_n \ge 1. Then define, for 1\le i\le m,

b_i =

# \{ j \in \{1, 2, ... , n\}: a_j \ge i\}, so

b_i

is the number of terms

a_j

of the given sequence for which a_j \ge i. Similarly, we define, for 1\le j \le n,

c_j=

# \{ i \in \{1, 2, ... , m\}: b_i \ge j\} , thus

c_j

is the number of terms bi in the given sequence for which b_i \ge j. E.g.: If

a

is the sequence

5, 3, 3, 2, 1, 1

then

b

is the sequence

6, 4, 3, 1, 1

. (a) Prove that

a_j = c_j

for 1 \le j \le n. (b) Prove that for 1\le k \le m:

\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j

.

4

1

\sqrt{1 + 12n^2} in N => 2 + 2\sqrt{1 + 12n^2} = m^2

Let

n

be positive integer such that

\sqrt{1 + 12n^2}

is an integer. Prove that

2 + 2\sqrt{1 + 12n^2}

is the square of an integer.

1

in Z_+ : f(f(f(n))) + f(f(n)) + f(n) = 3n

Find all funtions

f : Z_{>0} \to Z_{>0}

that satisfy

f(f(f(n))) + f(f(n)) + f(n) = 3n

for all

n \in Z_{>0}

.

2

1

Julian and Johan play a game with 2n cards, always a winner

Julian and Johan are playing a game with an even number of cards, say

2n

cards, (

n \in Z_{>0}

). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

5

1

midpoint of CQ lies on \Gamma, right triangle and semicircle related

Let

\vartriangle ABC

be a right triangle with

\angle B = 90^o

and

|AB| > |BC|

, and let

\Gamma

be the semicircle with diameter

AB

that lies on the same side as

C

. Let

P

be a point on

\Gamma

such that

|BP| = |BC|

and let

Q

be on

AB

such that

|AP| = |AQ|

. Prove that the midpoint of

CQ

lies on

\Gamma

.

2008 Dutch IMO TST

Subcontests

\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j, sequences

\sqrt{1 + 12n^2} in N => 2 + 2\sqrt{1 + 12n^2} = m^2

in Z_+ : f(f(f(n))) + f(f(n)) + f(n) = 3n

Julian and Johan play a game with 2n cards, always a winner

midpoint of CQ lies on \Gamma, right triangle and semicircle related

2008 Dutch IMO TST

Subcontests

\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j, sequences

\sqrt{1 + 12n^2} in N =&gt; 2 + 2\sqrt{1 + 12n^2} = m^2

in Z_+ : f(f(f(n))) + f(f(n)) + f(n) = 3n

Julian and Johan play a game with 2n cards, always a winner

midpoint of CQ lies on \Gamma, right triangle and semicircle related

\sqrt{1 + 12n^2} in N => 2 + 2\sqrt{1 + 12n^2} = m^2